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*English*
*Pages 766
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*Year 2010*

Table of contents :

Contents

Foreword

Scope of this volume

List of symbols and abbreviations used in this volume

1.1. Historical introduction

1.2. General introduction to the subgroups of space groups

1.3. Computer checking of the subgroup data

1.4. The mathematical background of the subgroup tables

1.5. Remarks on Wyckoff positions

1.6. Relating crystal structures by group–subgroup relations

1.7. The Bilbao Crystallographic Server

2.1. Guide to the subgroup tables and graphs

2.2. Tables of maximal subgroups of the plane groups

2.3. Tables of maximal subgroups of the space groups

2.4. Graphs for translationengleiche subgroups

2.5. Graphs for klassengleiche subgroups

3.1. Guide to the tables

3.2. Tables of the relations of the Wyckoff positions

Appendix. Differences in the presentation of Parts 2 and 3

Subject index

Contents

Foreword

Scope of this volume

List of symbols and abbreviations used in this volume

1.1. Historical introduction

1.2. General introduction to the subgroups of space groups

1.3. Computer checking of the subgroup data

1.4. The mathematical background of the subgroup tables

1.5. Remarks on Wyckoff positions

1.6. Relating crystal structures by group–subgroup relations

1.7. The Bilbao Crystallographic Server

2.1. Guide to the subgroup tables and graphs

2.2. Tables of maximal subgroups of the plane groups

2.3. Tables of maximal subgroups of the space groups

2.4. Graphs for translationengleiche subgroups

2.5. Graphs for klassengleiche subgroups

3.1. Guide to the tables

3.2. Tables of the relations of the Wyckoff positions

Appendix. Differences in the presentation of Parts 2 and 3

Subject index

- Author / Uploaded
- H. Wondratschek
- U. Müller (Eds)

- Similar Topics
- Chemistry
- Physical Chemistry

INTERNATIONAL TABLES FOR CRYSTALLOGRAPHY

Volume A1 SYMMETRY RELATIONS BETWEEN SPACE GROUPS

Edited by ¨ LLER H. WONDRATSCHEK AND U. MU Second Edition

Published for

THE INTERNATIONAL UNION OF C RYSTALLOG RAPHY by

2010

Dedicated to Paul Niggli and Carl Hermann

In 1919, Paul Niggli (1888– 1953) published the first compilation of space groups in a form that has been the basis for all later space-group tables, in particular for the first volume of the trilingual series Internationale Tabellen zur Bestimmung von Kristallstrukturen (1935), for International Tables for X-ray Crystallography Volume I (1952) and for International Tables for Crystallography Volume A (1983). The tables in his book Geometrische Kristallographie des Diskontinuums (1919) contained the lists of the Punktlagen, now known as Wyckoff positions. He was a great universal geoscientist, his work covering all fields from crystallography to petrology. Carl Hermann (1898–1961) published among his seminal works four famous articles in the series Zur systematischen Strukturtheorie I to IV in Z. Kristallogr. 68 (1928) and 69 (1929). The first article contained the background to the Hermann–Mauguin space-group symbolism. The last article was fundamental to the theory of subgroups of space groups and forms the basis of the maximalsubgroup tables in the present volume. In addition, he was the editor of the first volume of the trilingual series Internationale Tabellen zur Bestimmung von Kristallstrukturen (1935) and one of the founders of n-dimensional crystallography, n > 3.

Contributing authors M. I. Aroyo: Departamento de Fı´sica de la Materia Condensada, Facultad de Ciencia y Tecnologı´a, Universidad del Paı´s Vasco, Apartado 644, E-48080 Bilbao, Spain. [Computer production of Parts 2 and 3, 1.1, 1.7, 2.1, 2.2, 2.3]

P. B. Konstantinov: Institute for Nuclear Research and Nuclear Energy, 72 Tzarigradsko Chaussee, BG-1784 Sofia, Bulgaria. [Computer production of Parts 2 and 3]

Y. Billiet‡: De´partement de Physique, UFR des Sciences et Techniques, Universite´ de Bretagne Occidentale, F-29200 Brest, France. [2.1.5, 2.2, 2.3]

E. B. Kroumova: Departamento de Fı´sica de la Materia Condensada, Facultad de Ciencia y Tecnologı´a, Universidad del Paı´s Vasco, Apartado 644, E-48080 Bilbao, Spain. [Computer production of Parts 2 and 3]

C. Capillas: Departamento de Fı´sica de la Materia Condensada, Facultad de Ciencia y Tecnologı´a, Universidad del Paı´s Vasco, Apartado 644, E-48080 Bilbao, Spain. [1.7]

U. Mu¨ller: Fachbereich Chemie, Philipps-Universita¨t, D-35032 Marburg, Germany. [Computer production of Parts 2 and 3, 1.1, 1.5, 1.6, 3, Appendix]

F. Ga¨hler: Fakulta¨t fu¨r Mathematik, Universita¨t Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany. [1.3]

G. Nebe: Lehrstuhl D fu¨r Mathematik, Rheinisch-Westfa¨lische Technische Hochschule, D-52062 Aachen, Germany. [1.4]

V. Gramlichx: Laboratorium fu¨r Kristallographie, Eidgeno¨ssische Technische Hochschule, Sonnegstrasse 5, ETH-Zentrum, CH-8092 Zu¨rich, Switzerland. [2.4, 2.5]

J. M. Perez-Mato: Departamento de Fı´sica de la Materia Condensada, Facultad de Ciencia y Tecnologı´a, Universidad del Paı´s Vasco, Apartado 644, E-48080 Bilbao, Spain. [1.7]

A. Kirov: Faculty of Physics, University of Sofia, bulvd. J. Boucher 5, BG-1164 Sofia, Bulgaria. [Computer production of Parts 2 and 3]

H.

Wondratschek: Laboratorium fu¨r Applikationen der Synchrotronstrahlung, Karlsruher Institut fu¨r Technologie (KIT), D-76128 Karlsruhe, Germany. [1.1, 1.2, 1.7, 2, Appendix]

‡ Present address: 8 place des Jonquilles, F-29860 Bourg-Blanc, France. x Present address: Guggachstrasse 24, CH-8057 Zu¨rich, Switzerland.

Acknowledgments and technicians at the Institut fu¨r Kristallographie, Universita¨t Karlsruhe, around 1970. M. I. Aroyo rechecked the graphs and transformed the hand-drawn versions into computer graphics. We are grateful to Dr L. L. Boyle, University of Kent, Canterbury, England, who read, commented on and improved all parts of the text, in particular the English. We thank Professors J. M. Perez-Mato and G. Madariaga, Universidad del Paı´s Vasco, Bilbao, for many helpful discussions on the content and the presentation of the data. M. I. Aroyo would like to note that most of his contribution to this volume was made during his previous appointment in the Faculty of Physics, University of Sofia, Bulgaria, and he is grateful to his former colleagues, especially Drs J. N. Kotzev and M. Mikhov, for their interest and encouragement during this time. We are indebted to Dr G. Nolze, Bundesanstalt fu¨r Materialforschung, Berlin, Germany, for checking the data of Chapter 3.2 using his computer program POWDER CELL. We also thank Mrs S. Funke and Mrs H. Sippel, Universita¨t Kassel, Germany, who rechecked a large part of the relations of the Wyckoff positions in this chapter. Mrs S. E. Barnes and the rest of the staff of the International Union of Crystallography in Chester, in particular Dr N. J. Ashcroft, took care of the successful technical production of this volume. We gratefully acknowledge the financial support received from various organizations which helped to make this volume possible: in the early stages of the project from the Deutsche Forschungsgemeinschaft, Germany, and more recently from the Alexander von Humboldt-Stiftung, Germany, the International Union for Crystallography, the Sofia University Research Fund and the Universidad del Paı´s Vasco, Bilbao.

Work on Parts 1 and 2 of this volume was spread out over a long period of time and there have been a number of colleagues and friends who have helped us in the preparation of this book. We are particularly grateful to Professor Th. Hahn, RWTH Aachen, Germany, not only for providing the initial impetus for this work but also for his constant interest and support. His constructive proposals on the arrangement and layout of the tables of Part 2 improved the presentation of the data and his stimulating comments contributed considerably to the content and the organization of Part 1, particularly Chapter 1.2. Chapter 1.1 contains a contribution by Professor Y. Billiet, Bourg-Blanc, France, concerning isomorphic subgroups. Nearly all contributors to this volume, but particularly Professors J. Neubu¨ser and W. Plesken, RWTH Aachen, and Professor M. I. Aroyo commented on Chapter 1.2, correcting the text and giving valuable advice. Section 1.2.7 was completely reworked after intensive discussions with Professor V. Janovec, University of Liberec, Czech Republic, making use of his generously offered expertise in the field of domains. We thank Professor H. Ba¨rnighausen, Universita¨t Karlsruhe, Germany, for discussions and advice on Chapter 1.6. In the late 1960s and early 1970s, J. Neubu¨ser and his team at RWTH Aachen calculated the basic lattices of non-isomorphic subgroups by computer. The results now form part of the content of the tables of Chapters 2.2 and 2.3. The team provided a great deal of computer output which was used for the composition of earlier versions of the present tables and for their checking by hand. The typing and checking of the original tables was done with great care and patience by Mrs R. Henke and many other members of the Institut fu¨r Kristallographie, Universita¨t Karlsruhe. The graphs of Chapters 2.4 and 2.5 were drawn and checked by Professor W. E. Klee, Dr R. Cruse and numerous students

vi

Contents PAGE

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Computer production of Parts 2 and 3 (P. Konstantinov, A. Kirov, E. B. Kroumova, M. I. Aroyo and U. Mu¨ller) .. .. .. .. .. ..

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List of symbols and abbreviations used in this volume .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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Foreword (Th. Hahn)

Scope of this volume (M. I. Aroyo, U. Mu¨ller and H. Wondratschek)

PART 1. SPACE GROUPS AND THEIR SUBGROUPS .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

2

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2

1.1.2. Symmetry and crystal-structure determination .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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1.1. Historical introduction (M. I. Aroyo, U. Mu¨ller and H. Wondratschek) 1.1.1. The fundamental laws of crystallography

1.1.3. Development of the theory of group–subgroup relations .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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1.2. General introduction to the subgroups of space groups (H. Wondratschek) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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1.2.1. General remarks .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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1.2.2. Mappings and matrices .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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1.2.3. Groups .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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1.2.4. Subgroups .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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1.2.7. Application to domain structures .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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1.2.8. Lemmata on subgroups of space groups .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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1.1.4. Applications of group–subgroup relations

1.2.5. Space groups

1.2.6. Types of subgroups of space groups

1.3. Computer checking of the subgroup data (F. Ga¨hler) 1.3.1. Introduction

1.3.2. Basic capabilities of the Cryst package

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1.4.3. Groups .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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1.3.3. Computing maximal subgroups 1.3.4. Description of the checks

1.4. The mathematical background of the subgroup tables (G. Nebe) 1.4.1. Introduction

1.4.2. The affine space

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1.4.5. Maximal subgroups

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1.4.6. Quantitative results

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1.4.7. Qualitative results

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1.5.2. Derivative structures and phase transitions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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1.6. Relating crystal structures by group–subgroup relations (U. Mu¨ller) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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1.6.5. Handling cell transformations .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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1.4.4. Space groups

1.4.8. Minimal supergroups

1.5. Remarks on Wyckoff positions (U. Mu¨ller)

1.5.1. Crystallographic orbits and Wyckoff positions

1.5.3. Relations between the positions in group–subgroup relations

1.6.1. Introduction

1.6.2. The symmetry principle in crystal chemistry 1.6.3. Ba¨rnighausen trees

1.6.4. The different kinds of symmetry relations among related crystal structures

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CONTENTS .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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1.6.6. Comments concerning phase transitions and twin domains 1.6.7. Exercising care in the use of group–subgroup relations

1.7. The Bilbao Crystallographic Server (M. I. Aroyo, J. M. Perez-Mato, C. Capillas and H. Wondratschek) 1.7.1. Introduction

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1.7.3. Group–subgroup and group–supergroup relations between space groups .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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1.7.2. Databases and retrieval tools

1.7.4. Relations of Wyckoff positions for a group–subgroup pair of space groups

PART 2. MAXIMAL SUBGROUPS OF THE PLANE GROUPS AND SPACE GROUPS .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

2.1. Guide to the subgroup tables and graphs (H. Wondratschek and M. I. Aroyo)

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2.1.1. Contents and arrangement of the subgroup tables .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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2.1.3. I Maximal translationengleiche subgroups (t-subgroups) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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2.1.4. II Maximal klassengleiche subgroups (k-subgroups)

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2.1.2. Structure of the subgroup tables

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2.1.5. Series of maximal isomorphic subgroups (Y. Billiet) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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2.2. Tables of maximal subgroups of the plane groups (Y. Billiet, M. I. Aroyo and H. Wondratschek)

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2.3. Tables of maximal subgroups of the space groups (Y. Billiet, M. I. Aroyo and H. Wondratschek)

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2.4. Graphs for translationengleiche subgroups (V. Gramlich and H. Wondratschek) .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

433

2.1.6. The data for minimal supergroups

2.1.7. Derivation of the minimal supergroups from the subgroup tables 2.1.8. The subgroup graphs

2.4.1. Graphs of the translationengleiche subgroups with a cubic summit

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2.4.2. Graphs of the translationengleiche subgroups with a tetragonal summit

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2.4.3. Graphs of the translationengleiche subgroups with a hexagonal summit

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2.4.4. Graphs of the translationengleiche subgroups with an orthorhombic summit 2.5. Graphs for klassengleiche subgroups (V. Gramlich and H. Wondratschek)

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2.5.1. Graphs of the klassengleiche subgroups of monoclinic and orthorhombic space groups 2.5.2. Graphs of the klassengleiche subgroups of tetragonal space groups 2.5.3. Graphs of the klassengleiche subgroups of trigonal space groups 2.5.4. Graphs of the klassengleiche subgroups of hexagonal space groups

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2.5.5. Graphs of the klassengleiche subgroups of cubic space groups .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

462

PART 3. RELATIONS BETWEEN THE WYCKOFF POSITIONS 3.1. Guide to the tables (U. Mu¨ller) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

466

3.1.1. Arrangement of the entries .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

466

3.1.2. Cell transformations .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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3.1.4. Nonconventional settings of orthorhombic space groups .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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3.1.3. Origin shifts

3.1.5. Conjugate subgroups

3.1.6. Monoclinic and triclinic subgroups

3.2. Tables of the relations of the Wyckoff positions (U. Mu¨ller)

Appendix. Differences in the presentation of Parts 2 and 3 (U. Mu¨ller and H. Wondratschek) Subject index

viii

Foreword By Th. Hahn

subgroup types and for IIc only the subgroups of lowest index were given. All these data were presented in the form known in 1983, and this involved certain omissions and shortcomings in the presentation, e.g. no Wyckoff positions of the subgroups and no conjugacy relations were given. Meanwhile, both the theory of subgroups and its application have made considerable progress, and the present Volume A1 is intended to fill the gaps left in Volume A and present the ‘complete story’ of the sub- and supergroups of space groups in a comprehensive manner. In particular, all maximal subgroups of types I, IIa and IIb are listed individually with the appropriate transformation matrices and origin shifts, whereas for the infinitely many maximal subgroups of type IIc expressions are given which contain the complete characterization of all isomorphic subgroups for any given index. In addition, the relations of the Wyckoff positions for each group–subgroup pair of space groups are listed for the first time in the tables of Part 3 of this volume. In the second edition of Volume A1 (2010), additional aspects of group–subgroup relations are included; in particular procedures for the derivation of the minimal supergroups of the space groups are described. Now the minimal supergroups can be calculated from the data for the maximal subgroups to the full extent, with the exception of the low-symmetry (triclinic and monoclinic) space groups. Two new chapters on trees of group– subgroup relations (Ba¨rnighausen trees) and on the Bilbao Crystallographic Server bring these tools closer to the user. Volume A1 is thus a companion to Volume A, and the editors of both volumes have cooperated closely on problems of symmetry for many years. I wish Volume A1 the same acceptance and success that Volume A has enjoyed.

Symmetry and periodicity are among the most fascinating and characteristic properties of crystals by which they are distinguished from other forms of matter. On the macroscopic level, this symmetry is expressed by point groups, whereas the periodicity is described by translation groups and lattices, and the full structural symmetry of crystals is governed by space groups. The need for a rigorous treatment of space groups was recognized by crystallographers as early as 1935, when the first volume of the trilingual series Internationale Tabellen zur Bestimmung von Kristallstrukturen appeared. It was followed in 1952 by Volume I of International Tables for X-ray Crystallography and in 1983 by Volume A of International Tables for Crystallography (fifth edition 2002). As the depth of experimental and theoretical studies of crystal structures and their properties increased, particularly with regard to comparative crystal chemistry, polymorphism and phase transitions, it became apparent that not only the space group of a given crystal but also its ‘descent’ and ‘ascent’, i.e. its sub- and supergroups, are of importance and have to be derived and listed. This had already been done in a small way in the 1935 edition of Internationale Tabellen zur Bestimmung von Kristallstrukturen with the brief inclusion of the translationengleiche subgroups of the space groups (see the first volume, pp. 82, 86 and 90). The 1952 edition of International Tables for X-ray Crystallography did not contain sub- and supergroups, but in the 1983 edition of International Tables for Crystallography the full range of maximal subgroups was included (see Volume A, Section 2.2.15): translationengleiche (type I) and klassengleiche (type II), the latter subdivided into ‘decentred’ (IIa), ‘enlarged unit cell’ (IIb) and ‘isomorphic’ (IIc) subgroups. For types I and IIa, all subgroups were listed individually, whereas for IIb only the

ix

Scope of this volume By Mois I. Aroyo, Ulrich Mu¨ller and Hans Wondratschek

In this second edition all misprints and errors found up to now have been corrected and the number of illustrating examples has been increased. In addition, a few changes and extensions have been introduced to facilitate the use of the volume and to extend its range:

Group–subgroup relations between space groups, the primary subject of this volume, are an important tool in crystallographic, physical and chemical investigations of solids. These relations are complemented by the corresponding relations between the Wyckoff positions of the group–subgroup pairs. The basis for these tables was laid by the pioneering papers of Carl Hermann in the late 1920s. Some subgroup data were made available in Internationale Tabellen zur Bestimmung von Kristallstrukturen (1935), together with a graph displaying the symmetry relations between the crystallographic point groups. Since then, the vast number of crystal structures determined and improvements in experimental physical methods have directed the interest of crystallographers, physicists and chemists to the problems of structure classification and of phase transitions. Methods of computational mathematics have been developed and applied to the problems of crystallographic group theory, among them to the group–subgroup relations. When the new series International Tables for Crystallography began to appear in 1983, the subgroup data that were then available were included in Volume A. However, these data were incomplete and their description was only that which was available in the late 1970s. This is still the case in the present (fifth) edition of Volume A. The subgroup data for the space groups are now complete and form the basis of this volume. After introductory chapters on group-theoretical aspects of space groups, on group–subgroup relations and on the underlying mathematical background, this volume provides the reader with:

(7) The subgroup tables, in particular those of the isomorphic subgroups, have been homogenized. (8) The data for the minimal supergroups are sufficient to derive all minimal supergroups starting from the subgroup data, with the exception of the translationengleiche supergroups of triclinic and monoclinic space groups. The procedures by which this derivation can be achieved are described in detail. The supergroup data are useful for the prediction of hightemperature phase transitions, including the search for new ferroelectric and/or ferroelastic materials, for the treatment of the problem of overlooked symmetry in structure determination and for the study of phase transitions in which a hypothetical parent phase plays an important role. (9) A new chapter is devoted to the construction of family trees connecting crystal structures (Ba¨rnighausen trees). In a Ba¨rnighausen tree the relations between the Wyckoff positions occupied in the different crystal structures are accompanied by the relations between the corresponding group–subgroup pairs of space groups. Such trees display the additional degrees of freedom for the structural parameters of the low-symmetry phases: (a) the possibility of distortions due to reduction of site symmetries; (b) chemical variations (atomic substitutions) allowed for atomic positions that have become symmetryindependent.

(1) Complete listings of all maximal non-isomorphic subgroups for each space group, not just by type but individually, including their general positions or their generators, their conjugacy relations and transformations to their conventional settings. (2) Listings of the maximal isomorphic subgroups with index 2, 3 or 4 individually in the same way as for non-isomorphic subgroups. (3) Listings of all maximal isomorphic subgroups as members of infinite series, but with the same information as for the nonisomorphic subgroups. (4) The listings of Volume A for the non-isomorphic supergroups for all space groups. (5) Two kinds of graphs for all space groups displaying their types of translationengleiche subgroups and their types of non-isomorphic klassengleiche subgroups. (6) Listings of all the Wyckoff positions for each space group with their splittings and/or site-symmetry reductions if the symmetry is reduced to that of a maximal subgroup. These data include the corresponding coordinate transformations such that the coordinates in the subgroup can be obtained directly from the coordinates in the original space group.

Ba¨rnighausen trees visualize in a compact manner the structural relations between different polymorphic modifications involved in a phase transition and enable the comparison of crystal structures and their classification into crystal-structure types. (10) A new chapter is dedicated to the Bilbao Crystallographic Server, http://www.cryst.ehu.es/. The server offers freely accessible crystallographic databases and computer programs, in particular those related to the contents of this volume. The available computer tools permit the studies of general group–subgroup relations between space groups and the corresponding Wyckoff positions. The data in this volume are indispensable for a thorough analysis of phase transitions that do not involve drastic structural changes: the group–subgroup relations indicate the possible symmetry breaks that can occur during a phase transition and are essential for determining the symmetry of the driving mechanism and the related symmetry of the resulting phase. The group– subgroup graphs describing the symmetry breaks provide information on the possible symmetry modes taking part in the transition and allow a detailed analysis of domain structures and twins.

x

Computer production of Parts 2 and 3 By Preslav Konstantinov, Asen Kirov, Eli B. Kroumova, Mois I. Aroyo and Ulrich Mu¨ller

The different types of data in the LATEX 2" files were either keyed by hand or computer-generated. The preparation of the data files of Part 2 can be summarized as follows:

The tables of this volume were produced electronically using the LATEX 2" typesetting system [Lamport (1994). A Document Preparation System. 2nd ed. Reading: Addison-Wesley], which has the following advantages:

(i) Headline, origin: hand-keyed. (ii) Generators: hand-keyed. (iii) General positions: created by a program from a set of generators. The algorithm uses the well known generating process for space groups based on their solvability property, cf. Section 8.3.5 of IT A. (iv) Maximal subgroups: hand-keyed. The data for the subgroup generators (or general-position representatives for the cases of translationengleiche subgroups and klassengleiche subgroups with ‘loss of centring translations’), for transformation matrices and for conjugacy relations between subgroups were checked by specially designed computer programs. (v) Minimal supergroups: created automatically from the data for maximal subgroups.

(1) correcting and modifying the layout and the data are easy; (2) correcting or updating these for further editions of this volume should also be simple; (3) the cost of production for the first edition and later editions should be kept low. A separate data file was created for every space group in each setting listed in the tables. These files contained only the information about the subgroups and supergroups, encoded using specially created LATEX 2" commands and macros. These macros were defined in a separate package file which essentially contained the algorithm for the layout. Keeping the formatting information separate from the content as much as possible allowed us to change the layout by redefining the macros without changing the data files. This was done several times during the production of the tables. The data files are relatively simple and only a minimal knowledge of LATEX 2" is required to create and revise them should it be necessary later. A template file was used to facilitate the initial data entry by filling blank spaces and copying pieces of text in a text editor. It was also possible to write computer programs to extract the information from the data files directly. Such programs were used for checking the data in the files that were used to typeset the volume. The data prepared for Part 2 were later converted into a more convenient, machine-readable format so that they could be used in the database of the Bilbao Crystallographic Server at http://www.cryst.ehu.es/. The final composition of all plane-group and space-group tables of maximal subgroups and minimal supergroups was done by a single computer job. References in the tables from one page to another were automatically computed. The run takes 1 to 2 minutes on a modern workstation. The result is a PostScript or pdf file which can be fed to most laser printers or other modern printing/typesetting equipment. The resulting files were also used for the preparation of the fifth edition of International Tables for Crystallography Volume A (2002) (abbreviated as IT A). Sections of the data files of Part 2 of the present volume were transferred directly to the data files for Parts 6 and 7 of IT A to provide the subgroup and supergroup information listed there. The formatting macros were rewritten to achieve the layout used in IT A.

The electronic preparation of the subgroup tables of Part 2 was carried out on various Unix- and Windows-based computers in Sofia, Bilbao, Stuttgart and Karlsruhe. The development of the computer programs and the layout macros in the package file was done in parallel by different members of the team. Th. Hahn (Aachen) contributed to the final arrangement of the data. The tables of Part 3 have a different layout, and a style file of their own was created for their production. Again, separate data files were prepared for every space group, containing only the information concerning the subgroups. The macros of the style file were developed by U. Mu¨ller, who also hand-keyed all files over the course of seven years. Most of the data of Part 2 were checked using computer programs developed by F. Ga¨hler (cf. Chapter 1.3) and A. Kirov. The relations of the Wyckoff positions (Part 3) were checked by G. Nolze (Berlin) with the aid of his computer program POWDER CELL [Nolze (1996). POWDER CELL. Computer program for the calculation of X-ray powder diagrams. Bundesanstalt fu¨r Materialforschung, Berlin]. In addition, all relations were cross-checked with the program WYCKSPLIT [Kroumova et al. (1998). J. Appl. Cryst. 31, 646; http://www. cryst.ehu.es/cryst/wpsplit.html], with the exception of the positions of high multiplicities of some cubic space groups with subgroup indices > 50, which could not be handled by the program.

xi

List of symbols and abbreviations used in this volume By Mois I. Aroyo, Ulrich Mu¨ller and Hans Wondratschek (1) Points and point space

(4) Groups

P, Q, R, X O An ; An ; Pn En ; En x; y; z; or xi x X~ x~ x~ i x0

G R H; U M M P; S; V; Z T ðGÞ; T ðRÞ A E

x0i

points origin n-dimensional affine space n-dimensional Euclidean point space point coordinates column of point coordinates image point column of coordinates of an image point coordinates of an image point column of coordinates in a new coordinate system (after basis transformation) coordinates in a new coordinate system

F I N O

(2) Vectors and vector space a, b, c; or ai r, x o a, b, c , , ; or j r ri (a)T Vn

NG ðHÞ NE ðHÞ NA ðHÞ PG ; PH SG ðXÞ; SH ðXÞ

basis vectors of the space vectors, position vectors zero vector (all coefficients zero) lattice lengths of basis vectors parameters angles between basis vectors column of vector coefficients vector coefficients row of basis vectors n-dimensional vector space

a; b ; g ; h; m; t e

2; 21 ; m; 1; . . . i or [i]

group; space group space group (Chapter 1.4) subgroups of G maximal subgroup of G (Chapter 1.4) Hermann’s group (Chapters 1.2, 1.7, 2.1) groups or sets of group elements, e.g. cosets group of all translations of G; R group of all affine mappings = affine group group of all isometries (motions) = Euclidean group factor group trivial group, consisting of the unit element e only normal subgroup group of all orthogonal mappings = orthogonal group normalizer of H in G Euclidean normalizer of H affine normalizer of H point groups of the space groups G, H site-symmetry groups of point X in the space groups G, H group elements unit element symmetry operations index of H in G

(5) Symbols used in the tables p n, n0 , n00 , n000 q, r, u, v, w a, b, c a0 , b0 , c 0 x, y, z t(1, 0, 0), t(0, 1, 0), . . .

(3) Mappings and their matrices and columns A, W (3 3) matrices T A matrix A transposed I (3 3) unit matrix Aik, Wik coefficients (A, a), (W, w) matrix–column pairs W augmented matrix x; x~ ; t augmented columns P, P transformation matrices A; I ; W mappings w column of the translation part of a mapping wi coefficients of the translation part of a mapping fundamental matrix and its coefficients G, Gik det( . . . ) determinant of a matrix tr( . . . ) trace of a matrix

prime number > 1 arbitrary positive integer numbers arbitrary integer numbers in the given range basis vectors of the space group basis vectors of the subgroup or supergroup point coordinates in the space group generating translations

(6) Abbreviations HM symbol IT A PCA k-subgroup t-subgroup

xii

Hermann–Mauguin symbol International Tables for Crystallography Volume A parent-clamping approximation klassengleiche subgroup translationengleiche subgroup

International Tables for Crystallography (2011). Vol. A1, Chapter 1.1, pp. 2–6.

1.1. Historical introduction By Mois I. Aroyo, Ulrich Mu¨ller and Hans Wondratschek

concepts of unit cell and of translation symmetry were thus introduced implicitly. The classiﬁcation of the 14 underlying lattices of translations was completed by Auguste Bravais (1850). In the second half of the nineteenth century, interest turned to the derivation and classiﬁcation of inﬁnite regular systems of points or ﬁgures, in particular by Leonhard Sohncke and Evgraf Stepanovich Fedorov. Sohncke (1879), Fedorov (1891), Arthur Schoenﬂies (1891) and later William Barlow (1894) turned to the investigation of the underlying groups, the space groups and plane groups. The derivation and classiﬁcation of these groups were completed in the early 1890s. It was a plausible hypothesis that the structures of crystals were related to combinations of regular systems of atoms (Haag, 1887) and that the symmetry of a crystal structure should be a space group, but both conjectures were speculations at that time with no experimental proof. This also applies to the atom packings described by Sohncke and Barlow, such as a model of the NaCl structure (which they did not assign to any substance).

1.1.1. The fundamental laws of crystallography The documentation concerning group-theoretical aspects of group–subgroup relations and the rising importance of these relations in three-dimensional crystallography is scattered widely in the literature. This short review, therefore, cannot be exhaustive and may even be unbalanced if the authors have missed essential sources. Not included here is the progress made in the general theory of crystallographic groups, e.g. in higher dimensions, which is connected with the names of the mathematicians and physicists Ascher, Brown, Janner, Janssen, Neubu¨ser, Plesken, Souvignier, Zassenhaus and others. This volume A1 of International Tables for Crystallography, abbreviated IT A1, is concerned with objects belonging to the ‘classical’ theory of crystallographic groups. For a long time, the objective of crystallography, as implied by the word itself, was the description of crystals which were found in nature or which grew from solutions of salts or from melts. Crystals often display more-or-less planar faces which are symmetrically equivalent or can be made so by parallel shifting of the faces along their normals such that they form a regular body. The existence of characteristic angles between crystal faces was observed by Niels Stensen in 1669. The importance of the shape of crystals and their regularity was supported by observations on the cleavage of crystals. It was in particular this regularity which attracted mineralogists, chemists, physicists and eventually mathematicians, and led to the establishment of the laws by which this regularity was governed. The law of symmetry and the law of rational indices were formulated by Rene´ Just Hau¨y around 1800. These studies were restricted to the shape of macroscopic crystals and their physical properties, because only these were accessible to measurements until the beginning of the twentieth century. Later in the nineteenth century, interest turned to ‘regular systems’ of points, for which the arrangement of points around every point is the same. These were studied by Wiener (1863) and Sohncke (1874). For such sets, there are isometric mappings of any point onto any other point which are such that they map the whole set onto itself. The primary aim was the classiﬁcation and listing of such regular systems, while the groups of isometries behind these systems were of secondary importance. These regular systems were at ﬁrst the sets of crystal faces, of face normals of crystal faces and of directions in crystals which are equivalent with respect to the physical properties or the symmetry of the external shape of the crystal. The listing and the classiﬁcation of these ﬁnite sets was based on the law of rational indices and resulted in the derivation of the 32 crystal classes of point groups by Moritz Ludwig Frankenheim in 1826, Johann Friedrich Christian Hessel in 1830 and Axel V. Gadolin in 1867, cited by Burckhardt (1988). The list of this classiﬁcation contained crystal classes that had not been observed in any crystal and included group–subgroup relations implicitly. Crystal cleavage led Hau¨y in 1784 to assume small parallelepipeds as building blocks of crystals. In 1824, Ludwig August Seeber explained certain physical properties of crystals by placing molecules at the vertices of the parallelepipeds. The Copyright © 2011 International Union of Crystallography

1.1.2. Symmetry and crystal-structure determination In 1895, Wilhelm Ro¨ntgen, then at the University of Wu¨rzburg, discovered what he called X-rays. Medical applications emerged the following year, but it was 17 years later that Max von Laue suggested at a scientiﬁc discussion in Munich that a crystal should be able to act as a diffraction grating for X-rays. Two young physicists, Walther Friedrich and Paul Knipping, successfully performed the experiment in May 1912 with a crystal of copper sulfate. In von Laue’s opinion, the experiment was so important that he later publicly donated one third of his 1914 Nobel prize to Friedrich and Knipping. The discovery immediately aroused the curiosity of father William Henry Bragg and son William Lawrence Bragg. The son’s experiments in Cambridge, initially with NaCl and KCl, led to the development of the Bragg equation and to the ﬁrst crystal-structure determinations of diamond and simple inorganic materials. Since then, the determination of crystal structures has been an ever-growing enterprise. The diffraction of X-rays by crystals is partly determined by the space group and partly by the relative arrangement of the atoms, i.e. by the atomic coordinates and the lattice parameters. The presentations by Fedorov and Schoenﬂies of the 230 space groups were not yet appropriate for use in structure determinations with X-rays. The breakthrough came with the fundamental book of Paul Niggli (1919), who described the space groups geometrically by symmetry elements and point positions and provided the ﬁrst tables of what are now called Wyckoff positions. Niggli emphasized the importance of the multiplicity and site symmetry of the positions and demonstrated with examples the meaning of the reﬂection conditions. Niggli’s book pointed the way. The publication of related tables by Ralph W. G. Wyckoff (1922) included diagrams of the unit cells with special positions and symmetry elements. Additional tables by Astbury & Yardley (1924) listed ‘abnormal

2

1.1. HISTORICAL INTRODUCTION spacings’ for the space groups, i.e. the reﬂection conditions. These tables made the concepts and the data of geometric crystallography widely available and were the basis for the series Internationale Tabellen zur Bestimmung von Kristallstrukturen (1935) (abbreviated as IT 35), International Tables for X-ray Crystallography, Vol. I (1952, 1965, 1969) (abbreviated as IT 52) and International Tables for Crystallography, Vol. A (1983 and subsequent editions 1987, 1992, 1995, 2002, 2005) (abbreviated as IT A). Group–subgroup relations were used in the original derivation of the space groups and plane groups. However, in the ﬁrst decades of crystal-structure determinations, the derivation of geometric data (atomic coordinates) was of prime importance and the group-theoretical information in the publications was small, although implicitly present. With the growing number of crystal structures determined, however, it became essential to understand the rules and laws of crystal chemistry, to classify the incomprehensible set of structures into crystal-structure types, to develop methods for ordering the structure types in a systematic way, to show relations among them and to ﬁnd common underlying principles. To this end, different approaches were presented over time. By 1926, the number of crystal structures was already large enough for Viktor Moritz Goldschmidt to formulate basic principles of packing of atoms and ions in inorganic solids (Goldschmidt, 1926). Shortly afterwards, Linus Pauling (1928, 1929) formulated his famous rules about ionic radii, valence bonds, coordination polyhedra and the joining of these polyhedra. Later, Wilhelm Biltz (1934) focused attention on the volume requirements of atoms. Many other important factors determining crystal structures, such as chemical bonding, molecular shape, valenceelectron concentration, electronic band structures, crystal-orbital overlap populations and others, have been the subject of subsequent studies. Each one of these aspects can serve as an ordering principle in crystal chemistry, giving insights from a different point of view. For the aspects mentioned above, symmetry considerations are only secondary tools or even unimportant, and group–subgroup relations hardly play a role. Although symmetry is indispensable for the description of a speciﬁc crystal structure, for a long time crystal symmetry and the group–subgroup relations involved did not attract much attention as possible tools for working out the relations between crystal structures. This has even been the case for most textbooks on solid-state chemistry and physics. The lack of symmetry considerations is almost a characteristic feature of many of these books. There is a reason for this astonishing fact: the necessary group-theoretical material only became available in a useful form in 1965, namely as a listing of the maximal subgroups of all space groups by Neubu¨ser & Wondratschek. However, for another 18 years this material was only distributed among interested scientists before it was ﬁnally included in the 1983 edition of IT A. And yet, even in the printing of 2005, the listing of the subgroups in Volume A is incomplete. It is this present Volume A1 which contains the complete listing.

(I) a nomenclature of the space-group types, a predecessor of the Hermann–Mauguin nomenclature; (II) a method for the derivation of the 230 space-group types which is related to the nomenclature in (I); (III) the derivation of the 75 types of rod groups and the 80 types of layer groups; and (IV) the subgroups of space groups. In paper (IV), Hermann introduced two distinct kinds of subgroups. The translationengleiche subgroups of a space group G have retained all translations of G but belong to a crystal class of lower symmetry than the crystal class of G (Hermann used the term zellengleiche instead of translationengleiche, see footnote 10 in Section 1.2.6). The klassengleiche subgroups are those which belong to the same crystal class as G, but have lost translations compared with G. General subgroups are those which have lost translations as well as crystal-class symmetry. Hermann proved the theorem, later called Hermann’s theorem, that any general subgroup is a klassengleiche subgroup of a uniquely determined translationengleiche subgroup M of G. In particular, this implies that a maximal subgroup of G is either a translationengleiche subgroup or a klassengleiche subgroup of G. Because of the strong relation (homomorphism) between a space group G and its point group P G, the set of translationengleiche subgroups of G is in a one-to-one correspondence with the set of subgroups of the point group P G. The crystallographic point groups are groups of maximal order 48 with well known group–subgroup relations and with not more than 96 subgroups. Thus, the maximal translationengleiche subgroups of any space group G can be obtained easily by comparison with the subgroups of its point group P G. The kind of derivation of the space-group types by H. Heesch (1930) also gives access to translationengleiche subgroups. In IT 35, the types of the translationengleiche subgroups were listed for each space group [for a list of corrections to these data, see Ascher et al. (1969)]. A graph of the group–subgroup relations between the crystallographic point groups can also be found in IT 35; the corresponding graphs for the space groups were published by Ascher (1968). In these lists and graphs the subgroups are given only by their types, not individually. The group–subgroup relations between the space groups were ﬁrst applied in Volume 1 of Strukturbericht (1931). In this volume, a crystal structure is described by the coordinates of the atoms, but the space-group symmetry is stated not only for spherical particles but also for molecules or ions with lower symmetry. Such particles may reduce the site symmetry and with it the space-group symmetry to that of a subgroup. In addition, the symmetry reduction that occurs if the particles are combined into larger structural units is stated. The listing of these detailed data was discontinued both in the later volumes of Strukturbericht and in the series Structure Reports. Meanwhile, experience had shown that there is no point in assuming a lower symmetry of the crystal structure if the geometrical arrangement of the centres of the particles does not indicate it. With time, not only the classiﬁcation of the crystal structures but also a growing number of investigations of (continuous) phase transitions increased the demand for data on subgroups of space groups. Therefore, when the Executive Committee of the International Union of Crystallography decided to publish a new series of International Tables for Crystallography, an extension of the subgroup data was planned. Stimulated and strongly supported by the mathematician J. Neubu¨ser, the systematic derivation of the subgroups of the plane groups and the space groups

1.1.3. Development of the theory of group–subgroup relations The systematic survey of group–subgroup relations of space groups started with the fundamental publication by Carl Hermann (1929). This paper is the last in a series of four publications, dealing with

3

1. SPACE GROUPS AND THEIR SUBGROUPS began. The listing was restricted to the maximal subgroups of each space group, because any subgroup of a space group can be obtained by a chain of maximal subgroups. The derivation by Neubu¨ser & Wondratschek started in 1965 with the translationengleiche subgroups of the space groups, because the complete set of these (maximally 96) subgroups could be calculated by computer. All klassengleiche subgroups of indices 2, 3, 4, 6, 8 and 9 were also obtained by computer. As the index of a maximal non-isomorphic subgroup of a space group is restricted to 2, 3 or 4, all maximal non-isomorphic subgroups of all space groups were contained in the computer outputs. First results and their application to relations between crystal structures are found in Neubu¨ser & Wondratschek (1966). In the early tables, the subgroups were only listed by their types. For International Tables, an extended list of maximal non-isomorphic subgroups was prepared. For each space group the maximal translationengleiche subgroups and those maximal klassengleiche subgroups for which the reduction of the translations could be described as ‘loss of centring translations’ of a centred lattice are listed individually. For the other maximal klassengleiche subgroups, i.e. those for which the conventional unit cell of the subgroup is larger than that of the original space group, the description by type was retained, because the individual subgroups of this kind were not completely known in 1983. The deﬁciency of such a description becomes clear if one realizes that a listed subgroup type may represent 1, 2, 3, 4 or even 8 individual subgroups. In the present Volume A1, all maximal non-isomorphic subgroups are listed individually, in Chapter 2.2 for the plane groups and in Chapters 2.3 and 3.2 for the space groups. In addition, graphs for the translationengleiche subgroups (Chapter 2.4) and for the klassengleiche subgroups (Chapter 2.5) supplement the tables. After several rounds of checking by hand and after comparison with other listings, e.g. those by H. Zimmermann (unpublished) or by Neubu¨ser and Eick (unpublished), intensive computer checking of the hand-typed data was carried out by F. Ga¨hler as described in Chapter 1.3. The mathematician G. Nebe describes general viewpoints and new results in the theory of subgroups and supergroups of space groups in Chapter 1.4. The maximal isomorphic subgroups are a special subset of the maximal klassengleiche subgroups. Maximal isomorphic subgroups are treated separately because each space group G has an inﬁnite number of maximal isomorphic subgroups and, in contrast to non-isomorphic subgroups, there is no limit for the index of a maximal isomorphic subgroup of G. An isomorphic subgroup of a space group seems to have ﬁrst been described in a crystal–chemical relation when the crystal structure of Sb2ZnO6 (structure type of tapiolite, Ta2FeO6) was determined by Bystro¨m et al. (1941): ‘If no distinction is drawn between zinc and antimony, this structure appears as three cassiterite-like units stacked end-on-end’ (Wyckoff, 1965). The space group of Sb2ZnO6 is a maximal isomorphic subgroup of index 3 with c0 ¼ 3c of the space group P42 =mnm, No. 136, of cassiterite SnO2 (rutile type, TiO2). The ﬁrst systematic study attempting to enumerate all isomorphic subgroups (not just maximal ones) for each space-group type was by Billiet (1973). However, the listing was incomplete and, moreover, in the case of enantiomorphic pairs of spacegroup types, only those with the same space-group symbol (called isosymbolic subgroups) were taken into account. Sayari (1976) derived the conventional bases for all maximal isomorphic subgroups of all plane groups. The general laws of

number theory which underlie these results for plane-group types p4, p3 and p6 and space-group types derived from point groups 4, 4, 4=m, 3, 3, 6, 6 and 6=m were published by Mu¨ller & Brelle (1995). Bertaut & Billiet (1979) suggested a new analytical approach for the derivation of all isomorphic subgroups of space and plane groups. Because of the inﬁnite number of maximal isomorphic subgroups, only a few representatives of lowest index are listed in IT A with their lattice relations but without origin speciﬁcation, cf. IT A (2005), Section 2.2.15.2. Part 13 of IT A (Billiet & Bertaut, 2005) is fully devoted to isomorphic subgroups, cf. also Billiet (1980) and Billiet & Sayari (1984). In this volume, all maximal isomorphic subgroups are listed as members of inﬁnite series, where each individual subgroup is speciﬁed by its index, its generators, its basis and the coordinates of its conventional origin as parameters. The relations between a space group and its subgroups become more transparent if they are considered in connection with their normalizers in the afﬁne group A and the Euclidean group E (Koch, 1984). Even the corresponding normalizers of Hermann’s group M play a role in these relations, cf. Wondratschek & Aroyo (2001). In addition to subgroup data, supergroup data are listed in IT A. If H is a maximal subgroup of G, then G is a minimal supergroup of H. In IT A, the type of a space group G is listed as a minimal non-isomorphic supergroup of H if H is listed as a maximal non-isomorphic subgroup of G. Thus, for each space group H one can ﬁnd in the tables the types of those groups G for which H is listed as a maximal subgroup. The supergroup data of IT A1 are listed also not individually but in such a way that most supergroups may be obtained by inversion from the subgroup data. The procedure is described in Sections 2.1.6 and 2.1.7. 1.1.4. Applications of group–subgroup relations Phase transitions. In 1937, Landau introduced the idea of the order parameter for the description of second-order phase transitions (Landau, 1937). Landau theory has turned out to be very useful in the understanding of phase transitions and related phenomena. A second-order transition can only occur if there is a group–subgroup relation between the space groups of the two crystal structures. Often only the space group of one phase is known (usually the high-temperature phase) and subgroup relations help to eliminate many groups as candidates for the unknown space group of the other phase. Landau & Lifshitz (1980) examined the importance of group–subgroup relations further and formulated two theorems regarding the index of the group–subgroup pair. The signiﬁcance of the subgroup data in second-order phase transitions was also pointed out by Ascher (1966, 1967), who formulated the maximal-subgroup rule: ‘The symmetry group of a phase that arises in a ferroelectric transition is a maximal polar subgroup of the group of the high-temperature phase.’ There are analogous applications of the maximalsubgroup rule (with appropriate modiﬁcations) to other types of continuous transitions. The group-theoretical aspects of Landau theory have been worked out in great detail with major contributions by Birman (1966a,b), Cracknell (1975), Stokes & Hatch (1988), Tole´dano & Tole´dano (1987) and many others. For example, Landau theory gives additional criteria based on thermodynamic arguments for second-order phase transitions. The general statements are reformulated into group-theoretical rules which permit a phasetransition analysis without the tedious algebraic treatment

4

1.1. HISTORICAL INTRODUCTION oped by Ba¨rnighausen (1980). The work became more widely known through a number of courses taught in Germany, France and Italy from 1976 to 2008. For a script of the 1992 course, see Chapuis (1992). The script of the 2008 course is available at http:// www.crystallography.fr/mathcryst/gargnano2008.htm. The basic ideas can also be found in the textbook by Mu¨ller (2006). For a review see Mu¨ller (2004). Details are presented in Chapter 1.6. Prediction of crystal structures. Ba¨rnighausen trees (see Chapter 1.6) may even include structures as yet unknown, i.e. the symmetry relations can also serve to predict new structure types that are derived from the aristotype, cf. Section 1.6.4.7. In addition, the number of such structure types can be calculated for each space group of the tree. Setting up a Ba¨rnighausen tree not only requires one to ﬁnd the group–subgroup relations between the space groups involved. It also requires there to be an immediate correspondence between the atomic positions of the crystal structures considered. For a given structure, each atomic position belongs to a certain Wyckoff position of the space group. Upon transition to a subgroup, the Wyckoff position will or will not split into different Wyckoff positions of the subgroup. With the growing number of applications of group–subgroup relations there had been an increasing demand for a list of the relations of the Wyckoff positions for every group–subgroup pair. These listings are accordingly presented in Part 3 of this volume.

involving high-order polynomials. The necessity of having complete subgroup data for the space groups for the successful implementation of these rules was stated by Deonarine & Birman (1983): ‘ . . . there is a need for tables yielding for each of the 230 three-dimensional space groups a complete lattice of decomposition of all its subgroups.’ The group–subgroup relations between space groups are fundamental for the so-called symmetry-mode analysis of phase transitions that allows the determination of the contributions of the primary and secondary order parameters to the structural distortion characterizing the transition (Aroyo & Perez-Mato, 1998). In treating successive phase transitions within Landau theory, Levanyuk & Sannikov (1971) introduced the idea of a hypothetical parent phase whose symmetry group is a supergroup of the observed (initial) space group. Moreover, the detection of pseudosymmetries is necessary for the prediction of highertemperature phase transitions, cf. Kroumova et al. (2002) and references therein. In a number of cases, transitions between two phases with no group–subgroup relations between their space groups have been analysed using common subgroups. For examples, see Sections 1.6.4.5 and 1.7.3.1.5. Domain structures and twinning. Domain-structure analysis (Janovec & Prˇı´vratska´, 2003; Janovec et al., 2003) is another aspect of phase-transition problems where group–subgroup relations between space groups play an essential role. The mutual orientations of twin domains resulting from phase transitions involving a group–subgroup relation can be derived. This is also an aid in the crystal-structure determination of twinned crystals. Domain structures are considered in Sections 1.2.7 and 1.6.6. Topotactic reactions. Bernal (1938) observed that in the solidstate reaction MnðOHÞ2 ! MnOOH ! MnO2 the starting and the product crystal had the same orientation. Chemical reactions in the solid state are now called topotactic reactions if the crystal orientations of the reactant and the product(s) are related (Giovanoli & Leuenberger, 1969; Lotgering, 1959). Frequently, like in the case of phase transitions, the space groups of the reactant and the product(s) are connected by a group–subgroup relation which allows one to explain the domain structure of the product crystals. cf. Sections 1.6.2 and 1.6.6. Overlooked symmetry. Since powerful diffractometers and computer programs for crystal-structure determination have become widespread and easy to handle, the number of crystalstructure determinations with a wrongly assigned space group has been increasing (Baur & Kassner, 1992; Clemente, 2005; Marsh et al., 2002; Marsh & Clemente, 2007). Frequently, space groups with too low a symmetry have been chosen. In most cases, the correct space group is a supergroup of the space group that has been assigned. A criterion for the correct assignment of the space group is given by Fischer & Koch (1983). Computer packages for treating the problem can be made more efﬁcient if the possible supergroups are known. Twinning can also lead to a wrong space-group assignment if it is not recognized, as a twinned crystal can feign a higher or lower symmetry. The true space group of the correct structure is usually a supergroup or subgroup of the space group that has been assumed (Nespolo & Ferraris, 2004). Solution of the phase problem. Group–subgroup relations can be of help in solving the phase problem in the crystal-structure determination of proteins (Di Costanzo et al., 2003). Relations between crystal structures. Working out relations between different crystal structures with the aid of crystallographic group–subgroup relations was systematically devel-

References Aroyo, M. I. & Perez-Mato, J. M. (1998). Symmetry-mode analysis of displacive phase transitions using International Tables for Crystallography. Acta Cryst. A54, 19–30. Ascher, E. (1966). Role of particular maximal subgroups in continuous phase transitions. Phys. Lett. 20, 352–354. Ascher, E. (1967). Symmetry changes in continuous transitions. A simpliﬁed theory applied to V3 Si. Chem. Phys. Lett. 1, 69–72. Ascher, E. (1968). Lattices of Equi-translation Subgroups of the Space Groups. Geneva: Battelle Institute. Ascher, E., Gramlich, V. & Wondratschek, H. (1969). Corrections to the sections ‘Untergruppen’ of the space groups in Internationale Tabellen zur Bestimmung von Kristallstrukturen (1935) Vol. I. Acta Cryst. B25, 2154–2156. Astbury, W. T. & Yardley, K. (1924). Tabulated data for the examination of the 230 space-groups by homogeneous X-rays. Philos. Trans. R. Soc. London, 224, 221–257. ¨ ber die geometrischen Eigenschaften homogener Barlow, W. (1894). U starrer Strukturen. Z. Kristallogr. Mineral. 23, 1–63. Ba¨rnighausen, H. (1980). Group–subgroup relations between space groups: a useful tool in crystal chemistry. MATCH Commun. Math. Chem. 9, 139–175. Baur, W. H. & Kassner, D. (1992). The perils of Cc: comparing the frequencies of falsely assigned space groups with their general population. Acta Cryst. B48, 356–369. Bernal, J. D. (1938). Conduction in solids and diffusion and chemical change in solids. Geometrical factors in reactions involving solids. Trans. Faraday Soc. 34, 834–839. Bertaut, E. F. & Billiet, Y. (1979). On equivalent subgroups and supergroups of the space groups. Acta Cryst. A35, 733–745. Billiet, Y. (1973). Les sous-groupes isosymboliques des groupes spatiaux. Bull. Soc. Fr. Mine´ral. Cristallogr. 96, 327–334. Billiet, Y. (1980). The subgroups of ﬁnite index of the space groups: determination via conventional coordinate systems. MATCH Commun. Math. Chem. 9, 127–190. Billiet, Y. & Bertaut, E. F. (2005). Isomorphic subgroups of space groups. International Tables for Crystallography, Vol. A, Space-Group Symmetry, edited by Th. Hahn, Part 13. Dordrecht: Kluwer Academic Publishers. Billiet, Y. & Sayari, A. (1984). Les sous-groupes isomorphes d’un group d’espace de type p4. I. De´termination univoque. Acta Cryst. A40, 624– 631. Biltz, W. (1934). Raumchemie der festen Stoffe. Leipzig: L. Voss.

5

1. SPACE GROUPS AND THEIR SUBGROUPS search. Results for materials with Pba2 and Pmc21 symmetry. Acta Cryst. B58, 921–933. Landau, L. D. (1937). Theory of phase transitions. Zh. Eksp. Teoret. Fiz. 7, pp. 19–32, 627–636. (In Russian.) (German: Phys. Z. Sowjetunion, 11, pp. 20–47, 545–555.) Landau, L. D. & Lifshitz, E. M. (1980). Statistical Physics, Part 1, 3rd ed., pp. 459–471. London: Pergamon. (Russian: Statisticheskaya Fizika, chast 1. Moskva: Nauka, 1976; German: Lehrbuch der theoretischen Physik, 6. Auﬂ., Bd. 5, Teil 1, S. 436–447. Berlin: Akademie-Verlag, 1984). Levanyuk, A. P. & Sannikov, D. G. (1971). Phenomenological theory of dielectric anomalies in ferroelectric materials with several phase transitions at temperatures close together. Sov. Phys. JETP, 11, 600– 604. Lotgering, F. K. (1959). Topotactical reactions with ferrimagnetic oxides having hexagonal crystal structures – I. J. Inorg. Nucl. Chem. 9, 113– 123. Marsh, R. E. & Clemente, D. A. (2007). A survey of crystal structures published in the Journal of the American Chemical Society. Inorg. Chim. Acta, 360, 4017–4024. Marsh, R. E., Kapon, M., Hu, S. & Herbstein, F. H. (2002). Some 60 new space-group corrections. Acta Cryst. B58, 62–77. Mu¨ller, U. (2004). Kristallographische Gruppe–Untergruppe-Beziehungen und ihre Anwendung in der Kristallchemie. Z. Anorg. Allg. Chem. 630, 1519–1537. Mu¨ller, U. (2006). Inorganic Structural Chemistry, 2nd ed., pp. 212–225. Chichester: Wiley. (German: Anorganische Strukturchemie, 6. Auﬂ., 2008, S. 308–327. Wiesbaden: Teubner.) ¨ ber isomorphe Untergruppen von Mu¨ller, U. & Brelle, A. (1995). U 4=m, 3, 3, 6, 6 und 6=m. Acta Raumgruppen der Kristallklassen 4, 4, Cryst. A51, 300–304. Nespolo, M. & Ferraris, G. (2004). Applied geminography – symmetry analysis of twinned crystals and deﬁnition of twinning by reticular polyholohedry. Acta Cryst. A60, 89–95. Neubu¨ser, J. & Wondratschek, H. (1966). Untergruppen der Raumgruppen. Krist. Tech. 1, 529–543. Niggli, P. (1919). Geometrische Kristallographie des Diskontinuums. Leipzig: Gebru¨der Borntraeger. Reprint (1973) Wiesbaden: Dr M. Saendig. Pauling, L. (1928). The coordination theory of the structure of ionic compounds. Probleme der modernen Physik (Sommerfeld-Festschrift), p. 11. Leipzig: S. Hirzel. Pauling, L. (1929). The principles determining the structures of complex ionic crystals. J. Am. Chem. Soc. 51, 1010–1026. Sayari, A. (1976). Contribution a` l’e´tude des groupes d’espace bidimensionnels, de´rivation des sous-groupes et des surstructures. Thesis, University of Tunis, Tunisia. Schoenﬂies, A. M. (1891). Krystallsysteme und Krystallstruktur. Leipzig: Teubner. Reprint (1984): Springer. Sohncke, L. (1874). Die regelma¨ssigen ebenen Punktsysteme von unbegrenzter Ausdehnung. J. Reine Angew. Math. 77, 47–101. Sohncke, L. (1879). Entwickelung einer Theorie der Krystallstruktur. Leipzig: Teubner. Stokes, H. T. & Hatch, D. M. (1988). Isotropy Subgroups of the 230 Crystallographic Space Groups. Singapore: World Scientiﬁc. Strukturbericht 1913–1928 (1931). Edited by P. P. Ewald & C. Hermann. Leipzig: Akademische Verlagsgesellschaft. Continued by Structure Reports. Tole´dano, J.-C. & Tole´dano, P. (1987). The Landau Theory of Phase Transitions. Singapore: World Scientiﬁc. Wiener, C. (1863). Grundzu¨ge der Weltordnung. I. Atomlehre, p. 82. Leipzig, Heidelberg: Wintersche Verlagsbuchhandlung. Wondratschek, H. & Aroyo, M. I. (2001). The application of Hermann’s group M in group–subgroup relations between space groups. Acta Cryst. A57, 311–320. Wyckoff, R. W. G. (1922). The Analytical Expression of the Results of the Theory of Space Groups. Washington: Carnegie Institution. Wyckoff, R. W. G. (1965). Crystal Structures, 2nd ed., Vol. 3, pp. 361–362. New York: Interscience.

Birman, J. L. (1966a). Full group and subgroup methods in crystal physics. Phys. Rev. 150, 771–782. Birman, J. L. (1966b). Simpliﬁed theory of symmetry change in second– order phase transitions: application to V3 Si. Phys. Rev. Lett. 17, 1216– 1219. Bravais, A. (1850). Me´moire sur les syste`mes forme´s par les points distribue´s re´gulie`rement sur un plan ou dans l’espace. J. Ecole Polytech. 19, 1–128. (English: Memoir 1, Crystallographic Society of America, 1949.) Burckhardt, J. J. (1988). Die Symmetrie der Kristalle. Basel: Birkha¨user. Bystro¨m, A., Ho¨k, B. & Mason, B. (1941). The crystal structure of zinc metaantimonate and similar compounds. Ark. Kemi Mineral. Geol. 15B, 1–8. Chapuis, G. C. (1992). Symmetry relationships between crystal structures and their practical applications. Modern Perspectives in Inorganic Chemistry, edited by E. Parthe´, pp. 1–16. Dordrecht: Kluwer Academic Publishers. Clemente, D. A. (2005). A survey of the 8466 structures reported in Inorganica Chimica Acta: 52 space group changes and their consequences. Inorg. Chim. Acta, 358, 1725–1748. Cracknell, A. P. (1975). Group Theory in Solid State Physics. New York: Taylor and Francis Ltd/Pergamon. Deonarine, S. & Birman, J. L. (1983). Group–subgroup phase transitions, Hermann’s space group decomposition theorem, and chain subduction criterion in crystals. Phys. Rev. B, 27, 4261–4273. Di Costanzo, L., Forneris, F., Geremia, S. & Randaccio, L. (2003). Phasing protein structures using the group–subgroup relation. Acta Cryst. D59, 1435–1439. Fedorov, E. (1891). Symmetry of regular systems and ﬁgures. Zap. Mineral. Obshch. (2), 28, 1–46. (In Russian.) (English: Symmetry of Crystals, American Crystallographic Association, 1971.) Fischer, W. & Koch, E. (1983). On the equivalence of point conﬁgurations due to Euclidean normalizers (Cheshire groups) of space groups. Acta Cryst. A39, 907–915. ¨ ber die Oxidation von Giovanoli, D. & Leuenberger, U. (1969). U Manganoxidhydroxid. Helv. Chim. Acta, 52, 2333–2347. Goldschmidt, V. M. (1926). Untersuchungen u¨ber den Bau und Eigenschaften von Krystallen. Skr. Nor. Vidensk. Akad. Oslo Mat.Nat. Kl. 1926 No. 2 and 1927 No. 8. Haag, F. (1887). Die regula¨ren Krystallko¨rper. Programm des Ko¨niglichen Gymnasiums in Rottweil zum Schlusse des Schuljahres 1886–1887. Heesch, H. (1930). Zur systematischen Strukturtheorie. II. Z. Kristallogr. 72, 177–201. Hermann, C. (1929). Zur systematischen Strukturtheorie. IV. Untergruppen. Z. Kristallogr. 69, 533–555. International Tables for Crystallography (1983). Vol. A, Space-Group Symmetry, edited by Th. Hahn, 1st ed. Dordrecht: Kluwer Academic Publishers. International Tables for Crystallography (2005). Vol. A, Space-Group Symmetry, edited by Th. Hahn, 5th ed. Dordrecht: Kluwer Academic Publishers. International Tables for X-ray Crystallography (1952, 1965, 1969). Vol. I, Symmetry Groups, edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press. Internationale Tabellen zur Bestimmung von Kristallstrukturen (1935). 1. Bd. Edited by C. Hermann. Berlin: Borntraeger. (In German, English and French.) Janovec, V., Hahn, Th. & Klapper, H. (2003). Twinning and domain structures. International Tables for Crystallography, Vol. D, edited by A. Authier, Physical Properties of Crystals, ch. 3.2. Dordrecht: Kluwer Academic Publishers. Janovec, V. & Prˇı´vratska´, J. (2003). Domain structures. International Tables for Crystallography, Vol. D, edited by A. Authier, Physical Properties of Crystals, ch. 3.4. Dordrecht: Kluwer Academic Publishers. Koch, E. (1984). The implications of normalizers on group–subgroup relations between space groups. Acta Cryst. A40, 593–600. Kroumova, E., Aroyo, M. I. & Pe´rez-Mato, J. M. (2002). Prediction of new displacive ferroelectrics through systematic pseudosymmetry

6

references

International Tables for Crystallography (2011). Vol. A1, Chapter 1.2, pp. 7–24.

1.2. General introduction to the subgroups of space groups By Hans Wondratschek

1.2.1. General remarks

applies to isometries. Afﬁne mappings are important for the classiﬁcation of crystallographic symmetries, cf. Section 1.2.5.2.

The performance of simple vector and matrix calculations, as well as elementary operations with groups, are nowadays common practice in crystallography, especially since computers and suitable programs have become widely available. The authors of this volume therefore assume that the reader has at least some practical experience with matrices and groups and their crystallographic applications. The explanations and deﬁnitions of the basic terms of linear algebra and group theory in these ﬁrst sections of this introduction are accordingly short. Rather than replace an elementary textbook, these ﬁrst sections aim to acquaint the reader with the method of presentation and the terminology that the authors have chosen for the tables and graphs of this volume. The concepts of groups, their subgroups, isomorphism, coset decomposition and conjugacy are considered to be essential for the use of the tables and for their practical application to crystal structures; for a deeper understanding the concept of normalizers is also necessary. Frequently, however, an ‘intuitive feeling’ obtained by practical experience may replace a full comprehension of the mathematical meaning. From Section 1.2.6 onwards, the presentation will be more detailed because the subjects are more specialized (but mostly not more difﬁcult) and are seldom found in textbooks.

Deﬁnition 1.2.2.1.2. A mapping is called a symmetry operation of an object if (1) it is an isometry, (2) it maps the object onto itself.

Instead of ‘maps the object onto itself’, one frequently says ‘leaves the object invariant (as a whole)’. This does not mean that each point of the object is mapped onto itself; rather, the object is mapped in such a way that an observer cannot distinguish the states of the object before and after the mapping. Deﬁnition 1.2.2.1.3. A symmetry operation of a crystal pattern is & called a crystallographic symmetry operation. The symmetry operations of a macroscopic crystal are also crystallographic symmetry operations, but they belong to another kind of mapping which will be discussed in Section 1.2.5.4. There are different types of isometries which may be crystallographic symmetry operations. These types are described and discussed in many textbooks of crystallography and in mathematical, physical and chemical textbooks. They are listed here without further treatment. Fixed points are very important for the characterization of isometries.

1.2.2. Mappings and matrices 1.2.2.1. Crystallographic symmetry operations

Deﬁnition 1.2.2.1.4. A point P is a ﬁxed point of a mapping if it is mapped onto itself, i.e. the image point P~ is the same as the & original point P: P~ ¼ P.

A crystal is a ﬁnite block of an inﬁnite periodic array of atoms in physical space. The inﬁnite periodic array is called the crystal pattern. The ﬁnite block is called the macroscopic crystal.1 Periodicity implies that there are translations which map the crystal pattern onto itself. Geometric mappings have the property that for each point P of the space, and thus of the object, there is a ~ the image point. The mapping is uniquely determined point P, reversible if each image point P~ is the image of one point P only. Translations belong to a special category of mappings which leave all distances in the space invariant (and thus within an object and between objects in the space). Furthermore, a mapping of an object onto itself (German: Deckoperation) is the basis of the concept of geometric symmetry. This is expressed by the following two deﬁnitions.

The set of all ﬁxed points of an isometry may be the whole space, a plane in the space, a straight line, a point, or the set may be empty (no ﬁxed point). The following kinds of isometries exist: (1) The identity operation, which maps each point of the space onto itself. It is a symmetry operation of every object and, although trivial, is indispensable for the group properties which are discussed in Section 1.2.3. (2) A translation t which shifts every object. A translation is characterized by its translation vector t and has no ﬁxed point: if x is the column of coordinates of a point P, then the coordinates x~ of the image point P~ are x~ ¼ x þ t. If a translation is a symmetry operation of an object, the object extends inﬁnitely in the directions of t and t. A translation preserves the ‘handedness’ of an object, e.g. it maps any righthand glove onto a right-hand one and any left-hand glove onto a left-hand one. (3) A rotation is an isometry that leaves one line ﬁxed pointwise. This line is called the rotation axis. The degree of rotation about this axis is described by its rotation angle ’. In particular, a rotation is called an N-fold rotation if the rotation angle is ’ ¼ k 360 =N, where k and N are relatively prime integers. A rotation preserves the ‘handedness’ of any object. (4) A screw rotation is a rotation coupled with a translation parallel to the rotation axis. The rotation axis is now called

Deﬁnition 1.2.2.1.1. A mapping is called a motion, a rigid motion or an isometry if it leaves all distances invariant (and thus all angles, as well as the size and shape of an object). In this volume & the term ‘isometry’ is used. An isometry is a special kind of afﬁne mapping. In an afﬁne mapping, parallel lines are mapped onto parallel lines; lengths and angles may be distorted but quotients of lengths on the same line are preserved. In Section 1.2.2.3, the description of afﬁne mappings is discussed, because this type of description also 1 A real single crystal is still different from a macroscopic crystal. There are dislocations, point defects like vacancies, interstitial atoms or replacements of atoms, and the atoms are never at rest but vibrate. Therefore, the macroscopic crystal is a more-or-less strongly idealized model of the real crystal.

Copyright © 2011 International Union of Crystallography

&

7

1. SPACE GROUPS AND THEIR SUBGROUPS

(5)

(6)

(7)

(8)

the screw axis. The translation vector is called the screw vector. A screw rotation has no ﬁxed points. The screw axis is invariant as a whole under the screw rotation but not pointwise. An N-fold rotoinversion is an N-fold rotation coupled with inversion through a point on the rotation axis. This point is called the centre of the rotoinversion. For N 6¼ 2 it is the only ﬁxed point. The axis of the rotation is invariant as a whole under the rotoinversion and is called its rotoinversion axis. A rotoinversion changes the handedness by its inversion component: it maps any right-hand glove onto a left-hand one and vice versa. Performed twice it results in a rotation. Special rotoinversions are those for N ¼ 1 and N ¼ 2, which are dealt with separately. The inversion can be considered as a onefold rotoinversion (N ¼ 1). The ﬁxed point is called the inversion centre or the centre of symmetry. A twofold rotoinversion (N ¼ 2) is called a reﬂection or a reﬂection through a plane. It is an isometry which leaves a plane perpendicular to the twofold rotoinversion axis ﬁxed pointwise. This plane is called the reﬂection plane or mirror plane and it intersects the rotation axis in the centre of the rotoinversion. Its orientation is described by the direction of its normal vector, i.e. of the rotation axis. For a twofold rotoinversion, neither the rotation nor the inversion are symmetry operations themselves. As for any other rotoinversion, the reﬂection changes the handedness of an object. A glide reﬂection is a reﬂection through a plane coupled with a translation parallel to this plane. The translation vector is called the glide vector. A glide reﬂection changes the handedness and has no ﬁxed point. The former reﬂection plane is now called the glide plane. Under a glide reﬂection, the glide plane is invariant as a whole but not pointwise.

The primary aim of the choice of the crystallographic coordinate system is to describe the crystal pattern and its set of all symmetry operations in a simple way. This aim holds in particular for the inﬁnitely many symmetry translations of the crystal pattern, which form its translation group. Secondary to this aim are equality of the lengths of, and right angles between, the basis vectors. The vector t belonging to the translation t is called a translation vector or a lattice vector. The set of all translation vectors of the crystal pattern is called its vector lattice L. Both the translation group and the vector lattice are useful tools for describing the periodicity of the crystals. For the description of a vector lattice several kinds of bases are in use. Orthonormal bases are not the most convenient, because the coefﬁcients of the lattice vectors may then be any real number. The coefﬁcients of the lattice vectors are more transparent if the basis vectors themselves are lattice vectors. Deﬁnition 1.2.2.2.1. A basis which consists of lattice vectors of a crystal pattern is called a lattice basis or a crystallographic basis. & Referred to a lattice basis, each lattice vector t 2 L is a linear combination of the basis vectors with rational coefﬁcients. One can even select special bases such that the coefﬁcients of all lattice vectors are integers. Deﬁnition 1.2.2.2.2. A crystallographic basis is called a primitive & basis if every lattice vector has integer coefﬁcients. A fundamental feature of vector lattices is that for any lattice in a dimension greater than one an inﬁnite number of primitive bases exists. With certain rules, the choice of a primitive basis can be made unique (reduced bases). In practice, however, the conventional bases are not always primitive; the choice of a conventional basis is determined by the matrix parts of the symmetry operations, cf. Section 1.2.5.1.

Symmetry operations of crystal patterns may belong to any of these isometries. The set of all symmetry operations of a crystal pattern has the following properties: performing two (and thus more) symmetry operations one after the other results in another symmetry operation. Moreover, there is the identity operation in this set, i.e. an operation that leaves every point of the space and thus of the crystal pattern ﬁxed. Finally, for any symmetry operation of an object there is an ‘inverse’ symmetry operation by which its effect is reversed. These properties are necessary for the application of group theory, cf. Section 1.2.3.

1.2.2.3. The description of mappings The instruction for the calculation of the coordinates of X~ from the coordinates of X is simple for an afﬁne mapping and thus for an isometry. The equations are x~ ¼ W11 x þ W12 y þ W13 z þ w1 y~ ¼ W21 x þ W22 y þ W23 z þ w2

ð1:2:2:1Þ

z~ ¼ W31 x þ W32 y þ W33 z þ w3 ; 1.2.2.2. Coordinate systems and coordinates

where the coefﬁcients Wik and wj are constant. These equations can be written using the matrix formalism: 10 1 0 1 0 1 0 x~ x W11 W12 W13 w1 @ y~ A ¼ @ W21 W22 W23 [email protected] y A þ @ w2 A: ð1:2:2:2Þ W31 W32 W33 w3 z~ z

To describe mappings analytically, one introduces a coordinate system fO; a; b; cg, consisting of three linearly independent (i.e. not coplanar) basis vectors a; b; c (or a1 ; a2 ; a3 ) and an origin O. For the plane (two-dimensional space) an origin and two linearly independent (i.e. not parallel) basis vectors a; b (or a1 ; a2 ) are chosen. Referred to this coordinate system, each point P can be described by three (or two for the plane) coordinates x; y; z (or x1 ; x2 ; x3 ). An object, for example a crystal, can now be described by a continuous or discontinuous function of the coordinates such as the electron density or the coordinates of the centres of the atoms. A mapping can be regarded as an instruction of how to calculate the coordinates x~ ; y~ ; z~ of the image point X~ from the coordinates x; y; z of the original point X. In contrast to the practice in physics and chemistry, a nonCartesian coordinate system is usually chosen in crystallography.

This matrix equation is usually abbreviated by x~ ¼ W x þ w; where 0 1 0 1 0 1 0 x~ x w1 W11 x~ ¼@ y~ A; x ¼@ y A; w ¼@ w2 A and W ¼@ W21 z~ z w3 W31

8

ð1:2:2:3Þ

W12 W22 W32

1 W13 W23 A: W33

1.2. GENERAL INTRODUCTION TO THE SUBGROUPS OF SPACE GROUPS Deﬁnition 1.2.2.3.1. The matrix W is called the linear part or matrix part, the column w is the translation part or column part of & a mapping.

matrix which can be combined with the analogous matrices of other mappings and can be inverted. These matrices are called augmented matrices and are designated by open-face letters in this volume: 1 0 0 1 0 1 W11 W12 W13 w1 x x~ B C ByC B W21 W22 W23 w2 C B y~ C C ~ B C B C B W¼B C x ¼ B z~ C; x ¼ B C: @zA @ A B W31 W32 W33 w3 C A @ 1 1 0 0 0 1

In equations (1.2.2.1) and (1.2.2.3), the coordinates are mixed with the quantities describing the mapping, designated by the letters Wik and wj or W and w. Therefore, one prefers to write equation (1.2.2.3) in the form x~ ¼ ðW ; wÞ x or x~ ¼ fW j wg x:

ð1:2:2:4Þ

The symbols (W ; w) and {W j w} which describe the mapping referred to the chosen coordinate system are called the matrix– column pair and the Seitz symbol. The formulae for the combination of afﬁne mappings and for the inverse of an afﬁne mapping (regular matrix W ) are obtained by

ð1:2:2:7Þ In order to write equation (1.2.2.3) as x~ ¼ Wx with the augmented matrices W, the columns x~ and x also have to be extended to the augmented columns x and x~ . Equations (1.2.2.5) and (1.2.2.6) then become 1

x~ ¼ W 1 x þ w1 ; x~~ ¼ W 2 x~ þ w2 ¼ W 3 x þ w3 x~~ ¼ W 2 ðW 1 x þ w1 Þ þ w2 ¼ W 2 W 1 x þ W 2 w1 þ w2 :

W3 ¼ W2 W1 and ðWÞ

¼ ðW1 Þ:

ð1:2:2:8Þ

The vertical and horizontal lines in the matrix have no mathematical meaning. They are simply a convenience for separating the matrix part from the column part and from the row ‘0 0 0 1’, and could be omitted. Augmented matrices are very useful when writing down general formulae which then become more transparent and more elegant. However, the matrix–column pair formalism is, in general, advantageous for practical calculations. For the augmented columns of vector coefﬁcients, see Section 1.2.2.6.

From x~ ¼ Wx þ w, it follows that W 1 x~ ¼ x þ W 1 w or x ¼ W 1 x~ W 1 w: Using matrix–column pairs, this reads ðW 3 ; w3 Þ ¼ ðW 2 ; w2 Þ ðW 1 ; w1 Þ ¼ ðW 2 W 1 ; W 2 w1 þ w2 Þ ð1:2:2:5Þ and x ¼ ðW ; wÞ1 x~ ¼ ðW 0 ; w0 Þ~x or ðW 0 ; w0 Þ ¼ ðW ; wÞ1 ¼ ðW 1 ; W 1 wÞ:

1.2.2.5. Isometries

ð1:2:2:6Þ

Isometries are special afﬁne mappings, as in Deﬁnition 1.2.2.1.1. The matrix W of an isometry has to fulﬁl conditions which depend on the coordinate basis. These conditions are:

One ﬁnds from equations (1.2.2.5) and (1.2.2.6) that the linear parts of the matrix–column pairs transform as one would expect:

(1) A basis a1 ; a2 ; a3 is characterized by the scalar products ðaj ak Þ of its basis vectors or by its lattice parameters a; b; c; ; and . Here a; b; c are the lengths of the basis vectors a1 ; a2 ; a3 and ; and are the angles between a2 and a3 , a3 and a1 , a1 and a2 , respectively. The metric matrix M [called G in International Tables for Crystallography Volume A (2005) (abbreviated as IT A), Chapter 9.1] is the ð3 3Þ matrix which consists of the scalar products of the basis vectors: 0 1 a b cos a c cos a2 M ¼ @ b a cos b2 b c cos A: c a cos c b cos c2

(1) the linear part of the product of two matrix–column pairs is the product of the linear parts, i.e. if ðW 3 ; w3 Þ ¼ ðW 2 ; w2 Þ ðW 1 ; w1 Þ then W 3 ¼ W 2 W 1 ; (2) the linear part of the inverse of a matrix–column pair is the inverse of the linear part, i.e. if ðX; xÞ ¼ ðW ; wÞ1, then X ¼ W 1 . [This relation is included in the ﬁrst one: from ðW ; wÞ ðX; xÞ ¼ ðW X; Wx þ wÞ ¼ ðI; oÞ follows X ¼ W 1. Here I is the unit matrix and o is the column consisting of zeroes]. These relations will be used in Section 1.2.5.4. For the column parts, equations (1.2.2.5) and (1.2.2.6) are less convenient: ð1Þ w3 ¼ W 2 w1 þ w2 ;

ð2Þ w0 ¼ W 1 w:

If W is the matrix part of an isometry, referred to the basis (a1 ; a2 ; a3 ), then W must fulﬁl the condition W T M W ¼ M, where W T is the transpose of W. (2) For the determinant of W, detðW Þ ¼ 1 must hold; detðW Þ ¼ þ1 for the identity, translations, rotations and screw rotations; detðW Þ ¼ 1 for inversions, reﬂections, glide reﬂections and rotoinversions. (3) For the trace, trðW Þ ¼ W11 þ W22 þ W33 ¼ ð1 þ 2 cos ’Þ holds, where ’ is the rotation angle; the + sign applies if detðW Þ ¼ þ1 and the sign if detðW Þ ¼ 1.

Because of the inconvenience of these relations, it is often preferable to use ‘augmented’ matrices, by which one can describe the combination of afﬁne mappings and the inverse mapping by the equations of the usual matrix multiplication. These matrices are introduced in the next section. 1.2.2.4. Matrix–column pairs and (n + 1) (n + 1) matrices It is natural to combine the matrix part and the column part describing an afﬁne mapping to form a ð3 4Þ matrix, but such matrices cannot be multiplied by the usual matrix multiplication and cannot be inverted. However, if one supplements the ð3 4Þ matrix by a fourth row ‘0 0 0 1’, one obtains a ð4 4Þ square

Algorithms for the determination of the kind of isometry from a given matrix–column pair and for the determination of the matrix–column pair for a given isometry can be found in IT A, Part 11 or in Hahn & Wondratschek (1994).

9

1. SPACE GROUPS AND THEIR SUBGROUPS 1.2.2.6. Vectors and vector coefﬁcients

equations (1.2.2.1) to (1.2.2.3), and the column of vector coefﬁcients is v, see Section 1.2.2.6. A new coordinate system may be introduced with the basis ða01 ; a02 ; a03 ÞT and the origin O0 . Referred to the new coordinate system, the column of coordinates of the point P is x0, the symmetry operation is described by 0 W 0 and w0 and !0the column of vector coefﬁcients is v . Let p ¼ O O be the column of coefﬁcients for the vector from the old origin O to the new origin O0 and let 0 1 P11 P12 P13 P ¼ @ P21 P22 P23 A ð1:2:2:9Þ P31 P32 P33

In crystallography, vectors and their coefﬁcients as well as points and their coordinates are used for the description of crystal structures. Vectors represent translation shifts, distance and Patterson vectors, reciprocal-lattice vectors etc. With respect to a given basis a vector has three coefﬁcients. In contrast to the coordinates of a point, these coefﬁcients do not change if the origin of the coordinate system is shifted. In the usual description by columns, the vector coefﬁcients cannot be distinguished from the point coordinates, but in the augmented-column description the difference becomes visible: the vector from the point P to the point Q has the coefﬁcients v1 ¼ q1 p1, v2 ¼ q2 p2 , v3 ¼ q3 p3 , 1 1. Thus, the column of the coefﬁcients of a vector is not augmented by ‘1’ but by ‘0’. Therefore, when the point P is mapped onto the point P~ by x~ ¼ W x þ w according to ! equation (1.2.2.3), ! then the vector v ¼ P Q is mapped onto the ~ by transforming its coefﬁcients by v~ ¼ W v, vector v~ ¼ P~ Q because the coefﬁcients wj are multiplied by the number ‘0’ augmenting the column v ¼ ðvj Þ. Indeed, the distance vector ! v ¼ P Q is not changed when the whole space is mapped onto itself by a translation.

be the matrix of a basis change, i.e. the matrix that relates the new basis ða01 ; a02 ; a03 ÞT to the old basis ða1 ; a2 ; a3 ÞT according to 0 1 P11 P12 P13 T T T ða01 ; a02 ; a03 Þ ¼ ða1 ; a2 ; a3 Þ P ¼ ða1 ; a2 ; a3 Þ @P21 P22 P23 A: P31 P32 P33 ð1:2:2:10Þ Then the following equations hold:

Remarks (1) The difference in transformation behaviour between the point coordinates x and the vector coefﬁcients v is not visible in the equations where the symbols x and v are used, but is obvious only if the columns are written in full, viz 0 1 0 1 x1 v1 B x2 C B v2 C B C B C B x3 C and B v3 C: @ A @ A 1

x0 ¼ P 1 x P 1 p or x ¼ P x0 þ p;

ð1:2:2:11Þ

W 0 ¼ P 1 W P or W ¼ P W 0 P 1 ;

ð1:2:2:12Þ

w0 ¼ P 1 ðw þ ðW IÞ pÞ or w ¼ P w0 ðW IÞ p: ð1:2:2:13Þ For the columns of vector coefﬁcients v and v0 , the following holds: v0 ¼ P 1 v or v ¼ P v0 ;

0

ð1:2:2:14Þ

i.e. an origin shift does not change the vector coefﬁcients. These equations read in the augmented-matrix formalism

(2) The transformation behaviour of the vector coefﬁcients is also apparent if the vector is understood to be a translation vector and the transformation behaviour of the translation is considered as in the last paragraph of the next section. (3) The transformation v~ ¼ W v is called an orthogonal mapping if W is the matrix part of an isometry.

x0 ¼ P1 x; W 0 ¼ P1 WP; v0 ¼ P1 v:

ð1:2:2:15Þ

For the difference in the transformation behaviour of point coordinates and vector coefﬁcients, see the remarks at the end of Section 1.2.2.6. A vector v can be regarded as a translation vector; its translation is then described by (I, v), i.e. W ¼ I; w ¼ v. It can be shown using equation (1.2.2.13) that the translation and thus the translation vector are not changed under an origin shift, (I; p), because ðI; vÞ0 ¼ ðI; vÞ holds. Moreover, under a general coordinate transformation the origin shift is not effective: in equation (1.2.2.13) only v0 ¼ P 1 v remains because of the equality W ¼ I.

1.2.2.7. Origin shift and change of the basis It is in general advantageous to refer crystallographic objects and their symmetries to the most appropriate coordinate system. The best coordinate system may be different for different steps of the calculations and for different objects which have to be considered simultaneously. Therefore, a change of the origin and/ or the basis are frequently necessary when treating crystallographic problems. Here the formulae for the inﬂuence of an origin shift and a change of basis on the coordinates, on the matrix–column pairs of mappings and on the vector coefﬁcients are only stated; the equations are derived in detail in IT A Chapters 5.1 and 5.2, and in Hahn & Wondratschek (1994). Let a coordinate system be given with a basis ða1 ; a2 ; a3 ÞT and an origin O.2 Referred to this coordinate system, the column of coordinates of a point P is x; the matrix and column parts describing a symmetry operation are W and w according to

1.2.3. Groups Group theory is the proper tool for studying symmetry in science. A group is a set of elements with a law of composition, by which to any two elements a third element is uniquely assigned. The symmetry group of an object is the set of all isometries (rigid motions) which map that object onto itself. If the object is a crystal, the isometries which map it onto itself (and also leave it invariant as a whole) are the crystallographic symmetry operations. There is a huge amount of literature on group theory and its applications. The book Introduction to Group Theory (Ledermann, 1976; Ledermann & Weir, 1996) is recommended. The book Symmetry of Crystals. Introduction to International Tables for Crystallography, Vol. A by Hahn & Wondratschek (1994)

2 In this volume, point coordinates and vector coefﬁcients are thought of as columns in matrix multiplication. Therefore, columns are considered to be ‘standard’. These ‘columns’ are not marked, even if they are written in a row. To comply with the rules of matrix multiplication, rows are also introduced. These rows of symbols (e.g. vector coefﬁcients of reciprocal space, i.e. Miller indices, or a set of basis vectors of direct space) are ‘transposed relative to columns’ and are, therefore, marked ðh; k; l ÞT or ða; b; cÞT , even if they are written in a row.

10

1.2. GENERAL INTRODUCTION TO THE SUBGROUPS OF SPACE GROUPS relation g k g j ¼ g j g k is fulﬁlled for all pairs of elements of a group G, then G is called a commutative or an Abelian group. For groups of higher order, it is usually inappropriate and for groups of inﬁnite order it is impossible to list all elements of a group. The following deﬁnition nearly always reduces the set of group elements to be listed explicitly to a small set.

Table 1.2.3.1. Multiplication table of a group The group elements g 2 G are listed at the top of the table and in the same sequence on the left-hand side; the unit element ‘e’ is listed ﬁrst. The table is thus a square array. The product g k g j of any pair of elements is listed at the intersection of the kth row and the jth column. It can be shown that each group element is listed exactly once in each row and once in each column of the table. In the row of an element g 2 G, the unit element 2 e appears in the column of g 1. If ðg Þ ¼ e, i.e. g ¼ g 1 , e appears on the main diagonal. The multiplication table of an Abelian group is symmetric about the main diagonal. G

e

a

b

c

...

e a b c

e a b c

a a2 ba ca

b ab b2 cb

c ac bc c2

... ... ... ... .. .

.. .

.. .

.. .

.. .

.. .

Deﬁnition 1.2.3.1.1. A set S ¼ fg p ; g q ; . . .g 2 G such that every element of G can be obtained by composition of the elements of S and their inverses is called a set of generators of G. The elements & g i 2 S are called generators of G. A group is cyclic if it consists of the unit element e and all powers of one element g : Cðg Þ ¼ f . . . g 3 ; g 2 ; g 1 ; e ; g 1 ; g 2 ; g 3 ; . . . g:

describes a way in which the data of IT A can be interpreted by means of matrix algebra and elementary group theory. It may also help the reader of this volume.

If there is an integer number n > 0 with g n ¼ e and n is the smallest number with this property, then the group Cðg Þ has the ﬁnite order n. Let g k with 0 < k < n be the inverse element of g k where n is the order of g. Because g k ¼ g n g k ¼ g nk ¼ g m with n ¼ m þ k, the elements of a cyclic group of ﬁnite order can all be written as positive powers of the generator g . Otherwise, if such an integer n does not exist, the group Cðg Þ is of inﬁnite order and the positive powers g k are different from the negative ones g m . In the same way, from any element g j 2 G its cyclic group Cðg j Þ can be generated even if G is not cyclic itself. The order of this group Cðg j Þ is called the order of the element g j.

1.2.3.1. Some properties of symmetry groups The geometric symmetry of any object is described by a group G. The symmetry operations g j 2 G are the group elements, and the set fg j 2 Gg of all symmetry operations fulﬁls the group postulates. [A ‘symmetry element’ in crystallography is not a group element of a symmetry group but is a combination of a geometric object with that set of symmetry operations which leave the geometric object invariant, e.g. an axis with its threefold rotations or a plane with its glide reﬂections etc., cf. Flack et al. (2000).] Groups will be designated by upper-case calligraphic script letters G, H etc. Group elements are represented by lowercase italic sans serif letters g ; h etc. The result g r of the composition of two elements g j ; g k 2 G will be called the product of g j and g k and will be written g r ¼ g k g j . The ﬁrst operation is the right factor because the point coordinates or vector coefﬁcients are written as columns on which the matrices of the symmetry operations are applied from the left side. The law of composition in the group is the successive application of the symmetry operations. The group postulates are shown to hold for symmetry groups:

1.2.3.2. Group isomorphism and homomorphism A ﬁnite group G of small order may be conveniently visualized by its multiplication table, group table or Cayley table. An example is shown in Table 1.2.3.1. The multiplication tables can be used to deﬁne one of the most important relations between two groups, the isomorphism of groups. This can be done by comparing the multiplication tables of the two groups. Deﬁnition 1.2.3.2.1. Two groups are isomorphic if one can arrange the rows and columns of their multiplication tables such that these tables are equal, apart from the names or symbols of the & group elements.

(1) The closure, i.e. the property that the composition of any two symmetry operations results in a symmetry operation again, is always fulﬁlled for geometric symmetries: if g j 2 G and g k 2 G, then g j g k ¼ g r 2 G also holds. (2) The associative law is always fulﬁlled for the composition of geometric mappings. If g j ; g k ; g m 2 G, then ðg j g k Þ g m ¼ g j ðg k g m Þ ¼ g q for any triplet j; k; m. Therefore, the parentheses are not necessary, one can write g j g k g m ¼ g q. In general, however, the sequence of the symmetry operations must not be changed. Thus, in general g j g k g m 6¼ g j g m g k . (3) The unit element or neutral element e 2 G is the identity operation which maps each point onto itself, i.e. leaves each point invariant. (4) The isometry which reverses a given symmetry operation g 2 G is also a symmetry operation of G and is called the inverse symmetry operation g 1 of g. It has the property g g 1 ¼ g 1 g ¼ e .

Multiplication tables are useful only for groups of small order. To deﬁne ‘isomorphism’ for arbitrary groups, one can formulate the relations expressed by the multiplication tables in a more abstract way. The ‘same multiplication table’ for the groups G and G 0 means that there is a reversible mapping g q !g 0q of the elements 0 g q 2 G and g 0q 2 G 0 such that ðg j g k Þ ¼ g 0j g 0k holds for any pair of indices j and k. In words: Deﬁnition 1.2.3.2.2. Two groups G and G 0 are isomorphic if there is a reversible mapping of G onto G 0 such that for any pair of elements of G the image of the product is equal to the product of & the images. Isomorphic groups have the same order. By isomorphism the set of all groups is classiﬁed into isomorphism types or isomorphism classes of groups. Such a class is often called an abstract group. The isomorphism between the space groups and the corresponding matrix groups makes an analytical treatment of crystallographic symmetry possible. Moreover, the isomorphism of

The number of elements of a group G is called its order jGj. The order of a group may be ﬁnite, for example 24 for the symmetry operations of a regular tetrahedron, or inﬁnite, for example for any space group because of its inﬁnite set of translations. If the

11

1. SPACE GROUPS AND THEIR SUBGROUPS different space groups allows one to classify the inﬁnite number of space groups into a ﬁnite number of isomorphism types of space groups, which is one of the bases of crystallography, see Section 1.2.5. Isomorphism provides a very strong relation between groups: the groups are identical in their group-theoretical properties. One can weaken this relation by omitting the condition of reversibility of the mapping. One then admits that more than one element of the group G is mapped onto the same element of G 0. This concept leads to the deﬁnition of homomorphism.

1.2.6.2. Moreover, the relations between a space group and its maximal subgroups are particularly transparent, cf. Lemma 1.2.8.1.3.

1.2.4.2. Coset decomposition and normal subgroups Let H < G be a subgroup of G of order jHj. Because H is a proper subgroup of G there must be elements g q 2 G that are not elements of H. Let g 2 2 G be one of them. Then the set of elements g 2 H ¼ fg 2 hj j hj 2 Hg4 is a subset of elements of G with the property that all its elements are different and that the sets H and g 2 H have no element in common. Thus, the set g 2 H also contains jHj elements of G. If there is another element g 3 2 G which belongs neither to H nor to g 2 H, one can form another set g 3 H ¼ fg 3 h j j h j 2 Hg. All elements of g 3 H are different and none occurs already in H or in g 2 H. This procedure can be continued until each element g r 2 G belongs to one of these sets. In this way the group G can be partitioned, such that each element g 2 G belongs to exactly one of these sets.

Deﬁnition 1.2.3.2.3. A mapping of a group G onto a group G 0 is called homomorphic, and G 0 is called a homomorphic image of the group G, if for any pair of elements of G the image of the product is equal to the product of the images and if any element of G 0 is the image of at least one element of G. The relation of G and G 0 is called a homomorphism. More formally: For the & mapping G onto G 0, ðg j g k Þ0 ¼ g 0j g 0k holds. The formulation ‘mapping onto’ implies that each element g 0 2 G 0 occurs among the images of the elements g 2 G at least

Deﬁnition 1.2.4.2.1. The partition just described is called a decomposition (G : H) into left cosets of the group G relative to the group H. The sets g p H; p ¼ 1; . . . ; i are called left cosets, because the elements hj 2 H are multiplied with the new elements from the left-hand side. The procedure is called a decomposition into right cosets Hg s if the elements h j 2 H are multiplied with the new elements g s from the right-hand side. The elements g p or g s are called the coset representatives. The number & of cosets is called the index i ¼ jG : Hj of H in G.

once.3 The very important concept of homomorphism is discussed further in Lemma 1.2.4.4.3. The crystallographic point groups are homomorphic images of the space groups, see Section 1.2.5.4.

1.2.4. Subgroups 1.2.4.1. Deﬁnition There may be sets of elements g k 2 G that do not constitute the full group G but nevertheless fulﬁl the group postulates for themselves.

Remarks (1) The group H ¼ g 1 H with g 1 ¼ e is the ﬁrst coset for both kinds of decomposition. It is the only coset which forms a group by itself. (2) All cosets have the same length, i.e. the same number of elements, which is equal to jHj, the order of H. (3) The index i is the same for both right and left decompositions. In IT A and in this volume, the index is frequently designated by the symbol ½i. (4) A coset does not depend on its representative element; starting from any of its elements will result in the same coset. The right cosets may be different from the left ones and the representatives of the right and left cosets may also differ. (5) If the order jGj of G is inﬁnite, then either the order jHj of H or the index i ¼ jG : Hj of H in G or both are inﬁnite. (6) The coset decomposition of a space group G relative to its translation subgroup T (G) is fundamental in crystallography, cf. Section 1.2.5.4.

Deﬁnition 1.2.4.1.1. A subset H of elements of a group G is called a subgroup H of G if it fulﬁls the group postulates with respect to & the law of composition of G. Remarks (1) The group G is considered to be one of its own subgroups. If subgroups Hj are discussed where G is included among the subgroups, we write Hj G or G Hj . If G is excluded from the set fHj g of its subgroups, we write Hj < G or G > Hj . A subgroup Hj < G is called a proper subgroup of G. (2) In a relation G H or G > H, G is called a supergroup of H. The symbols ; ; < and > are used for supergroups in the same way as they are used for subgroups, cf. Section 2.1.6. (3) A subgroup of a ﬁnite group is ﬁnite. A subgroup of an inﬁnite group may be ﬁnite or inﬁnite. (4) A subset K of elements g k 2 G which does not necessarily form a group is designated by the symbol K G.

From its deﬁnition and from the properties of the coset decomposition mentioned above, one immediately obtains the fundamental theorem of Lagrange (for another formulation, see Chapter 1.4):

Deﬁnition 1.2.4.1.2. A subgroup H < G is a maximal subgroup if no group Z exists for which H < Z < G holds. If H is a maximal & subgroup of G, then G is a minimal supergroup of H.

Lemma 1.2.4.2.2. Lagrange’s theorem: Let G be a group of ﬁnite order jGj and H < G a subgroup of G of order jHj. Then jHj is a divisor of jGj and the equation jHj i ¼ jGj holds where & i ¼ jG : Hj is the index of H in G.

This deﬁnition is very important for the tables of this volume, as only maximal subgroups of space groups are listed. If all maximal subgroups are known for any given space group, then any general subgroup H < G can be obtained by a (ﬁnite) chain of maximal subgroups between G and H, see Section

A special situation exists when the left and right coset decompositions of G relative to H result in the partition of G into the same cosets:

3 In mathematics, the term ‘homomorphism’ includes mappings of a group G into a group G 0, i.e. mappings in which not every g 0 2 G 0 is the image of some element of g 2 G. The term ‘homomorphism onto’ deﬁned above is also known as an epimorphism, e.g. in Ledermann (1976) and Ledermann & Weir (1996). In the older literature the term ‘multiple isomorphism’ can also be found.

4

The formulation g 2 H ¼ fg 2 hj j hj 2 Hg means: ‘g 2 H is the set of the products g 2 hj of g 2 with all elements h j 2 H.’

12

1.2. GENERAL INTRODUCTION TO THE SUBGROUPS OF SPACE GROUPS g p H ¼ H g p for all 1 p i:

1.2.4.4. Factor groups and homomorphism

ð1:2:4:1Þ

For the following deﬁnition, the ‘product of sets of group elements’ will be used:

Subgroups H that fulﬁl equation (1.2.4.1) are called ‘normal subgroups’ according to the following deﬁnition:

Deﬁnition 1.2.4.4.1. Let G be a group and Kj ¼ fg j1 ; . . . ; g jn g, Kk ¼ fg k1 ; . . . ; g km g be two arbitrary sets of its elements which are not necessarily groups themselves. Then the product Kj Kk of Kj and Kk is the set of all products Kj Kk ¼ & fg jp g kq j g jp 2 Kj ; g kq 2 Kk g.5

Deﬁnition 1.2.4.2.3. A subgroup H < G is called a normal subgroup or invariant subgroup of G, H / G, if equation (1.2.4.1) is & fulﬁlled. The relation H / G always holds for jG : Hj ¼ 2, i.e. subgroups of index 2 are always normal subgroups. The subgroup H contains half of the elements of G, whereas the other half of the elements forms ‘the other’ coset. This coset must then be the right as well as the left coset.

The coset decomposition of a group G relative to a normal subgroup H / G has a property which makes it particularly useful for displaying the structure of a group. Consider the coset decomposition with the cosets S j and S k of a group G relative to its subgroup H < G. In general the product S j S k of two cosets, cf. Deﬁnition 1.2.4.4.1, will not be a coset again. However, if and only if H / G is a normal subgroup of G, the product of two cosets is always another coset. This means that for the set of all cosets of a normal subgroup H / G there exists a law of composition for which the closure is fulﬁlled. One can show that the other group postulates are also fulﬁlled for the cosets and their multiplication if H / G holds: there is a neutral element (which is H), for each coset g H ¼ H g the coset g 1 H ¼ H g 1 forms the inverse element and for the coset multiplication the associative law holds.

1.2.4.3. Conjugate elements and conjugate subgroups In a coset decomposition, the set of all elements of the group G is partitioned into cosets which form classes in the mathematical sense of the word, i.e. each element of G belongs to exactly one coset. Another equally important partition of the group G into classes of elements arises from the following deﬁnition: Deﬁnition 1.2.4.3.1. Two elements g j ; g k 2 G are called conjugate & if there is an element g q 2 G such that g 1 q gj gq ¼ gk.

Deﬁnition 1.2.4.4.2. Let H / G. The cosets of the decomposition of the group G relative to the normal subgroup H / G form a group with respect to the composition law of coset multiplication. This group is called the factor group G=H. Its order is jG : Hj, i.e. & the index of H in G.

Remarks (1) Deﬁnition 1.2.4.3.1 partitions the elements of G into classes of conjugate elements which are called conjugacy classes of elements. (2) The unit element always forms a conjugacy class by itself. (3) Each element of an Abelian group forms a conjugacy class by itself. (4) Elements of the same conjugacy class have the same order. (5) Different conjugacy classes may contain different numbers of elements, i.e. have different ‘lengths’.

A factor group F ¼ G=H is not necessarily isomorphic to a subgroup Hj < G. Factor groups are indispensable for an understanding of the homomorphism of one group onto the other. The relations between a group G and its homomorphic image are very strong and are expressed by the following lemma:

Not only the individual elements of a group G but also the subgroups of G can be classiﬁed in conjugacy classes.

Lemma 1.2.4.4.3. Let G 0 be a homomorphic image of the group G. Then the set of all elements of G that are mapped onto the unit element e 0 2 G 0 forms a normal subgroup X of G. The group G 0 is isomorphic to the factor group G=X and the cosets of X in G are mapped onto the elements of G 0. The normal subgroup X is called the kernel of the mapping; it forms the unit element of the factor group G=X. A homomorphic image of G exists for any & normal subgroup of G.

Deﬁnition 1.2.4.3.2. Two subgroups Hj ; Hk < G are called conjugate if there is an element g q 2 G such that g 1 q Hj g q ¼ Hk g & holds. This relation is often written Hj q ¼ Hk . Remarks (1) The ‘trivial subgroup’ I (consisting only of the unit element of G) and the group G itself each form a conjugacy class by themselves. (2) Each subgroup of an Abelian group forms a conjugacy class by itself. (3) Subgroups in the same conjugacy class are isomorphic and thus have the same order. (4) Different conjugacy classes of subgroups may contain different numbers of subgroups, i.e. have different lengths.

The most important homomorphism in crystallography is the relation between a space group G and its homomorphic image, the point group P, where the kernel is the subgroup T (G) of all translations of G, cf. Section 1.2.5.4.

1.2.4.5. Normalizers The concept of the normalizer N G ðHÞ of a group H < G in a group G is very useful for the considerations of the following sections. The length of the conjugacy class of H in G is determined by this normalizer. Let H < G and hj 2 H. Then h1 j Hh j ¼ H holds because H is a group. If H / G, then g 1 k Hg k ¼ H for any g k 2 G. If H is not a

Equation (1.2.4.1) can be written gp H ¼ g 1 for all p; 1 p i: p H g p or H ¼ H

ð1:2:4:2Þ

Using conjugation, Deﬁnition 1.2.4.2.3 can be formulated as Deﬁnition 1.2.4.3.3. A subgroup H of a group G is a normal subgroup H / G if it is identical with all of its conjugates, i.e. if its & conjugacy class consists of the one subgroup H only.

5

The right-hand side of this equation is the set of all products g r ¼ g jp g kq , where g jp runs through all elements of Kj and g kq through all elements of Kk. Each element g r is taken only once in the set.

13

1. SPACE GROUPS AND THEIR SUBGROUPS normal subgroup of G, there may nevertheless be elements g p 2 G; g p 62 H for which g 1 p Hg p ¼ H holds. We consider the set of all elements g p 2 G that have this property.

number of afﬁne space-group types for n = 6 is 28 927 922 (Plesken & Schulz, 2000). 7052 of these afﬁne space-group types split into enantiomorphic pairs of space groups, such that there are 28 934 974 crystallographic space-group types (Souvignier, 2003). One of the characteristics of a space group is its translation group. Any space group G is an inﬁnite group because the number of its translations is already inﬁnite. The set of all translations of G forms the inﬁnite translation subgroup T ðGÞ / G with the composition law of performing one translation after the other, represented by the multiplication of matrix–column pairs. The group T ðGÞ is a normal subgroup of G of ﬁnite index. The vector lattice L, cf. Section 1.2.2.2, forms a group with the composition law of vector addition. This group is isomorphic to the group T (G). The matrix–column pairs of the symmetry operations of a space group G are mostly referred to the conventional coordinate system. Its basis is chosen as a lattice basis and in such a way that the matrices for the linear parts of the symmetry operations of G are particularly simple. The origin is chosen such that as many coset representatives as possible can be selected with their column coefﬁcients to be zero, or such that the origin is situated on a centre of inversion. This means (for details and examples see Section 8.3.1 of IT A):

Deﬁnition 1.2.4.5.1. The set of all elements g p 2 G that map the gp subgroup H < G onto itself by conjugation, H ¼ g 1 p Hg p ¼ H , forms a group N G ðHÞ, called the normalizer of H in G, where & H / N G ðHÞ G. Remarks (1) The group H < G is a normal subgroup of G, H / G, if and only if N G ðHÞ ¼ G. (2) Let N G ðHÞ ¼ fg p g. One can decompose G into right cosets relative to N G ðHÞ. All elements g p g r of a right coset N G ðHÞ g r of this decomposition ðG : N G ðHÞÞ transform H 1 1 into the same subgroup g 1 r g p H g p g r ¼ g r H g r < G, which is thus conjugate to H in G by g r . (3) The elements of different cosets of ðG : N G ðHÞÞ transform H into different conjugates of H. The number of cosets of N G ðHÞ is equal to the index iN ¼ jG : N G ðHÞj of N G ðHÞ in G. Therefore, the number NH of conjugates in the conjugacy class of H is equal to the index iN and is thus determined by the order of N G ðHÞ. From N G ðHÞ H, iN ¼ jG : N G ðHÞj jG : Hj ¼ i follows. This means that the number of conjugates of a subgroup H < G cannot exceed the index i ¼ jG : Hj. (4) If H < G is a maximal subgroup of G, then either N G ðHÞ ¼ G and H / G is a normal subgroup of G or N G ðHÞ ¼ H and the number of conjugates is equal to the index i ¼ jG : Hj. (5) For the normalizers of the space groups, see the corresponding part of Section 1.2.6.3.

(1) The basis is always chosen such that all matrix coefﬁcients are 0 or 1. (2) If possible, the basis is chosen such that all matrices have main diagonal form; then six of the nine coefﬁcients are 0 and three are 1. (3) If (2) is not possible, the basis is chosen such that the matrices are orthogonal. Again, six coefﬁcients are 0 and three are 1. (4) If (3) is not possible, the basis is hexagonal. At least ﬁve of the nine matrix coefﬁcients are 0 and at most four are 1. (5) The conventional basis chosen according to these rules is not always primitive, cf. the ﬁrst example of Section 8.3.1 of IT A. If the conventional basis is primitive, then the lattice is also called primitive; if the conventional basis is not primitive, then the lattice referred to this (non-primitive) basis is called a centred lattice. (6) The matrix parts of a translation and of an inversion are

1.2.5. Space groups 1.2.5.1. Space groups and their description The set of all symmetry operations of a three-dimensional crystal pattern, i.e. its symmetry group, is the space group of this crystal pattern. In a plane, the symmetry group of a twodimensional crystal pattern is its plane group. In the following, the term ‘space group’ alone will be used and the plane groups are included because they are the space groups of twodimensional space. A crystal pattern is a periodic array. This means that there are translations among its symmetry operations. The translations of crystals are small (up to a few hundred a˚ngstro¨ms) but cannot be arbitrarily short because of the ﬁnite size of the particles in crystal structures. One thus deﬁnes for any ﬁnite integer n:

0

1 I ¼ @0 0

0 1 0

0 1 0 1 0 A and I ¼ @ 0 1 0

0 1 0

1 0 0 A; 1

i.e. the unit matrix and the negative unit matrix. They are independent of the basis. (7) The basis vectors of a crystallographic basis are lattice vectors. This makes the description of the lattice and of the symmetry operations of a crystal pattern independent of the actual metrics of the lattice, for example independent of temperature and pressure. It also means that the description of the symmetry may be the same for space groups of the same type, cf. Section 1.2.5.3. (8) The origin is chosen at a point of highest site symmetry which is left invariant by as many symmetry operations as possible. The column parts wk of these symmetry operations are wk ¼ o, i.e. the columns consist of zeroes only.

Deﬁnition 1.2.5.1.1. A group G of isometries in n-dimensional space is called an n-dimensional space group if (1) G contains n linearly independent translations. (2) There is a minimum length > 0 such that the length r of any & translation vector is at least r ¼ . Condition (2) is justiﬁed because crystal structures contain atoms of ﬁnite size, and it is necessary to avoid inﬁnitely small translations as elements of space groups. Several fundamental properties would not hold without this condition, such as the existence of a lattice of translation vectors and the restriction to only a few rotation angles. In this volume, only the dimensions n ¼ 2 and n ¼ 3 will be dealt with. However, the space groups (more precisely, the spacegroup types, cf. Section 1.2.5.3) and other crystallographic items are also known for dimensions n = 4, 5 and 6. For example, the

It is obviously impossible to list all elements of an inﬁnite group individually. One can deﬁne the space group by a set of generators, because the number of necessary generators for any space group is ﬁnite: theoretically, up to six generators might be

14

1.2. GENERAL INTRODUCTION TO THE SUBGROUPS OF SPACE GROUPS treated in the next sections. The point groups and the translation groups of the space groups can also be classiﬁed in a similar way. Only the classiﬁcation of the point groups is treated in this chapter. For a more detailed treatment and for the classiﬁcation of the lattices, the reader is referred to Chapter 1.4 of this volume, to Part 8 of IT A or to Brown et al. (1978).

necessary but in practice up to ten generators are chosen for a space group. In IT A and in this volume, the set of the conventional generators is listed in the block ‘Generators selected’. The unit element is taken as the ﬁrst generator; the generating translations follow and the generation is completed with the generators of the non-translational symmetry operations. The rules for the choice of the conventional generators are described in IT A, Section 8.3.5. The description by generators is particularly important for this volume because many of the maximal subgroups in Chapters 2.2 and 2.3 are listed by their generators. These generators are chosen such that the generation of the general position can follow a composition series, cf. Ledermann (1976) and Ledermann & Weir (1996). This procedure allows the generation by a short program or even by hand. For details see IT A, Section 8.3.5; in Table 8.3.5.2 of IT A an example for the generation of a space group along these lines is displayed. There are four ways to describe a space group in IT A:

1.2.5.3. Space groups and space-group types We ﬁrst consider the classiﬁcation of the space groups into types. A more detailed treatment may be found in Section 8.2.1 of IT A. In practice, a common way is to look for the symmetry of the space group G and to compare this symmetry with that of the diagrams in the tables of IT A. With the exception of some double descriptions,6 there is exactly one set of diagrams which displays the symmetry of G, and G belongs to that space-group type which is described in this set. From those diagrams the Hermann–Mauguin symbol, abbreviated as HM symbol, the Schoenﬂies symbol and the spacegroup number are taken. A rigorous deﬁnition is:

(i) A set of generators is the ﬁrst way in which the space-group types G, cf. Section 1.2.5.3, are described in IT A. This way is also used in the tables of this volume. (ii) By the matrices of the coset representatives of ðG : T ðGÞÞ in the general position. These matrices are not written in full but in a shorthand notation, cf. Section 8.1.5 or Chapter 11.1 of IT A. This kind of description is used for translationengleiche maximal subgroups in Chapters 2.2 and 2.3 of this volume, but in a slightly modiﬁed way, cf. Section 2.1.3. (iii) In a visual way by diagrams of the symmetry elements (not symmetry operations!) of G within a unit cell and its surroundings. (iv) Also in a visual way by depicting the general-position points, again within a unit cell and its surroundings.

Deﬁnition 1.2.5.3.1. Two space groups belong to the same afﬁne & space-group type if and only if they are isomorphic.7 This deﬁnition refers to a rather abstract property which is of great mathematical but less practical value. In crystallography another deﬁnition is more appropriate which results in exactly the same space-group types as are obtained by isomorphism. It starts from the description of the symmetry operations of a space group by matrix–column pairs or, as will be formulated here, by augmented matrices. For this one refers each of the space groups to one of its lattice bases. Deﬁnition 1.2.5.3.2. Two space groups G and G 0 belong to the same afﬁne space-group type if for a lattice basis and an origin of G, a lattice basis and an origin of G 0 can also be found so that the groups of augmented matrices fWg describing G and fW 0 g & describing G 0 are identical.

1.2.5.2. Classiﬁcations of space groups There are an inﬁnite number of space groups because there are an inﬁnite number of known or conceivable crystals and crystal patterns. Indeed, because the lattice parameters depend on temperature and pressure, so do the lattice translations and the space group of a crystal. There is great interest in getting an overview of this vast number of space groups. To achieve this goal, one ﬁrst characterizes the space groups by their grouptheoretical properties and classiﬁes them into space-group types where the space groups of each type have certain properties in common. To get a better overview, one then classiﬁes the spacegroup types such that related types belong to the same ‘superclass’. This classiﬁcation is done in two ways (cf. Sections 1.2.5.4 and 1.2.5.5):

In this deﬁnition the coordinate systems are chosen such that the groups of augmented matrices agree. It is thus possible to describe the symmetry of all space groups of the same type by one (standardized) set of matrix–column pairs, as is done, for example, in the tables of IT A. In the subgroup tables of Chapters 2.2 and 2.3 it frequently happens that a subgroup H < G of a space group G is given by its matrix–column pairs referred to a nonconventional coordinate system. In this case, a transformation of the coordinate system can bring the matrix–column pairs to the standard form by which the space-group type may be determined. In the subgroup tables both the space-group type and the transformation of the coordinate system are listed. One can also use this procedure for the deﬁnition of the term ‘afﬁne space-group type’:

(1) ﬁrst into geometric crystal classes by the point group of the space group, and then into crystal systems; (2) into the arithmetic crystal classes of the space groups and then into Bravais ﬂocks and into lattice systems (not treated here, cf. IT A, Section 8.2.5); (3) all these classes: geometric and arithmetic crystal classes, crystal systems, Bravais ﬂocks and lattice systems are classiﬁed into crystal families.

Deﬁnition 1.2.5.3.3. Let two space groups G and G 0 be referred to lattice bases and represented by their groups of augmented matrices fWg and fW 0 g. The groups G and G 0 belong to the same 6

Monoclinic space groups are described in the settings ‘unique axis b’ and ‘unique axis c’; rhombohedral space groups are described in the settings ‘hexagonal axes’ and ‘rhombohedral axes’; and 24 space groups are described with two origins by ‘origin choice 1’ and ‘origin choice 2’. In each case, both descriptions lead to the same short Hermann–Mauguin symbol and space-group number. 7 The name ‘afﬁne space-group type’ stems from Deﬁnition 1.2.5.3.3. ‘Afﬁne space-group types’ have to be distinguished from ‘crystallographic space-group types’ which are deﬁned by Deﬁnition 1.2.5.3.4.

In reality, the tables in Chapters 2.2 and 2.3 and the graphs in Chapters 2.4 and 2.5 are tables and graphs for space-group types. The sequence of the space-group types in IT A and thus in this volume is determined by their crystal class, their crystal system and their crystal family. Therefore, these classiﬁcations are

15

1. SPACE GROUPS AND THEIR SUBGROUPS afﬁne space-group type if an augmented matrix P with linear part P, detðPÞ 6¼ 0, and column part p exists, for which fW 0 g ¼ P1 fWg P holds.

long-winded. However, occasionally the distinction is indispensable in order to avoid serious difﬁculties of comprehension. For example, the sentence ‘A space group is a proper subgroup of itself’ is incomprehensible, whereas the sentence ‘A space group and its proper subgroup belong to the same space-group type’ makes sense.

ð1:2:5:1Þ &

The afﬁne space-group types are classes in the mathematical sense of the word, i.e. each space group belongs to exactly one type. The derivation of these types reveals 219 afﬁne space-group types and 17 plane-group types. In crystallography one usually distinguishes 230 rather than 219 space-group types in a slightly ﬁner subdivision. The difference can best be explained using Deﬁnition 1.2.5.3.3. The matrix part P may have a negative determinant. In this case, a righthanded basis is converted into a left-handed one, and righthanded and left-handed screw axes are exchanged. It is a convention in crystallography to always refer the space to a righthanded basis and hence transformations with detðPÞ < 0 are not admitted.

1.2.5.4. Point groups and crystal classes If the point coordinates are mapped by an isometry and its matrix–column pair, the vector coefﬁcients are mapped by the linear part, i.e. by the matrix alone, cf. Section 1.2.2.6. Because the number of its elements is inﬁnite, a space group generates from one point an inﬁnite set of symmetry-equivalent points by its matrix–column pairs. Because the number of matrices of the linear parts is ﬁnite, the group of matrices generates from one vector a ﬁnite set of symmetry-equivalent vectors, for example the vectors normal to certain planes of the crystal. These planes determine the morphology of the ideal macroscopic crystal and its cleavage; the centre of the crystal represents the zero vector. When the symmetry of a crystal can only be determined by its macroscopic properties, only the symmetry group of the macroscopic crystal can be found. All its symmetry operations leave at least one point of the crystal ﬁxed, viz its centre of mass. Therefore, this symmetry group was called the point group of the crystal, although its symmetry operations are those of vector space, not of point space. Although misunderstandings are not rare, this name is still used in today’s crystallography for historical reasons.8 Let a conventional coordinate system be chosen and the elements g j 2 G be represented by the matrix–column pairs ðW j ; wj Þ, with the representation of the translations t k 2 T ðGÞ by the pairs ðI; t k Þ. Then the composition of ðW j ; wj Þ with all translations forms an inﬁnite set fðI; t k Þ ðW j ; wj Þ ¼ ðW j ; wj þ t k Þg of symmetry operations which is a right coset of the coset decomposition ðG : T ðGÞÞ. From this equation it follows that the elements of the same coset of the decomposition ðG : T ðGÞÞ have the same linear part. On the other hand, elements of different cosets have different linear parts if T ðGÞ contains all translations of G. Thus, each coset can be characterized by its linear part. It can be shown from equations (1.2.2.5) and (1.2.2.6) that the linear parts form a group which is isomorphic to the factor group G=T ðGÞ, i.e. to the group of the cosets.

Deﬁnition 1.2.5.3.4. If the matrix P is restricted by the condition detðPÞ > 0, eleven afﬁne space-group types split into two spacegroup types each, one with right-handed and one with lefthanded screw axes, such that the total number of types is 230. These 230 space-group types are called crystallographic spacegroup types. The eleven splitting space-group types are called pairs of enantiomorphic space-group types and the space groups & themselves are enantiomorphic pairs of space groups. The space groups of an enantiomorphic pair belong to different crystallographic space-group types but are isomorphic. As a consequence, in the lists of isomorphic subgroups H < G of the tables of Chapter 2.3, there may occur subgroups H with another conventional HM symbol and another space-group number than that of G, cf. Example 1.2.6.2.7. In such a case, G and H are members of an enantiomorphic pair of space groups and H belongs to the space-group type enantiomorphic to that of G. There are no enantiomorphic pairs of plane groups. The space groups are of different complexity. The simplest ones are the symmorphic space groups (not to be confused with ‘isomorphic’ space groups) according to the following deﬁnition: Deﬁnition 1.2.5.3.5. A space group G is called symmorphic if representatives g k of all cosets T ðGÞ g k can be found such that the & set fg k g of all representatives forms a group.

Deﬁnition 1.2.5.4.1. A group of linear parts, represented by a group of matrices W j , is called a point group P. If the linear parts are those of the matrix–column pairs describing the symmetry operations of a space group G, the group is called the point group P G of the space group G. The point groups that can belong to & space groups are called crystallographic point groups.

The group fg k g is ﬁnite and thus leaves a point F ﬁxed. In the standard setting of any symmorphic space group such a point F is chosen as the origin. Thus, the translation parts of the elements g k consist of zeroes only. If a space group is symmorphic then all space groups of its type are symmorphic. Therefore, one can speak of ‘symmorphic spacegroup types’. Symmorphic space groups can be recognized easily by their HM symbols: they contain an unmodiﬁed point-group symbol: rotations, reﬂections, inversions and rotoinversions but no screw rotations or glide reﬂections. There are 73 symmorphic space-group types of dimension three and 13 of dimension two; none of them show enantiomorphism. One frequently speaks of ‘the 230 space groups’ or ‘the 17 plane groups’ and does not distinguish between the terms ‘space group’ and ‘space-group type’. This is very often possible and is also done in this volume in order to make the explanations less

According to Deﬁnition 1.2.5.4.1, the factor group G=T ðGÞ is isomorphic to the point group P G. This property is exploited in the graphs of translationengleiche subgroups of space groups, cf. Chapter 2.4 and Section 2.1.8.2. All point groups in the following sections are crystallographic point groups. The maximum order of a crystallographic point group is 48 in three-dimensional space and 12 in two-dimensional space. 8 The term point group is also used for a group of symmetry operations of point space, which is better called a site-symmetry group and which is the group describing the symmetry of the surroundings of a point in point space.

16

1.2. GENERAL INTRODUCTION TO THE SUBGROUPS OF SPACE GROUPS As with space groups, there are also an inﬁnite number of crystallographic point groups which may be classiﬁed into a ﬁnite number of point-group types. This cannot be done by isomorphism because geometrically different point groups may be isomorphic. For example, point groups consisting of the identity with the inversion fI; Ig or with a twofold rotation fI; 2g or with a reﬂection through a plane fI; mg are all isomorphic to the (abstract) group of order 2. As for space groups, the classiﬁcation may be performed, however, referring the point groups to corresponding vector bases. As translations do not occur among the point-group operations, one may choose any basis for the description of the symmetry operations by matrices. One takes the basis of fW 0 g as given and transforms the basis of fW g to the basis corresponding to that of fW 0 g. This leads to the deﬁnition:

Deﬁnition 1.2.5.4.4. The set of all orthogonal mappings with matrices W which map a lattice L onto itself is called the point group of the lattice L or the holohedry of the lattice L. A crystal class of point groups P G is called a holohedral crystal class if it & contains a holohedry. There are seven holohedral crystal classes in the space: 1, 2=m, mmm, 4=mmm, 3m, 6=mmm and m3m. Their lattices are called triclinic, monoclinic, orthorhombic, tetragonal, rhombohedral, hexagonal and cubic, respectively. There are four holohedral crystal classes in the plane: 2, 2mm, 4mm and 6mm. Their twodimensional lattices (or nets) are called oblique, rectangular, square and hexagonal, respectively. The lattices can be classiﬁed into lattice types or Bravais types, mostly called Bravais lattices, or into lattice systems (called Bravais systems in editions 1 to 4 of IT A). These classiﬁcations are not discussed here because they are not directly relevant to the classiﬁcation of the space groups. This is because the lattice symmetry is not necessarily typical for the symmetry of its space group but may accidentally be higher. For example, the lattice of a monoclinic crystal may be accidentally orthorhombic (only for certain values of temperature and pressure). In Sections 8.2.5 and 8.2.7 of IT A the ‘typical lattice symmetry’ of a space group is deﬁned.

Deﬁnition 1.2.5.4.2. Two crystallographic point groups P G and P 0G 0 belong to the same point-group type or to the same crystal class of point groups if there is a real non-singular matrix P which maps a matrix group fW g of P G onto a matrix group fW 0 g of P 0G 0 & by the transformation fW 0 g ¼ P 1 fW g P. Point groups can be classiﬁed by Deﬁnition 1.2.5.4.2. Further space groups may be classiﬁed into ‘crystal classes of space groups’ according to their point groups: Deﬁnition 1.2.5.4.3. Two space groups belong to the same crystal class of space groups if their point groups belong to the same & crystal class of point groups.

1.2.5.5. Crystal systems and crystal families The example of P1 mentioned above shows that the point group of the lattice may be systematically of higher order than that of its space group. There are obviously point groups and thus space groups that belong to a holohedral crystal class and those that do not. The latter can be assigned to a holohedral crystal class uniquely according to the following deﬁnition:9

Whether two space groups belong to the same crystal class or not can be worked out from their standard HM symbols: one removes the lattice parts from these symbols as well as the constituents ‘1’ from the symbols of trigonal space groups and replaces all constituents for screw rotations and glide reﬂections by those for the corresponding pure rotations and reﬂections. The symbols obtained in this way are those of the corresponding point groups. If they agree, the space groups belong to the same crystal class. The space groups also belong to the same crystal class if the point-group symbols belong to the pair 42m and 4m2 or to the pair 62m and 6m2. There are 32 classes of three-dimensional crystallographic point groups and 32 crystal classes of space groups, and ten classes of two-dimensional crystallographic point groups and ten crystal classes of plane groups. The distribution into crystal classes classiﬁes space-group types – and thus space groups – and crystallographic point groups. It does not classify the inﬁnite set of all lattices into a ﬁnite number of lattice types, because the same lattice may belong to space groups of different crystal classes. For example, the same lattice may be that of a space group of type P1 (of crystal class 1) and that of a space group of type P1 (of crystal class 1). Nevertheless, there is also a deﬁnition of the ‘point group of a lattice’. Let a vector lattice L of a space group G be referred to a lattice basis. Then the linear parts W of the matrix–column pairs (W, w) of G form the point group P G. If (W, w) maps the space group G onto itself, then the linear part W maps the (vector) lattice L onto itself. However, there may be additional matrices which also describe symmetry operations of the lattice L. For example, the point group P G of a space group of type P1 consists of the identity 1 only. However, with any vector t 2 L, the negative vector t 2 L also belongs to L. Therefore, the lattice L is always centrosymmetric and has the inversion 1 as a symmetry operation independent of the symmetry of the space group.

Deﬁnition 1.2.5.5.1. A crystal class C of a space group G is either holohedral H or it can be assigned uniquely to H by the condition: any point group of C is a subgroup of a point group of H but not a subgroup of a holohedral crystal class H 0 of smaller order. The set of all crystal classes of space groups that are assigned to the same holohedral crystal class is called a crystal system of space & groups. The 32 crystal classes of space groups are classiﬁed into seven crystal systems which are called triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal and cubic. There are four crystal systems of plane groups: oblique, rectangular, square and hexagonal. Like the space groups, the crystal classes of point groups are classiﬁed into the seven crystal systems of point groups. Apart from accidental lattice symmetries, the space groups of different crystal systems have lattices of different symmetry. As an exception, the hexagonal primitive lattice occurs in both hexagonal and trigonal space groups as the typical lattice. Therefore, the space groups of the trigonal and the hexagonal crystal systems are more related than space groups from other different crystal systems. Indeed, in different crystallographic schools the term ‘crystal system’ was used for different objects. One sense of the term was the ‘crystal system’ as deﬁned above, while another sense of the old term ‘crystal system’ is now called 9 This assignment does hold for low dimensions of space up to dimension 4. A dimension-independent deﬁnition of the concepts of ‘crystal system’ and ‘crystal family’ is found in IT A, Chapter 8.2, where the classiﬁcations are treated in more detail.

17

1. SPACE GROUPS AND THEIR SUBGROUPS a ‘crystal family’ according to the following deﬁnition [for deﬁnitions that are also valid in higher-dimensional spaces, see Brown et al. (1978) or IT A, Chapter 8.2]:

index can in principle be obtained from the data on maximal subgroups. A non-maximal subgroup H < G of ﬁnite index [i] is connected with the original group G through a chain H ¼ Z k < Z k1 < . . . < Z 1 < Z 0 ¼ G, where each group Z j < Z j1 is a maximal subgroup of Z j1, with the index ½ij ¼ jZ j1Q: Z j j, k j ¼ 1; . . . ; k. The number k is ﬁnite and the relation i ¼ j¼1 ij holds, i.e. the total index [i] is the product of the indices ij. According to Hermann (1929), the following types of subgroups of space groups have to be distinguished:

Deﬁnition 1.2.5.5.2. In three-dimensional space, the classiﬁcation of the set of all space groups into crystal families is the same as that into crystal systems with the one exception that the trigonal and hexagonal crystal systems are united to form the hexagonal crystal family. There is no difference between crystal systems and & crystal families in the plane.

Deﬁnition 1.2.6.2.1. A subgroup H of a space group G is called a translationengleiche subgroup or a t-subgroup of G if the set T ðGÞ of translations is retained, i.e. T ðHÞ ¼ T ðGÞ, but the number of cosets of G=T ðGÞ, i.e. the order P of the point group P G, is & reduced such that jG=T ðGÞj > jH=T ðHÞj.10

The partition of the space groups into crystal families is the most universal one. The space groups and their types, their crystal classes and their crystal systems are classiﬁed by the crystal families. Analogously, the crystallographic point groups and their crystal classes and crystal systems are classiﬁed by the crystal families of point groups. Lattices, their Bravais types and lattice systems can also be classiﬁed into crystal families of lattices; cf. IT A, Chapter 8.2.

The order of a crystallographic point group P G of the space group G is always ﬁnite. Therefore, the number of the subgroups of P G is also always ﬁnite and these subgroups and their relations are displayed in well known graphs, cf. Chapter 2.4 and Section 2.1.8 of this volume. Because of the isomorphism between the point group P G and the factor group G=T ðGÞ, the subgroup graph for the point group P G is the same as that for the t-subgroups of G, only the labels of the groups are different. For deviations between the point-group graphs and the actual space-group graphs of Chapter 2.4, cf. Section 2.1.8.2.

1.2.6. Types of subgroups of space groups 1.2.6.1. Introductory remarks Group–subgroup relations form an essential part of the applications of space-group theory. Let G be a space group and H < G a proper subgroup of G. All maximal subgroups H < G of any space group G are listed in Part 2 of this volume. There are different kinds of subgroups which are deﬁned and described in this section. The tables and graphs of this volume are arranged according to these kinds of subgroups. Moreover, for the different kinds of subgroups different data are listed in the subgroup tables and graphs. Let Gj and Hj be space groups of the space-group types G and H. The group–subgroup relation Gj > Hj is a relation between the particular space groups Gj and Hj but it can be generalized to the space-group types G and H. Certainly, not every space group of the type H will be a subgroup of every space group of the type G. Nevertheless, the relation Gj > Hj holds for any space group of G and H in the following sense: If Gj > Hj holds for the pair Gj and Hj , then for any space group Gk of the type G a space group Hk of the type H exists for which the corresponding relation Gk > Hk holds. Conversely, for any space group Hm of the type H a space group Gm of the type G exists for which the corresponding relation Gm > Hm holds. Only this property of the group– subgroup relations made it possible to compile and arrange the tables of this volume so that they are as concise as those of IT A.

Example 1.2.6.2.2 Consider a space group G of type P12=m1 referred to a conventional coordinate system. The translation subgroup T ðGÞ consists of all translations with translation vectors t ¼ ua þ vb þ wc, where u; v; w run through all integer numbers. The coset decomposition of ðG : T ðGÞÞ results in the four cosets T (G), T ðGÞ 20 , T ðGÞ m0 and T ðGÞ 10 , where the right operations are a twofold rotation 20 around the rotation axis passing through the origin, a reﬂection m0 through a plane containing the origin and an inversion 10 with the inversion point at the origin, respectively. The three combinations H1 ¼ T ðGÞ [ T ðGÞ 20 , H2 ¼ T ðGÞ [ T ðGÞ m0 and H3 ¼ T ðGÞ [ ðT GÞ 10 each form a translationengleiche maximal subgroup of G of index 2 with the space-group symbols P121, P1m1 and P1, respectively. Deﬁnition 1.2.6.2.3. A subgroup H < G of a space group G is called a klassengleiche subgroup or a k-subgroup if the set T (G) of all translations of G is reduced to T ðHÞ < T ðGÞ but all linear parts of G are retained. Then the number of cosets of the decompositions H=T ðHÞ and G=T ðGÞ is the same, i.e. jH=T ðHÞj ¼ jG=T ðGÞj. In other words: the order of the point & group P H is the same as that of P G. See also footnote 10.

1.2.6.2. Deﬁnitions and examples

For a klassengleiche subgroup H < G, the cosets of the factor group H=T ðHÞ are smaller than those of G=T ðGÞ. Because T (H) is still inﬁnite, the number of elements of each coset is inﬁnite but the index jT ðGÞ : T ðHÞj > 1 is ﬁnite. The number of k-subgroups of G is always inﬁnite.

‘Maximal subgroups’ have been introduced by Deﬁnition 1.2.4.1.2. The importance of this deﬁnition will become apparent in the corollary to Hermann’s theorem, cf. Lemma 1.2.8.1.3. In this volume only the maximal subgroups are listed for any plane and any space group. A maximal subgroup of a plane group is a plane group, a maximal subgroup of a space group is a space group. On the other hand, a minimal supergroup of a plane group or of a space group is not necessarily a plane group or a space group, cf. Section 2.1.6. If the maximal subgroups are known for each space group, then each non-maximal subgroup of a space group G with ﬁnite

10

German: zellengleiche means ‘with the same cell’; translationengleiche means ‘with the same translations’; klassengleiche means ‘of the same (crystal) class’. Of the different German declension endings only the form with terminal -e is used in this volume. The terms zellengleiche and klassengleiche were introduced by Hermann (1929). The term zellengleiche was later replaced by translationengleiche because of possible misinterpretations. In this volume they are sometimes abbreviated as t-subgroups and k-subgroups.

18

1.2. GENERAL INTRODUCTION TO THE SUBGROUPS OF SPACE GROUPS Example 1.2.6.2.4 Consider a space group G of the type C121, referred to a conventional coordinate system. The set T ðGÞ of all translations can be split into the set T i of all translations with integer coefﬁcients u, v and w and the set T f of all translations for which the coefﬁcients u and v are fractional. The set T i forms a group; the set T f is the other coset in the decomposition ðT ðGÞ : T i Þ and does not form a group. Let t C be the ‘centring translation’ with the translation vector 12 ða þ bÞ. Then T f can be written T i t C . Let 20 mean a twofold rotation around the rotation axis through the origin. There are altogether four cosets of the decomposition (G : T i ), which can be written now as T i, T f ¼ T i t C , T i 20 and T f 20 ¼ ðT i t C Þ 20 ¼ T i ðt C 20 Þ. The union T i [ ðT i t C Þ ¼ T G forms the translationengleiche maximal subgroup C1 (conventional setting P1) of G of index 2. The union T i [ ðT i 20 Þ forms the klassengleiche maximal subgroup P121 of G of index 2. The union T i [ ðT i ðt C 20 ÞÞ also forms a klassengleiche maximal subgroup of index 2. Its HM symbol is P121 1, and the twofold rotations 2 of the point group 2 are realized by screw rotations 21 in this subgroup because (t C 20 ) is a screw rotation with its screw axis running parallel to the b axis through the point 14 ; 0; 0. There are in fact these two k-subgroups of C121 of index 2 which have the group T i in common. In the subgroup table of C121 both are listed under the heading ‘Loss of centring translations’ because the conventional unit cell is retained while only the centring translations have disappeared. (Four additional klassengleiche maximal subgroups of C121 with index 2 are found under the heading ‘Enlarged unit cell’.) The group T i of type P1 is a non-maximal subgroup of C121 of index 4.

groups of index 2 for any space group of type P 1 which are obtained by seven different cell enlargements. If G belongs to a pair of enantiomorphic space-group types, then the isomorphic subgroups of G may belong to different crystallographic space-group types with different HM symbols and different space-group numbers. In this case, an inﬁnite number of subgroups belong to the crystallographic space-group type of G and another inﬁnite number belong to the enantiomorphic space-group type. Example 1.2.6.2.7 Space group P41, No. 76, has for any prime number p > 2 an isomorphic maximal subgroup of index p with the lattice parameters a; b; pc. This is an inﬁnite number of subgroups because there is an inﬁnite number of primes. The subgroups belong to the space-group type P41 if p ¼ 4n þ 1; they belong to the type P43 if p ¼ 4n þ 3. Deﬁnition 1.2.6.2.8. A subgroup of a space group is called general or a general subgroup if it is neither a translationengleiche nor a klassengleiche subgroup. It has lost translations as well as linear & parts, i.e. point-group symmetry. Example 1.2.6.2.9 The subgroup T g in Example 1.2.6.2.6 has lost all inversions of the original space group P1 as well as all translations with odd u. It is a general subgroup P1 of the space group P1 of index 4.

1.2.6.3. The role of normalizers for group–subgroup pairs of space groups

Deﬁnition 1.2.6.2.5. A klassengleiche or k-subgroup H < G is called isomorphic or an isomorphic subgroup if it belongs to the same afﬁne space-group type (isomorphism type) as G. If a subgroup is not isomorphic, it is sometimes called non& isomorphic.

In Section 1.2.4.5, the normalizer N G ðHÞ of a subgroup H < G in the group G was deﬁned. The equation H / N G ðHÞ G holds, i.e. H is a normal subgroup of N G ðHÞ. The normalizer N G ðHÞ, by its index in G, determines the number Nj ¼ jG : N G ðHÞj of subgroups Hj < G that are conjugate in the group G, cf. Remarks (2) and (3) below Deﬁnition 1.2.4.5.1. The group–subgroup relations between space groups become more transparent if one looks at them from a more general point of view. Space groups are part of the general theory of mappings. Particular groups are the afﬁne group A of all reversible afﬁne mappings, the Euclidean group E of all isometries, the translation group T of all translations and the orthogonal group O of all orthogonal mappings. Connected with any particular space group G are its group of translations T ðGÞ and its point group P G. In addition, the normalizers N A ðGÞ of G in the afﬁne group A and N E ðGÞ in the Euclidean group E are useful. They are listed in Section 15.2.1 of IT A. Although consisting of isometries only, N E ðGÞ is not necessarily a space group, see the second paragraph of Example 1.2.7.3.1. For the group–subgroup pairs H < G the following relations hold:

Isomorphic subgroups are special k-subgroups. The importance of the distinction between k-subgroups in general and isomorphic subgroups in particular stems from the fact that the number of maximal non-isomorphic k-subgroups of any space group is ﬁnite, whereas the number of maximal isomorphic subgroups is always inﬁnite, see Section 1.2.8. Example 1.2.6.2.6 Consider a space group G of type P1 referred to a conventional coordinate system. The translation subgroup T ðGÞ consists of all translations with translation vectors t ¼ ua þ vb þ wc, where u; v and w run through all integer numbers. There is an inversion 10 with the inversion point at the origin and also an inﬁnite number of other inversions, generated by the combinations of 10 with all translations of T ðGÞ. We consider the subgroup T g of all translations with an even coefﬁcient u and arbitrary integers v and w as well as the coset decomposition ðG : T g Þ. Let t a be the translation with the translation vector a. There are four cosets: T g , T g t a , T g 10 and T g ðt a 10 Þ. The union T g [ ðT g t a Þ forms the translationengleiche maximal subgroup T ðGÞ of index 2. The union T g [ ðT g 10 Þ forms an isomorphic maximal subgroup of index 2, as does the union T g [ ðT g ðt a 10 ÞÞ. There are thus two maximal isomorphic subgroups of index 2 which are obtained by doubling the a lattice parameter. There are altogether 14 isomorphic sub-

(1) T ðHÞ H N G ðHÞ G N E ðGÞ < E; (1a) H N G ðHÞ N E ðHÞ < E; (1b) N E ðHÞ N A ðHÞ < A; (2) T ðHÞ T ðGÞ < T < E; (3) T ðGÞ G N E ðGÞ N A ðGÞ < A. The subgroup H may be a translationengleiche or a klassengleiche or a general subgroup of G. In any case, the normalizer

19

1. SPACE GROUPS AND THEIR SUBGROUPS N G ðHÞ determines the length of the conjugacy class of H < G, but it is not feasible to list for each group–subgroup pair H < G its normalizer N G ðHÞ. Indeed, it is only necessary to list for any space group H its normalizer N E ðHÞ in the Euclidean group E of all isometries, as is done in IT A, Section 15.2.1. From such a list the normalizers for the group–subgroup pairs can be obtained easily, because for any chain of space groups H < G < E, the relations H NG ðHÞ G and H NG ðHÞ NE ðHÞ hold. The normalizer NG ðHÞ consists consequently of all those isometries of NE ðHÞ that are also elements of G, i.e. that belong to the intersection NE ðHÞ \ G, cf. the examples of Section 1.2.7.11 The isomorphism type of the Euclidean normalizer N E ðHÞ may depend on the lattice parameters of the space group (specialized Euclidean normalizer). For example, if the lattice of the space group P1 of a triclinic crystal is accidentally monoclinic at a certain temperature and pressure or for a certain composition in a continuous solid-solution series, then the Euclidean normalizer of this space group belongs to the spacegroup types P2=m or C2=m, otherwise it belongs to P1. Such a specialized Euclidean normalizer (here P2=m or C2=m) may be distinguished from the typical Euclidean normalizer (here P1), for which the lattice of H is not more symmetric than is required by the symmetry of H. The specialized Euclidean normalizers were ﬁrst listed in the 5th edition of IT A (2005), Section 15.2.1.

references. Domains are also considered in Section 1.6.6 of this volume. In this section, non-ferroelastic phase transitions are treated without any special assumption as well as ferroelastic phase transitions under the simplifying parent clamping approximation, abbreviated PCA, introduced by Janovec et al. (1989), see also IT D, Section 3.4.2.5. A transition is non-ferroelastic if the strain tensors (metric tensors) of the low-symmetry phase B have the same independent components as the strain tensor of the phase A.12 There are thus no spontaneous strain components which distort the lattices of the domains. In a ferroelastic phase transition the strain tensors of phase B have more independent components than the strain tensor of phase A. The additional strain components cause lattice strain. By the PCA the lattice parameters of phase B at the transition are adapted to those of phase A, i.e. to the lattice symmetry of phase A. Therefore, under the PCA the ferroelastic phases display the same behaviour as the non-ferroelastic phases. If in this section ferroelastic phase transitions are considered, the PCA is assumed to be applied. Under a non-ferroelastic phase transition or under the assumption of the PCA, the translations of the constituents of the phase B are translations of phase A and the space groups Hj of B are subgroups of the space group G of A, Hj < G. Under this supposition the domain structure formed may exhibit different chiralities and/or polarities of its domains with different spatial orientations of their symmetry elements. Nevertheless, each domain has the same speciﬁc energy and the lattice of each domain is part of the lattice of the parent structure A with space group G. The description of domain structures by their crystal structures is called the microscopic description, IT D, Section 3.4.2.1. In the continuum description, the crystals are treated as anisotropic continua, IT D, Section 3.4.2.1. The role of the space groups is then taken over by the point groups of the domains. The continuum description is used when one is essentially interested in the macroscopic physical properties of the domain structure. Different kinds of nomenclature are used in the discussion of domain structures. The basic concepts of domain and domain state are established in IT D, Section 3.4.2.1; the terms symmetry state and directional state are newly introduced here in the context of domain structures. All these concepts classify the domains and will be deﬁned in the next section and applied in different examples of phase transitions in Section 1.2.7.3.

1.2.7. Application to domain structures 1.2.7.1. Introductory remarks In this section, the basic group-theoretical aspects of this chapter are exempliﬁed using the topic of domain structures (transformation twins). Domain structures result from a displacive or order–disorder phase transition. A homogeneous single crystal phase A (parent or prototypic phase) is transformed to a crystalline phase B (daughter phase, distorted phase). In most cases phase B is inhomogeneous, consisting of homogeneous regions which are called domains. Deﬁnition 1.2.7.1.1. A connected homogeneous part of a domain structure or of a twinned crystal of phase B is called a domain. Each domain is a single crystal. The part of space that is occupied & by a domain is the region of that domain. The space groups Hj of phase B are conjugate subgroups of the space group G of phase A, Hj < G. The number of domains is not limited; they differ in their locations in space, in their orientations, in their sizes, in their shapes and in their space groups Hj < G which, however, all belong to the same space-group type H. The boundaries between the domains, called the domain walls, are assumed to be (inﬁnitely) thin. A deeper discussion of domain structures or transformation twins and their properties needs a much more detailed treatment, as is given in Volume D of International Tables for Crystallography (2003) (abbreviated as IT D) Part 3, by Janovec, Hahn & Klapper (Chapter 3.2), by Hahn & Klapper (Chapter 3.3) and by Janovec & Prˇı´vratska´ (Chapter 3.4) with more than 400

1.2.7.2. Domain states, symmetry states and directional states In order to describe what happens in a phase transition of the kind considered, a few notions are useful. If the domains of phase B have been formed from a single crystal of phase A, then they belong to a ﬁnite (small) number of domain states Bj with space groups Hj. The domain states have well deﬁned relations to the original crystal of phase A and its space group G. In order to describe these relations, the notion of crystal pattern is used. Any perfect (ideal) crystal is a ﬁnite block of the corresponding inﬁnite arrangement, the symmetry of which is a space group and thus contains translations. Here, this

11 For maximal subgroups, a calculation of the conjugacy classes is not necessary because these are indicated in the subgroup tables of Part 2 of this volume by braces to the left of the data sets for the low-index subgroups and by text for the series of isomorphic subgroups. For non-maximal subgroups, the conjugacy relations are not indicated but can be calculated in the way described here. They are also available online on the Bilbao Crystallographic Server, http://www. cryst.ehu.es/, under the program Subgroupgraph.

12

A phase transition is non-ferroelastic if the space groups G of the highsymmetry phase A and Hj of the low-symmetry phase B belong to the same crystal family, of which there are six: triclinic, monoclinic, orthorhombic, tetragonal, trigonal–hexagonal and cubic. In ferroelastic phase transitions the space groups G of A and Hj of B belong to different crystal families. Only then can spontaneous strain components occur. They are avoided by the assumption of the PCA.

20

1.2. GENERAL INTRODUCTION TO THE SUBGROUPS OF SPACE GROUPS (inﬁnite) periodic object is called a crystal pattern, cf. Section 1.2.2.1. Each domain belongs to a crystal pattern.

Domains of the same domain state always belong to the same symmetry state. Domains of different domain states may or may not belong to the same symmetry state. The number NS of symmetry states is determined by the normalizer N G ðH1 Þ of H1 in G. Let N G ðH1 Þ be the normalizer of the space group H1 in the space group G. Then G N G ðH1 Þ . H1 with the indices jG : H1 j ¼ ½ i and jG : N G ðH1 Þj ¼ ½ iS with iS i. By Lemma 1.2.7.2.2 the number N ¼ jG : H1 j of domain states is determined. For the number NS of symmetry states the following lemma holds:

Deﬁnition 1.2.7.2.1. The set of all domains which belong to the same crystal pattern forms a domain state Bj . The domains of one domain state occupy different regions of space and are part of the same crystal pattern: to each domain state Bj belongs a crystal & pattern. From the viewpoint of symmetry, different domains of the same domain state cannot be distinguished. The following considerations concern domain states and thus indirectly their constituents, the domains. The number of domain states that may be observed in the distorted phase B is limited and determined by the space groups G and Hj . The number of domains that belong to the same domain state is not limited. The diversity of the domains and their shapes is due to mechanical stresses, defects, electrical charges and nucleation phenomena, which strongly inﬂuence the kinetics of the phase transition; see, for example, Fig. 3.4.1.1 in IT D. A trivial domain structure is formed when phase B consists of one domain only, i.e. when it forms a single-domain structure. This is possible, for example, in nanocrystals (Chen et al., 1997) and in particular under an external electric ﬁeld or under external stress, and is stable under zero ﬁeld and zero stress (detwinning). For a phase transition of the type considered, the different domain states have the same a priori probability of appearing in the distorted phase. In reality not all of them may be observed and/or their relative frequencies and sizes may be rather different. In order to calculate the number of possible domain states, the space group G will be decomposed into cosets relative to its subgroup H1, which is the space group of the domain state B1,

Lemma 1.2.7.2.4. The number NS of symmetry states for the transition A!B with space groups G and Hj is jG : N G ðH1 Þj ¼ ½ iS ¼ NS N. To each symmetry state there belong ds ¼ N : NS domain states, i.e. ds NS ¼ N, cf. IT D, & Section 3.4.2.4.1. In a group–subgroup relation such as that between G and Hj , Hj has in general lost translations as well as non-translational symmetry operations. Thus, for the translation subgroups (the lattices) T G T Hj holds, and for the point groups P G P Hj holds. Then a uniquely determined space group Mj exists, called Hermann’s group, such that Mj has the translations of G and the point group of Hj, IT D, Example 3.2.3.32. The group Mj can thus be characterized as that translationengleiche subgroup of G which is simultaneously a klassengleiche supergroup of Hj . For general subgroups G > Mj > Hj , for translationengleiche subgroups Hj ¼ Mj < G and for klassengleiche subgroups Hj < Mj ¼ G holds. For the corresponding indices one ﬁnds by coset decomposition the following lemma, see IT D, equation (3.2.3.91). Lemma 1.2.7.2.5. The index i of Hj in G can be factorized into the point-group part iP and the lattice part iL : jG : Hj j ¼ i ¼ & jG : Mj j jMj : Hj j ¼ iP iL .

G ¼ H1 [ . . . [ g j H1 [ . . . [ g i H1 : The elements g 1 ¼ e ; . . . ; g j ; . . . g i of G are the coset representatives. The elements hr of H1 map the domain state B1 onto itself, hr B1 ¼ B1 , h r 2 H1 . The elements of the other cosets map B1 onto the other domain states, for example, g j B1 ¼ Bj .13 It follows:

In the continuum description the lattices are ignored and only the relation between the point groups is considered. Because Mj is a translationengleiche subgroup of G, the index of Mj in G is the same as the index iP of the point group of Hj in the point group of G. In many cases one is only interested in the orientation of the domain states of a domain structure in the space. These orientations are determined by the crystal patterns and may be taken also from the values of the components of the property tensors of the domain states. A classiﬁcation of the domain states according to their orientations into classes which shall be called directional states may be achieved in the following way.

Lemma 1.2.7.2.2. The number N of possible domain states of phase B is equal to the index i of H1 in G, N ¼ jG : H1 j ¼ ½ i . & Having determined the number of domain states, we turn to their space groups, i.e. to the space groups of their crystal patterns. From B1 ¼ hr B1 for any hr 2 H1, written as H1 B1 ¼ B1, and 1 Bj ¼ g j B1 or B1 ¼ g 1 j Bj , we conclude that g j Bj ¼ H1 B1 ¼ 1 1 H1 g j Bj or Bj ¼ g j H1 g j Bj . This means: the space group g j H1 g 1 j leaves the domain state Bj invariant and is a subgroup of G which is conjugate to H1 for any value of j. For the classiﬁcation of the domain states according to their space groups we now introduce the term symmetry state.

Deﬁnition 1.2.7.2.6. A directional state is a set of all domain states & that are parallel to each other. Parallel domain states (crystal patterns) have space groups with the same point group and the same lattice; they are distinguished only by the locations of the conventional origins. If B1 and Bj ¼ g j B1 are parallel, then the coset representative g j is a translation and Hj and H1 are klassengleiche subgroups of M1. Both subgroups have also the same translations because a translation g j commutes with all translations and thus leaves T ðH1 Þ invariant, see Examples 1.2.7.3.2 and 1.2.7.3.3. To determine the number of directional states, we consider the coset decomposition of G relative to Hermann’s group M1. Each coset of the decomposition G : M1 maps the domain state B1

Deﬁnition 1.2.7.2.3. A symmetry state is a set of all domain states & (crystal patterns) the space groups of which are identical. 13 The multiplication hr B1 etc. is the performance of a symmetry operation hr on the crystal pattern of the domain state B1 . Usually, the matrix of hr is applied to the coordinates of the atoms of the crystal pattern of B1. A symmetry operation of B1 maps the crystal pattern onto itself; another motion maps it onto another crystal pattern.

21

1. SPACE GROUPS AND THEIR SUBGROUPS onto a domain state Bj with another directional state which results in the following lemma:

Example 1.2.7.3.2 Let G ¼ Fm3m, No. 225, with lattice parameter a and H1 ¼ Pm3m, No. 221, with the same lattice parameter a. The relation H1 < G is klassengleich of index 4 and is found between the disordered and ordered modiﬁcations of the alloy AuCu3. In the disordered state, one Au and three Cu atoms occupy the positions of a cubic F lattice at random; in the ordered compound the Au atoms occupy the positions of a cubic P lattice whereas the Cu atoms occupy the centres of all faces of this cube. According to IT A, Table 15.2.1.4, the Euclidean normalizer of H1 is N E ðH1 Þ ¼ Im3m with lattice parameter a. The additional I-centring translations of N E ðH1 Þ are not translations of G and thus N G ðH1 Þ ¼ H1. There are four domain states, each one with its own distinct space group Hj ; j ¼ 1; . . . ; 4, and symmetry state. The shifts of the conventional origins of Bj relative to the origin of A are 0; 0; 0; 1 1 1 1 1 1 2 ; 2 ; 0; 2 ; 0; 2; and 0; 2 ; 2. These shifts do not show up in the macroscopic properties of the domains but in the mismatch at the boundaries (antiphase boundaries) where different domains (antiphase domains) meet. This may be observed, for example, by high-resolution transmission electron microscopy (HRTEM), IT D, Section 3.3.10.6.

Lemma 1.2.7.2.7. The number of directional states in the transition A!B with space groups G and H1 is jG : M1 j, i.e. the index & ½ iP of M1 in G. The number of directional states does not depend on the type of description, whether microscopic or continuum, and is thus the same for both. Therefore, the microscopic description of the directional state may form a bridge between both kinds of description. 1.2.7.3. Examples The terms just deﬁned shall be explained in a few examples. In Example 1.2.7.3.1 a translationengleiche transition is considered; i.e. H is a translationengleiche subgroup of G. Because Mj ¼ Hj, the relation between G and Hj is reﬂected by the relation between the space groups G and Mj and the results of the microscopic and continuum description correspond to each other. Example 1.2.7.3.1 Perovskite BaTiO3 exhibits a ferroelastic and ferroelectric phase transition from phase A with the cubic space group G ¼ Pm3m, No. 221, to phase B with tetragonal space groups of type H ¼ P4mm, No. 99. Several aspects of this transition are discussed in IT D. Let B1 be one of the domain states of phase B. Because the index jG : H1 j ¼ 6, there are six domain states, forming three pairs of domain states which point with their tetragonal c axes along the cubic x, y and z axes of G. Each pair consists of two antiparallel domain states of opposite polarization (ferroelectric domains), related by, for example, the symmetry plane perpendicular to the symmetry axes 4. The six domain states also form six directional states because Hj ¼ Mj (antiparallel domain states belong to different directional states). To ﬁnd the space groups of the domain states, the normalizers have to be determined. For the perovskite transition, the normalizer N G ðH1 Þ can be obtained from the Euclidean normalizer N E of P4mm in Table 15.2.1.4 of IT A, which is listed as P1 4=mmm. This Euclidean normalizer has continuous translations along the z direction (indicated by the P1 lattice part of the HM symbol) and is thus not a space group. However, all additional translations of P1 4=mmm are not elements of the space group G, and NG ðH1 Þ = ðNE ðH1 Þ \ GÞ = ðP1 4=mmm \ Pm3mÞ = P4=mmm is a subgroup of Pm3m with index 3 and with the lattice of Pm3m due to the PCA. There are thus three symmetry states, i.e. three subgroups of the type P4mm with their fourfold axes directed along the z, x and y directions of the cubic space group G. Two domain states (with opposite polar axes) belong to each of the three subgroups Hj of type P4mm.

In the next example there are two domain states and both belong to the same space group, i.e. to the same symmetry state. Example 1.2.7.3.3 There is an order–disorder transition of the alloy -brass, CuZn. In the disordered state the Cu and Zn atoms occupy at random the positions of a cubic I lattice with space group G ¼ Im3m, No. 229. In the ordered state, both kinds of atoms form a cubic primitive lattice P each, and one kind of atom occupies the centres of the cubes of the other, as in the CsCl crystal structure. Its space group is H ¼ Pm3m, No. 221, which is a subgroup of index [2] of G with the same cubic lattice parameter a. There are two domain states with their crystal structures shifted relative to each other by 12 ða þ b þ cÞ. The space group H is a normal subgroup of G and both domain states belong to the same symmetry state. Up to now, only examples with translationengleiche or klassengleiche transitions have been considered. Now we turn to the domain structure of a general transition, where H is a general subgroup of G. General subgroups are never maximal subgroups and are thus not listed in this volume, but have to be derived from the maximal subgroups of each single step of the group–subgroup chain between G and H. In the following Example 1.2.7.3.4, the chain has two steps. Example 1.2.7.3.4 -Gadolinium molybdate, Gd2(MoO4)3, is ferroelectric and ferroelastic. It was treated as an example from different points of view in IT D by Tole´dano (Section 3.1.2.5.2) and by Janovec & Prˇ´ıvratska´ (Example 3.4.2.6). The high-temperature phase A has space group G ¼ P421 m, No. 113, and basis vectors a, b and c. A phase transition to a low-temperature phase B occurs with space-group type H ¼ Pba2, No. 32, basis vectors a0 ¼ a b, b0 ¼ a þ b and c0 ¼ c. In addition to the reduction of the pointgroup symmetry the primitive unit cell is doubled. The PCA will be applied because the transition from the tetragonal to the orthorhombic crystal family would without the PCA allow a0 6¼ b0 for the lattice parameters. The index of H in G is ½ i ¼ jP421 m : Pba2j ¼ 4 such that there are four domain states. These relations are displayed in Fig. 1.2.7.1.

In Example 1.2.7.3.1, a phase transition was discussed which involves only translationengleiche group–subgroup relations and, hence, only directional relations between the domain states occur. Each domain state forms its own directional state. The following two examples treat klassengleiche transitions, i.e. H is a klassengleiche subgroup of G. Then Mj ¼ G and there is only one directional state: a translational domain structure, also called translation twin, is formed. The domain states of a translational domain structure differ in the origins of their space groups because of the loss of translations of the parent phase in the phase transition.

22

1.2. GENERAL INTRODUCTION TO THE SUBGROUPS OF SPACE GROUPS In this volume, only subgroups of ﬁnite index i are listed. However, the index i is not restricted, i.e. there is no number I with the property i < I for any i. Subgroups H < G with inﬁnite index are considered in International Tables for Crystallography, Volume E (2002). Lemma 1.2.8.1.2. Hermann’s theorem. For any group–subgroup chain G > H between space groups there exists a uniquely deﬁned space group M with G M H, where M is a translationengleiche subgroup of G and H is a klassengleiche subgroup & of M. The decisive point is that any group–subgroup chain between space groups can be split into a translationengleiche subgroup chain between the space groups G and M and a klassengleiche subgroup chain between the space groups M and H. It may happen that either G ¼ M or H ¼ M holds. In particular, one of these equations must hold if H < G is a maximal subgroup of G.

Fig. 1.2.7.1. Group–subgroup relations between the high (HT) and low temperature (LT) forms of gadolinium molybdate (Ba¨rnighausen tree as explained in Section 1.6.3).

Lemma 1.2.8.1.3. (Corollary to Hermann’s theorem.) A maximal subgroup of a space group is either a translationengleiche subgroup or a klassengleiche subgroup, never a general subgroup. &

To calculate the number of space groups Pba2, i.e. the number of symmetry states, one determines the normalizer of Pba2 in P421 m. From IT A, Table 15.2.1.3, one ﬁnds N E ðPba2Þ ¼ P1 4=mmm for the Euclidean normalizer of Pba2 under the PCA, which includes the condition a0 ¼ b0 . P1 4=mmm is a supergroup of P421 m. Thus, N G ðPba2Þ ¼ ðN E ðPba2Þ \ GÞ ¼ G and Pba2 is a normal subgroup. Therefore, under the PCA all four domain states belong to one space group Pba2, i.e. there is one symmetry state. Indeed, the tables of Volume A1 list only one subgroup of type Cmm2 under P421 m and only one subgroup Pba2 under Cmm2 with ½i ¼ 2 in both cases. Hermann’s group M, G > M > H is of space-group type Cmm2 with the point group mm2 of H and with the lattice of G. Because jG : Mj ¼ 2, there are two directional states which belong to the same space group. The directional state of B2 is obtained from that of B1 by, for example, the (lost) 4 operation of P421 m: the basis vector a of B2 is parallel to b of B1, b of B2 is parallel to a of B1 and c of B2 is antiparallel to c of B1. The other factor of 2 is caused by the loss of the centring translations of Cmm2, iT ¼ jM : Hj ¼ 2. Therefore, the domain state B3 is parallel to B1 but its origin is shifted with respect to that of B1 by a lost translation, for example, by t ð1; 0; 0Þ of G, which is t ð12 ; 12 ; 0Þ in the basis a0, b0 of Pba2. The same holds for the domain state B4 relative to B2.

The following lemma holds for space groups but not for arbitrary groups of inﬁnite order. Lemma 1.2.8.1.4. For any space group, the number of subgroups & with a given ﬁnite index i is ﬁnite. This number of subgroups can be further speciﬁed, see Chapter 1.4. Although for each index i the number of subgroups is ﬁnite, the number of all subgroups with ﬁnite index is inﬁnite because there is no upper limit for the number i. 1.2.8.2. Lemmata on maximal subgroups Even the set of all maximal subgroups of ﬁnite index is not ﬁnite, as can be seen from the following lemma. Lemma 1.2.8.2.1. The index i of a maximal subgroup of a space group is always of the form pn , where p is a prime number and n = & 1 or 2 for plane groups and n = 1, 2 or 3 for space groups. This lemma means that a subgroup of, say, index 6 cannot be maximal. Moreover, because of the inﬁnite number of primes, the set of all maximal subgroups of a given space group cannot be ﬁnite. An index of p2 ; p > 2; occurs for isomorphic subgroups of tetragonal space groups when the basis vectors are enlarged to pa, pb; for trigonal and hexagonal space groups, the enlargement pa, pb is allowed for p = 2 and for all or part of the primes p > 3. An index of p3 occurs for and only for isomorphic subgroups of cubic space groups with cell enlargements of pa, pb, pc (p > 2). There are even stronger restrictions for maximal nonisomorphic subgroups.

1.2.8. Lemmata on subgroups of space groups There are several lemmata on subgroups H < G of space groups G which may help in getting an insight into the laws governing group–subgroup relations of plane and space groups. They were also used for the derivation and the checking of the tables of Part 2. These lemmata are proved or at least stated and explained in Chapter 1.4. They are repeated here as statements, separated from their mathematical background, and are formulated for the three-dimensional space groups. They are valid by analogy for the (two-dimensional) plane groups.

Lemma 1.2.8.2.2. The index of a maximal non-isomorphic subgroup of a plane group is 2 or 3; for a space group the index is & 2, 3 or 4. This lemma can be speciﬁed further: Lemma 1.2.8.2.3. The index of a maximal non-isomorphic subgroup H is always 2 for oblique, rectangular and square plane groups and for triclinic, monoclinic, orthorhombic and tetragonal space groups G. The index is 2 or 3 for hexagonal plane groups and for trigonal and hexagonal space groups G. The index is 2, 3 & or 4 for cubic space groups G.

1.2.8.1. General lemmata Lemma 1.2.8.1.1. A subgroup H of a space group G is a space & group again, if and only if the index i ¼ jG : Hj is ﬁnite.

23

1. SPACE GROUPS AND THEIR SUBGROUPS There are also lemmata for the number of subgroups of a certain index. The most important are:

References Brown, H., Bu¨low, R., Neubu¨ser, J., Wondratschek, H. & Zassenhaus, H. (1978). Crystallographic Groups of Four-dimensional Space. New York: John Wiley & Sons. Chen, Ch.-Ch., Herhold, A. B., Johnson, C. S. & Alivisatos, A. P. (1997). Size dependence of structural metastability in semiconductor nanocrystals. Science, 276, 398–401. Flack, H. D., Wondratschek, H., Hahn, Th. & Abrahams, S. C. (2000). Symmetry elements in space groups and point groups. Addenda to two IUCr reports on the nomenclature of symmetry. Acta Cryst. A56, 96–98. Hahn, Th. & Wondratschek, H. (1994). Symmetry of Crystals. Introduction to International Tables for Crystallography, Vol. A. Soﬁa: Heron Press. Hermann, C. (1929). Zur systematischen Strukturtheorie. IV. Untergruppen. Z. Kristallogr. 69, 533–555. International Tables for Crystallography (2002). Vol. E, Subperiodic Groups, edited by V. Kopsky´ & D. B. Litvin. Dordrecht: Kluwer Academic Publishers. International Tables for Crystallography (2003). Vol. D, Physical Properties of Crystals, edited by A. Authier. Dordrecht: Kluwer Academic Publishers. International Tables for Crystallography (2005). Vol. A, Space-Group Symmetry, edited by Th. Hahn, 5th ed. Heidelberg: Springer. Janovec, V., Schranz, W., Warhanek, H. & Zikmund, Z. (1989). Symmetry analysis of domain structure in KSCN crystals. Ferroelectrics, 98, 171– 189. Ledermann, W. (1976). Introduction to Group Theory. London: Longman. (German: Einfu¨hrung in die Gruppentheorie, Braunschweig: Vieweg, 1977.) Ledermann, W. & Weir, A. J. (1996). Introduction to Group Theory, 2nd ed. Harlow: Addison-Wesley Longman. Plesken, W. & Schulz, T. (2000). Counting crystallographic groups in low dimensions. Exp. Math. 9, 407–411. Souvignier, B. (2003). Enantiomorphism of crystallographic groups in higher dimensions with results in dimensions up to 6. Acta Cryst. A59, 210–220.

Lemma 1.2.8.2.4. The number of subgroups of index 2 is 2N 1 with 0 N 6 for space groups and 0 N 4 for plane groups. The number of translationengleiche subgroups of index 2 is 2M 1 with 0 M 3 for space groups and 0 M 2 for & plane groups. Examples are: N ¼ 0 : 20 1 ¼ 0 subgroups of index 2 for p3, No. 13, and F23, No. 196; N ¼ 1 : 21 1 ¼ 1 subgroup of index 2 for p3m1, No. 14, and P3, No. 143; . . .; N ¼ 4 : 24 1 ¼ 15 subgroups of index 2 for p2mm, No. 6, and P1, No. 2; N ¼ 6 : 26 1 ¼ 63 subgroups of index 2 for Pmmm, No. 47. Lemma 1.2.8.2.5. The number of isomorphic subgroups of each space group is inﬁnite and this applies even to the number of & maximal isomorphic subgroups. Nevertheless, their listing is possible in the form of inﬁnite series. The series are speciﬁed by parameters. Lemma 1.2.8.2.6. For each space group, each maximal isomorphic subgroup H can be listed as a member of one of at most four series of maximal isomorphic subgroups. Each member is speci& ﬁed by a set of parameters. The series of maximal isomorphic subgroups are discussed in Section 2.1.5.

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International Tables for Crystallography (2011). Vol. A1, Chapter 1.3, pp. 25–26.

1.3. Computer checking of the subgroup data By Franz Ga¨hler

1.3.1. Introduction

Having expressed the element p as a product of generators, one simply replaces the factors in this product by their representative pre-images under H, which are known by construction, to obtain the representative pre-image g of p. To compute a generating set of the translation subgroup T G, one ﬁrst computes a set of deﬁning relations of the point group P for the given generators. For a ﬁnite group this is a standard task in GAP. These deﬁning relations are a set of inequivalent ways to express the identity element of P by the generators of P. Replacing the factors in such a deﬁning relation by their representative pre-images in G yields a pre-image of the identity of P, i.e. an element of T . The translations so obtained generate, together with the pure translation generators of G, the entire translation subgroup T G. We now have all the necessary information to test whether a given augmented matrix A is an element of a space group G. One ﬁrst tests whether the linear part M of A is an element of the point group P of G. If M 62 P, A is not an element of G. Otherwise, some pre-image G of M in G is computed. A is then an element of G if and only if A and G differ in their translation part by an element of the translation group T . With this membership test, we can determine whether two space groups are equal, or whether one is a subgroup of the other: a space group H is a subgroup of a space group G if all the generators of H are elements of G. Two space groups are equal if either of them is a subgroup of index 1 of the other.

Most of the data in Part 2 of this volume have been checked by a computer-algebra program. This program is built upon the package Cryst (Eick et al., 2001), formerly known as CrystGAP (Eick et al., 1997), which in turn is an extension of the computeralgebra program GAP (The GAP Group, 2002). GAP is a program system for computations involving general algebraic structures, in particular groups. Since space groups are inﬁnite groups, the generic algorithms for ﬁnite groups often cannot be used for space groups. The Cryst package provides the special methods and algorithms needed for working with space groups, thereby extending the ﬁeld of applicability of GAP. The algorithms used in Cryst have been described by Eick et al. (1997).

1.3.2. Basic capabilities of the Cryst package Before we describe in more detail the checks that have been performed, we brieﬂy summarize the capabilities of the Cryst package and how space groups are handled by Cryst. In Cryst, a space group is represented as a ﬁxed group of augmented matrices Wi, cf. Section 1.2.2.4. The group multiplication therefore coincides with matrix multiplication. The space group is deﬁned by an arbitrary (ﬁnite) set of generating elements. The space group consists of the set of all augmented matrices that can be obtained as products of the generating matrices. It is not important which generating elements are chosen. Different sets of generators can deﬁne the same space group. The representation of a space group as a group of augmented matrices implicitly requires the choice of an origin and a vector basis of Euclidean space. The Cryst package does not require any particular choice of origin or vector basis. In particular, it is not necessary to work with a lattice basis; in fact, any basis can be used. This has the advantage that one can continue to use the basis chosen for the parent group when passing to a subgroup. Space groups are conjugate by some afﬁne coordinate transformation if they differ only in their settings, i.e. by the choice of their origins and/or vector bases. Such an afﬁne coordinate transformation is also represented by an augmented matrix. Given a coordinate transformation V and a space group G, the transformed space group G 0 is simply generated by the transformed generators of G. If a space group G is given by a set of generators, Cryst ﬁrst needs to compute the point group P of G, and the translation subgroup T , represented by a (canonical) basis of the translation lattice (the basis of the translation lattice chosen by Cryst is canonical in the sense that the same basis is always chosen for a given T ). The point group P is simply generated by the linear parts of the generators of G. The mapping H, which sends each element of G to its linear part, is in fact a group homomorphism H: G!P. The point group P is ﬁnite and each point-group element p can easily be expressed as a product of generators of P, which are images of generators of G under H. This provides a way to compute, for any p 2 P, some representative pre-image g of p under H, i.e. some element g 2 G which is mapped onto p by H. Copyright © 2011 International Union of Crystallography

1.3.3. Computing maximal subgroups The Cryst package has built-in facilities for computing the maximal subgroups of a given index for any space group G. More precisely, given a prime number p, Cryst can compute conjugacyclass representatives of those maximal subgroups of G whose index in G is a power of p. The algorithms used for this task are described in Eick et al. (1997). Essentially, one determines the maximal subgroups of the (ﬁnite) factor group G=T p, where T p is the subgroup of those translations of G which are a p-fold multiple of an element of the full translation group T of G. After the maximal subgroups are obtained, the translations in T p are added back to the subgroups. From a representative H of a conjugacy class of subgroups, the list of all subgroups in the same conjugacy class is obtained by repeatedly conjugating H (and the subgroups obtained from it by conjugation) with generators of G, until no new conjugate subgroups are obtained. This is also the way of determining whether two subgroups are conjugate: one enumerates the groups in the conjugacy class of one of them, and checks whether the other is among them. The index of a subgroup H of G is easily computed as the product of the index of the point group of H in the point group of G and the index of the translation group of H in the translation group of G. For maximal subgroups, only one of these factors is different from 1. For klassengleiche subgroups, the two point groups are equal, whereas for translationengleiche subgroups the two translation subgroups are equal. Therefore, a maximal

25

1. SPACE GROUPS AND THEIR SUBGROUPS subgroup is easily identiﬁed either as a klassengleiche or a translationengleiche subgroup.

in the same order. Here, the preferred setting is the setting of the parent group, where applicable, and otherwise the preferred setting of the space-group type of the subgroup, if there is more than one setting in the tables. In a second step, a complete set of maximal subgroups of low index (2, 3 or 4) is computed afresh with the routines from the Cryst package. These subgroups are then divided into conjugacy classes, and classiﬁed as klassengleiche or translationengleiche subgroups. This list is then compared with the tabulated list of maximal subgroups. It was veriﬁed that each maximal subgroup of a given index was listed exactly once, that the classiﬁcation into conjugacy classes of subgroups was correct and that the subgroups were correctly identiﬁed as klassengleiche or as translationengleiche subgroups. All the tests described above concern the maximal subgroups of low index. Unfortunately, similar automatic tests could not be performed on the series of isomorphic subgroups. The subgroups in these series contain variable parameters, and Cryst can only deal with ﬁxed, concrete space groups without free parameters.

1.3.4. Description of the checks In order to avoid new errors being introduced in the typesetting process after the data have been checked and corrected, we carried out the computer checks directly on the LATEX 2" sources used for the production of this volume. The tables have been typeset with specially designed LATEX 2" macros, the primary purpose of which was to guarantee a homogeneous layout throughout the book. As a side effect, it was relatively easy to write a GAP program which parses the LATEX 2" sources, recognizes the macros, extracts the data (the arguments of the macros), brings the data into a form suitable for further processing with routines from the Cryst package and ﬁnally performs the various checks. In this way, the checks could be done fully automatically, and could be repeated after every modiﬁcation of the tabulated data. In the following, we describe the checks that have been carried out. We ﬁrst applied a number of tests individually to each of the tabulated maximal subgroups of low index: whether it is a subgroup, whether the index given is correct, whether the listed coordinate transformation maps the subgroup to the preferred setting of the given space-group type (and thus, whether the space-group type of the subgroup is correct) and whether the listed coordinate transformation maps the given generators of the subgroup to the standard generators of its space-group type,

References Eick, B., Ga¨hler, F. & Nickel, W. (1997). Computing maximal subgroups and Wyckoff positions of space groups. Acta Cryst. A53, 467–474. Eick, B., Ga¨hler, F. & Nickel, W. (2001). The ‘Cryst’ Package. Version 4.1. http://www.itap.physik.uni-stuttgart.de/~gaehler/gap/packages.html. The GAP Group (2002). GAP – Groups, Algorithms and Programming. Version 4.3. http://www.gap-system.org.

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International Tables for Crystallography (2011). Vol. A1, Chapter 1.4, pp. 27–40.

1.4. The mathematical background of the subgroup tables By Gabriele Nebe

1.4.1. Introduction

up to dimension 6 are used, for example, for the description of quasicrystals and incommensurate phases. For example, the more than 29 000 000 crystallographic groups up to dimension 6 can be parameterized, constructed and identiﬁed using the computer package [CARAT]: Crystallographic AlgoRithms And Tables, available from http://wwwb.math.rwth-aachen.de/carat/ index.html (for a description, see Opgenorth et al., 1998). As well as the points in point space, there are other objects, called ‘vectors’. The vector that connects the point P to the point ! Q is usually denoted by P Q. Vectors are usually visualized by arrows, where parallel arrows of the same length represent the same vector. Whereas the sum of two points P and Q is not deﬁned, one can add vectors. The sum v þ w of two vectors v and w is simply the sum of the two arrows. Similarly, multiplication of a vector v by a real number can be deﬁned. All the points in point space are equally good, but among the vectors one can be distinguished, the null vector o. It is characterized by the property that v þ o ¼ v for all vectors v. Although the notion of a vector seems to be more complicated than that of a point, we introduce vector spaces before giving a mathematical model for the point space, the so-called afﬁne space, which can be viewed as a certain subset of a higherdimensional vector space, where the addition of a point and a vector makes sense.

This chapter gives a brief introduction to the mathematics involved in the determination of the subgroups of space groups. To achieve this we have to detach ourselves from the geometric point of view in crystallography and introduce more abstract algebraic structures, such as coordinates, which are well known in crystallography and permit the formalization of symmetry operations, and also the abstract notion of a group, which allows us to apply general theorems to the concrete situation of (threedimensional) space groups. This algebraic point of view has the following advantages: (1) Geometric problems can be treated by algebraic calculations. These calculations can be dealt with by well established procedures. In particular, the use of computers and advanced programs enables one to solve even difﬁcult problems in a comparatively short time. (2) The mappings form groups in the mathematical sense of the word. This means that the very powerful methods of group theory may be applied successfully. (3) The procedures for the solution may be developed to a great extent independently of the dimension of the space. In Section 1.4.2, a basis is laid down which gives the reader an understanding of the algebraic point of view of the crystal space (or point space) and special mappings of this space onto itself. The set of these mappings is an example of a group. For a closer connection to crystallography, the reader may consult Section 8.1.1 of International Tables for Crystallography Volume A (2005) (abbreviated as IT A) or the book by Hahn & Wondratschek (1994). Section 1.4.3 gives an introduction to abstract groups and states the important theorems of group theory that will be applied in Section 1.4.4 to the most important groups in crystallography, the space groups. In particular, Section 1.4.4 treats maximal subgroups of space groups which have a special structure by the theorem of Hermann. In Section 1.4.5, we come back to abstract group theory stating general facts about maximal subgroups of groups. These general theorems allow us to calculate the possible indices of maximal subgroups of threedimensional space groups in Section 1.4.6. The next section, Section 1.4.7, deals with the very subtle question of when these maximal subgroups of a space group are isomorphic to this space group. In Section 1.4.8 minimal supergroups of space groups are treated brieﬂy.

1.4.2.2. Vector spaces We shall now exploit the advantage of being independent of the dimensionality. The following deﬁnitions are independent of the dimension by replacing the speciﬁc dimensions 2 for the plane and 3 for the space by an unspeciﬁed integer number n > 0. Although we cannot visualize four- or higher-dimensional objects, we can describe them in such a way that we are able to calculate with such objects and derive their properties. Algebraically, an n-dimensional (real) vector v can be represented by a column of n real numbers. The n-dimensional real vector space Vn is then 0 1 x1 B C Vn ¼ fx ¼ @ ... A j x1 ; . . . ; xn 2 Rg: xn (In crystallography n is normally 3.) The entries x1 ; . . . ; xn are called the coefﬁcients of the vector x. On Vn one can naturally deﬁne an addition, where the coefﬁcients of the sum of two vectors are the corresponding sums of the coefﬁcients of the vectors. To multiply a vector by a real number, one just multiplies all its coefﬁcients by this number. The null vector 0 1 0 . o ¼ @ .. A 2 Vn 0

1.4.2. The afﬁne space 1.4.2.1. Motivation The aim of this section is to give a mathematical model for the ‘point space’ (also known in crystallography as ‘direct space’ or ‘crystal space’) which contains the positions of atoms in crystals (the so-called ‘points’). This allows us in particular to describe the symmetry groups of crystals and to develop a formalism for calculating with these groups which has the advantage that it works in arbitrary dimensions. Such higher-dimensional spaces Copyright © 2011 International Union of Crystallography

can be distinguished, since v þ o ¼ v for all v 2 Vn . The identiﬁcation of a concrete vector space V with the vector space Vn can be done by choosing a basis of V. A basis of V is any

27

1. SPACE GROUPS AND THEIR SUBGROUPS tuple of n vectors B :¼ ða1 ; . . . ; an Þ such that every vector of V can be written uniquely as a linear combination of the basis vectors: V ¼ fx ¼ x1 a1 þ . . . þ xn an j x1 ; . . . ; xn 2 Rg. Whereas a vector space has many different bases, the number n of vectors of a basis is uniquely determined and is called the dimension of V. The isomorphism (see Section 1.4.3.4 for a deﬁnition of isomorphism) ’B between V and Vn maps the vector x ¼ x1 a1 þ . . . þ xn an 2 V to its coefﬁcient column 0 1 x1 B C x ¼ @ ... A 2 Vn

and cutting the c axis at x3 ¼ 1 in crystallographic notation. The points of A2 have coordinates 0 1 x1 B x2 C @ A; 1 which are the coefﬁcients of the vector from the origin to the point. This observation is generalized by the following: Deﬁnition 1.4.2.3.1.

xn

1 x1 B .. C B . C C An :¼ fB B xn C j xi 2 Rg A @ 1 0

with respect to the chosen basis B. The mapping ’B respects addition of vectors and multiplication of vectors with real numbers. Moreover, ’B is a bijective mapping, which means that for any coefﬁcient column x 2 Vn there is a unique vector x 2 V with ’B ðxÞ ¼ x. Therefore one can perform all calculations using the coefﬁcient columns. An important concept in mathematics is the automorphism group of an object. In general, if one has an object (here the vector space V) together with a structure (here the addition of vectors and the multiplication of vectors with real numbers), its automorphism group is the set of all one-to-one mappings of the object onto itself that preserve the structure. A bijective mapping ’ : V ! V of the vector space V into itself satisfying ’ðv þ wÞ ¼ ’ðvÞ þ ’ðwÞ for all v; w 2 V and ’ðxvÞ ¼ x’ðvÞ for all real numbers x 2 R and all vectors v 2 V is called a linear mapping and the set of all these linear mappings is the linear group of V. To know the image of x ¼ x1 a1 þ . . . þ xn an under a linear mapping ’ it sufﬁces to know the images of the basis vectors a1 ; . . . ; an under ’, since ’ðxÞ ¼ x1 ’ða1 Þ þ . . . þ xn ’ðan Þ. Writing the coefﬁcient columns of the images Pn of the basis vectors as columns of a matrix A [i.e. ’ðai Þ ¼ j¼1 aj Aji , i ¼ 1; . . . ; n], then the coefﬁcient column of ’ðxÞ with respect to the chosen basis B is just Ax. Note that the matrix of a linear mapping depends on the basis B of V. The matrix that corresponds to the composition of two linear mappings is the product of the two corresponding matrices. We have thus seen that the linear group of a vector space V of dimension n is isomorphic to the group of all invertible ðn nÞ matrices via the isomorphism ’B that associates to a linear mapping its corresponding matrix (with respect to the basis B). This means that one can perform all calculations with linear mappings using matrix calculations. In crystallography, the translation-vector space has an additional structure: one can measure lengths and angles between vectors. An n-dimensional real vector space with such an additional structure is called a Euclidean vector space, En . Its automorphism group is the set of all (bijective) linear mappings of En onto itself that preserve lengths and angles and is called the orthogonal group On of En. If one chooses the basis B ¼ ðe1 ; . . . ; en Þ to be the unit vectors (which are orthogonal vectors of length 1), then the isomorphism ’B above maps the orthogonal group On onto the set of all ðn nÞ matrices A with AT A ¼ I, the ðn nÞ unit matrix. T denotes the transposition operator, which maps columns to rows and rows to columns.

is an n-dimensional afﬁne space.

&

If 0

1 0 x1 B .. C B B . C B C B and Q ¼ B P¼B C B x @ n A @ 1

1 y1 .. C . C C 2 An ; yn C A 1

! then the vector P Q is deﬁned as the difference 1 0 y1 x1 .. C B C B . C B QP¼B C x y n A @ n 0 ! (computed in the vector space Vnþ1 ). The set of all P Q with P; Q 2 An forms an n-dimensional vector space which is called the underlying vector space ðAn Þ. Omitting the last coefﬁcient, we can identify 1 0 x1 B .. C B . C C ðAn Þ ¼ fB B xn Cj x1 ; . . . ; xn 2 Rg A @ 0 with Vn . As the coordinates already indicate, the sets An as well as ðAn Þ can be viewed as subsets of Vnþ1. Computed in Vnþ1 , the sum of two elements in ðAn Þ is again in ðAn Þ, since the last coefﬁcient of the sum is 0 þ 0 ¼ 0 and the sum of a point P 2 An and a vector v 2 Vn is again a point in An (since the last coordinate is 1 þ 0 ¼ 1), but the sum of two points does not make sense.

1.4.2.4. The afﬁne group The afﬁne group of geometry is the set of all mappings of the point space which fulﬁl the conditions (1) parallel straight lines are mapped onto parallel straight lines; (2) collinear points are mapped onto collinear points and the ratio of distances between them remains constant.

1.4.2.3. The afﬁne space In this section we build up a model for the ‘point space’. Let us ﬁrst assume n ¼ 2. Then the afﬁne space A2 may be imagined as an inﬁnite sheet of paper parallel, let us say, to the (a, b) plane

28

1.4. THE MATHEMATICAL BACKGROUND OF THE SUBGROUP TABLES In the mathematical model, the afﬁne group is the automorphism group of the afﬁne space and can be viewed as the set of all linear mappings of Vnþ1 that preserve An.

1.4.3. Groups 1.4.3.1. Groups The afﬁne group is only one example of the more general concept of a group. The following axiomatic deﬁnition sometimes makes it easier to examine general properties of groups.

Deﬁnition 1.4.2.4.1. The afﬁne group An is the subset of the set of all linear mappings ’ : Vnþ1 ! Vnþ1 with ’ðAn Þ ¼ An . The & elements of An are called afﬁne mappings.

Deﬁnition 1.4.3.1.1. A group ðG; Þ is a set G with a mapping G G ! G; ðg ; hÞ 7 !g h, called the composition law or multiplication of G, satisfying the following three axioms:

Since ’ is linear, it holds that ! ! ’ðP QÞ ¼ ’ðQ PÞ ¼ ’ðQÞ ’ðPÞ ¼ ’ðPÞ’ðQÞ:

(i) ðg hÞ k ¼ g ðh k Þ for all g ; h; k 2 G (associative law). (ii) There is an element e 2 G called the unit element of G with e g ¼ g e ¼ g for all g 2 G. (iii) For all g 2 G, there is an element g 1 2 G, called the inverse & of g, with g g 1 ¼ g 1 g ¼ e.

Hence an afﬁne mapping also maps ðAn Þ into itself. Since the ﬁrst n basis vectors of the chosen basis lie in ðAn Þ and the last one in An , it is clear that with respect to this basis the afﬁne mappings correspond to matrices of the form

W¼

Normally the symbol is omitted, hence the product g h is just written as g h and the set G is called a group. One should note that in particular property (i), the associative law, of a group is something very natural if one thinks of group elements as mappings. Clearly the composition of mappings is associative. In general, one can think of groups as groups of mappings as explained in Section 1.4.3.2. A subset of elements of a group G which themselves form a group is called a subgroup:

! w : 1

W oT

The linear mapping induced by ’ on ðAn Þ which is represented by the matrix W will be referred to as the linear part ’ of ’. The image ’ðPÞ of a point P with coordinates

x¼

x

!

1

Deﬁnition 1.4.3.1.2. A non-empty subset ; 6¼ U G is called a subgroup of G (abbreviated as U G) if g h1 2 U for all & g ; h 2 U.

2 An

The afﬁne group is an example of a group where is given by the composition of mappings. The unit element e 2 An is the identity mapping given by the matrix

can easily be found as

Wx ¼

Wx þ w

!

1

: I¼

If one has a way to measure lengths and angles (i.e. a Euclidean metric) on the underlying vector space ðAn Þ, one can compute the distance between P and Q 2 An as the length of the vector ! P Q and the angle determined by P, Q and R 2 An with vertex Q ! ! is obtained from cosðP; Q; RÞ ¼ cosðQ P; Q RÞ. In this case, An is the Euclidean point space, En . An afﬁne mapping of the Euclidean point space is called an isometry if its linear part is an orthogonal mapping of the Euclidean vector space ðAn Þ. The set of all isometries in An is called the Euclidean group and denoted by E n. Hence E n is the set of all distance-preserving mappings of En onto itself. The isometries are the afﬁne mappings with matrices of the form

W¼

W oT

w 1

T¼

! o ; 1

which also represents the translation by the vector o. The composition of two afﬁne mappings is again an afﬁne mapping and the inverse of an afﬁne mapping W has matrix W

1

¼

W 1 oT

! W 1 w : 1

Since the inverse of an isometry and the composition of two isometries are again isometries, the set of isometries E n is a subgroup of the afﬁne group An. The translation subgroup T n is a subgroup of E n. Any vector space Vn is a group with the usual vector addition as composition law. Therefore ðAn Þ is also a group.

!

Remarks (i) For every group G, the set feg consisting only of the unit element of G is a subgroup of G called the trivial subgroup I ¼ feg. (ii) If U is a subgroup of V and V is a subgroup of the group G, then U is a subgroup of G. (iii) If U and V are subgroups of the group G, then the intersection U \ V is also a subgroup of G. (iv) If S G is a subset of the group G, then the smallest subgroup of G containing S is denoted by

where the linear part W belongs to the orthogonal group of ðAn Þ. Special isometries are the translations, the isometries where the linear part is I, with matrix I oT

I oT

! w : 1

The group of all translations in E n is the translation subgroup of E n and is denoted by T n. Note that composition of two translations means addition of the translation vectors and T n is isomorphic to the translation vector space ðEn Þ.

hSi :¼

29

\ fU G j S Ug

1. SPACE GROUPS AND THEIR SUBGROUPS For a space group G and a point P in the point space, the stabilizer StabG ðPÞ is called the site-symmetry group of P with respect to G. The kernel K of the action of G on M is the intersection of the stabilizers of all elements in M,

and is called the subgroup generated by S. The elements of S are called the generators of this group. It is convenient not to list all the elements of a group G but just to give generators of G (this also applies to ﬁnite groups). Example 1.4.3.1.3 A well known group is the addition group of integers Z ¼ f0; 1; 1; 2; 2; . . .g where is normally denoted by + and the unit element e 2 Z is 0. The group Z is generated by f1g. Other generating sets are for example f1g or f2; 3g. Taking two integers a; b 2 Z which are divisible by some ﬁxed integer p 2 Z, then the sum a þ b and the negatives a and b are again divisible by p. Hence the set pZ of all integers divisible by p is a subgroup of Z. It is generated by fpg.

K :¼ fg 2 G j g m ¼ m for all m 2 Mg: M is called a faithful G-set and the action of G on M is also called & faithful if the kernel of the action is trivial, K ¼ feg. Note that any space group R acts faithfully on the point space. Remarks (i) If m1 ; m2 2 M, then their orbits are either equal or disjoint. For if there is an element g 1 m1 ¼ g 2 m2 , then by the 1 axioms of G-sets m1 ¼ em1 ¼ ðg 1 1 g 1 Þ m1 ¼ g 1 ðg 1 m1 Þ 1 1 ¼ g 1 ðg 2 m2 Þ ¼ ðg 1 g 2 Þ m2 , hence every element g m1 in the orbit of m1 is of the form g ðg 1 1 g 2 m2 Þ ¼ ðgg 1 g Þ m and therefore lies in the orbit of m 2 2 2. Hence 1 the set of orbits gives a partition of M into disjoint sets. If M is a ﬁnite set, then its order is the sum of the lengths of the different orbits. (ii) StabG ðmÞ is a subgroup of G, since for g 1 ; g 2 2 StabG ðmÞ, the 1 product ðg 1 g 1 2 Þ m ¼ g 1 ðg 2 mÞ ¼ g 1 m ¼ m. (iii) If m1 ¼ g m2, then StabG ðm1 Þ ¼ g StabG ðm2 Þg 1 ¼ fghg 1 j h 2 StabG ðm2 Þg.

Deﬁnition 1.4.3.1.4. The order jGj of a group G is the number of & elements in the set G. Most of the groups G in crystallography, for example Z, Vn , An , have inﬁnite order. Groups that are generated by one element are called cyclic. The cyclic group of order n is called Cycn . (We prefer to use three letters to denote the mathematical names of frequently occurring groups, since the more common symbol Cn could possibly cause confusion with the Schoenﬂies symbol Cn .) The group Vn is not generated by a ﬁnite set. These two groups Z and Vn have the property that for all elements g and h in the group it holds that g h ¼ h g . Hence these two groups are Abelian in the sense of the following:

Example 1.4.3.2.4 (Example 1.4.3.2.2 continued) (a) An is a transitive An -set. This is a mathematical expression of the fact that in point space no point is distinguished. (b) The An -set ðAn Þ decomposes into two orbits fog and fv 2 ðAn Þ j v 6¼ og. The kernel of the action of An on ðAn Þ is the translation subgroup T n. (c) ðAn Þ acts transitively on An . The kernel of the action only consists of the zero vector o.

Deﬁnition 1.4.3.1.5. The group ðG; Þ is called Abelian if g h ¼ & h g for all g ; h 2 G.

1.4.3.2. Actions of groups on sets The afﬁne group An is deﬁned via its action on the afﬁne space An . In general, the greatest signiﬁcance of groups is that they act on sets.

We now introduce some terminology for groups which is nicely formulated using G-sets.

Deﬁnition 1.4.3.2.1. Let G be a group. A non-empty set M is called a (left) G-set if there is a mapping G M ! M satisfying the following conditions:

Deﬁnition 1.4.3.2.5. The orbit of g 2 G under the action of the subgroup U G is the right coset U g ¼ fug j u 2 Ug (cf. IT A, Section 8.1.5). Analogously one deﬁnes a left coset as

(i) ðghÞ m ¼ g ðh mÞ for all g ; h 2 G and m 2 M. (ii) e m ¼ m for all m 2 M. If M is a G-set, one also says that G acts on M.

g U ¼ fgu j u 2 Ug &

and denotes the set of left cosets by G=U. If the number of left cosets (which is always equal to the number of right cosets) of U in G is ﬁnite, then this number is called the index ½G : U of U in G. If this number is inﬁnite, one & says that the index of U in G is inﬁnite.

Example 1.4.3.2.2 (a) The afﬁne space An is a G-set for the afﬁne group G ¼ An. (b) ðAn Þ is an An -set, where An acts via the linear parts. (c) ðAn Þ is also a group and acts on An by translations v P :¼ P þ v for v 2 ðAn Þ, P 2 An . (d) If U G is a subgroup of the group G, then G is a U-set where : U G ! G is the usual composition law. In particular, each group G is a G-set and hence every group G can be viewed as a group of mappings from G onto G.

Example 1.4.3.2.6 An is a coset of Vn in Vnþ1 , namely 0

1 0 B ... C B C An ¼ B C þ Vn : @0A

Deﬁnition 1.4.3.2.3. Let G be a group and M a G-set. If m 2 M, then the set G m :¼ fg mjg 2 Gg is called the orbit of m under G. The G-set M is called transitive if M ¼ G m for any m 2 M consists of a single orbit under G. If m 2 M then the stabilizer of m in G is StabG ðmÞ :¼ fg 2 G j g m ¼ mg.

1 If one thinks of A2 as an inﬁnite sheet of paper and puts uncountably many such sheets of paper (one for each real number) one onto the other, one gets the whole 3-space V3.

30

1.4. THE MATHEMATICAL BACKGROUND OF THE SUBGROUP TABLES Remark. The set of left cosets G=U is a left G-set with the operation g ðm UÞ :¼ ðgmÞ U for all g ; m 2 G. The kernel of the action is the intersection of all subgroups of GT that are conjugate to U and is called the core of U: core ðUÞ :¼ g 2G g U g 1 .

(iii) The group G itself and also the trivial subgroup feg G are always normal subgroups of G. Normal subgroups play an important role in the investigation of groups. If N / G is a normal subgroup, then the left coset g N equals the right coset N g for all g 2 G, because g N ¼ g ðg 1 N g Þ ¼ N g . The most important property of normal subgroups is that the set of left cosets of N in G forms a group, called the factor group G=N , as follows: The set of all products of elements of two left cosets of N again forms a left coset of N . Let g ; h 2 G. Then

We now assume that jGj is ﬁnite. Let U G be a subgroup of G. Then the set G is partitioned into left cosets of U, G ¼ g1 U [ . . . [ gi U, where i ¼ ½G : U is the index of U in G. Since the orders of the left cosets of U are all equal to the order of U, one gets Theorem 1.4.3.2.7. (Theorem of Lagrange.) Let U be a subgroup of the ﬁnite group G. Then

g N h N ¼ g ðh N h 1 Þh N ¼ gh N N ¼ gh N :

This deﬁnes a natural product on the set of left cosets of N in G which turns this set into a group. The unit element is eN . Hence the philosophy of normal subgroups is that they cut the group into pieces, where the two pieces G=N and N are again groups.

jGj ¼ jUj½G : U: In particular, the order of any subgroup of G and also the index of & any subgroup of G are divisors of the group order jGj. The G-set G=U is only a special case of a G-set. It is a transitive G-set. If M ¼ G m is a transitive G-set, then the mapping M ! G=StabG ðmÞ, g m 7 ! g StabG ðmÞ is a bijection (in fact an isomorphism of G-sets in the sense of Deﬁnition 1.4.3.4.1 below). Therefore, the number of elements of M, which is the length of the orbit of m under G, equals the index of the stabilizer of m in G, whence one gets the following generalization of the theorem of Lagrange:

Example 1.4.3.2.10. The group Z is Abelian. For any number p 2 Z, the set pZ is a subgroup of Z. Hence pZ is a normal subgroup of Z. The factor group Z=pZ inherits the multiplication from the multiplication in Z, since apZ pZ for all a 2 Z. If p is a prime number, then all elements 6¼ 0 þ pZ in Z=pZ have a multiplicative inverse, and therefore Z=pZ is a ﬁeld, the ﬁeld with p elements.

Theorem 1.4.3.2.8. Let G be a ﬁnite group and M be a G-set. Then

Proposition 1.4.3.2.11. Let N / G be a normal subgroup of the group G and U G. Then the set

jGj ¼ jG mjjStabG ðmÞj for all m 2 M.

N U ¼ UN :¼ fun j u 2 U; n 2 N g &

is a subgroup of G.

The point group G acts on the ﬁnite set M of ideal crystal faces. Then the length of the orbit (the number of equivalent crystal faces) times the order of the face-symmetry group is the order of the point group. Up to now, we have only considered the action of G upon G via multiplication. There is another natural action of G on itself via conjugation: G G ! G deﬁned by g m :¼ g mg 1 for all group elements g and elements m in the G-set G. The stabilizer of m is called the centralizer of m in G,

&

Proof. Let u 1 n1 , u 2 n2 2 UN . Then u 1 n 1 ðu 2 n 2 Þ

1

1 1 1 1 ¼ u 1 n1 n1 2 u 2 ¼ u 1 u 2 ðu 2 n 1 n 2 u 2 Þ ¼ un 2 UN ;

where u :¼ u 1 u 1 2 2 U, since U is a subgroup of G, and n :¼ 1 u 2 n 1 n 1 u 2 N , since N is a normal subgroup of G. QED 2 2

1.4.3.3. The Sylow theorems A nice application of the notion of G-sets are the three theorems of Sylow. By Theorem 1.4.3.2.7, the order of any subgroup U of a group G divides the order of G. But conversely, given a divisor d of jGj, one cannot predict the existence of a subgroup U of G with jUj ¼ d. If d ¼ p is a prime power that divides jGj, then the following theorem says that such a subgroup exists.

StabG ðmÞ ¼ CG ðmÞ ¼ fg 2 G j gmg 1 ¼ mg: If M G is a set of group elements, then the centralizer of M is the intersection of the centralizers of the elements in M: CG ðMÞ ¼ fg 2 G j gmg 1 ¼ m for all m 2 Mg:

Theorem 1.4.3.3.1. (Sylow) Let G be a ﬁnite group and p be a prime such that p divides the order of G. Then G possesses m subgroups of order p , where m > 0 satisﬁes m 1 ðmod pÞ. &

Deﬁnition 1.4.3.2.9. G also acts on the set U of all subgroups of G by conjugation, g U :¼ g U g 1 . The stabilizer of an element U 2 U is called the normalizer of U and denoted by N G ðUÞ. U is called a normal subgroup of G (denoted as U / G) if & N G ðUÞ ¼ G.

In particular this theorem implies that for every prime power that divides the order of the ﬁnite group G, the group G has a subgroup whose order is this prime power. This is not true for composite numbers. For instance, the alternating group Alt4 of order 12 (Hermann–Mauguin notation 23) has no subgroup of order 6. This group has three subgroups of order 2, a unique subgroup of order 4 = 22 and four subgroups of order 3. The group Cyc2 Sym3 (Hermann–Mauguin notation 3m) also has order 12 but seven subgroups of order 2, three subgroups of order 4 and a unique subgroup of order 3.

Remarks (i) Let U G. Then the index of the normalizer of U in G is the number of subgroups of G that are conjugate to U. Since U always normalizes itself [hence U is a subgroup of N G ðUÞ], the index of the normalizer divides the index of U. (ii) If G is Abelian, then the conjugation action of G is trivial, hence each subgroup of G is a normal subgroup.

31

1. SPACE GROUPS AND THEIR SUBGROUPS

Theorem 1.4.3.3.2. (Sylow) If jGj ¼ p s for some prime p not dividing s, then all subgroups of order p of G are conjugate in G. Such a subgroup U G of order jUj ¼ p is called a Sylow & p-subgroup.

ðwÞ ¼

I oT

w 1

!

is a homomorphism of the group ðAn Þ into An. The kernel of this homomorphism is fog and the image of the mapping is the translation subgroup T n of An. Hence the groups ðAn Þ and T n are isomorphic. The afﬁne group acts (as group of group automorphisms) on the normal subgroup T n / An via conjugation: g t :¼ gtg 1 . We have seen already in Example 1.4.3.2.4 (b) that it also acts (as a group of linear mappings) on ðAn Þ. The mapping is an isomorphism of An -sets.

Combining these two theorems with Theorem 1.4.3.2.8, one gets Sylow’s third theorem: Theorem 1.4.3.3.3. (Sylow) The number of Sylow p-subgroups of & G is 1 ðmod pÞ and divides the order of G. Proofs of the three theorems above can be found in Ledermann (1976), pp. 158–164, or in Ledermann & Weir (1996), pp. 155–161.

1.4.3.5. Isomorphism theorems

1.4.3.4. Isomorphisms

[cf. Ledermann (1976), pp. 68–73, or Ledermann & Weir (1996), pp. 85–92.]

If one wants to compare objects such as groups or G-sets, to be able to say when they should be considered equal, the concept of isomorphisms should be used:

Remark. If ’ is a homomorphism G ! H and N / H is a normal subgroup of H, then the pre-image ’1 ðN Þ :¼ fg 2 G j ’ðg Þ 2 N g is a normal subgroup of G. In particular, it holds that kerð’Þ / G. Hence the factor group G= kerð’Þ is a well deﬁned group. The following theorem says that this group is isomorphic to the image ’ðGÞ H of ’:

Deﬁnition 1.4.3.4.1. Let G and H be groups and M and N be G-sets. (a) A homomorphism ’ : G ! H is a mapping of the set G into the set H respecting the composition law i.e. ’ðghÞ ¼ ’ðg Þ’ðhÞ for all g ; h 2 G. If ’ is bijective, it is called an isomorphism and one says G is isomorphic to H (G ﬃ H). If e 2 H is the unit element of H, then the set of all preimages of e is called the kernel of ’: kerð’Þ :¼ fg 2 G j ’ðg Þ ¼ eg. An isomorphism ’ : G ! G is called an automorphism of G. (b) M and N are called isomorphic G-sets if there is a bijection ’ : M ! N with g ’ðmÞ ¼ ’ðg mÞ for all g 2 G; m 2 M. &

Theorem 1.4.3.5.1. (First isomorphism theorem.) Let ’ : G ! H be a homomorphism of groups. Then ’ : G= kerð’Þ ! ’ðGÞ H deﬁned by ’ðg kerð’ÞÞ ¼ ’ðg Þ is an isomorphism between the factor group G= kerð’Þ and the image group of ’, which is a & subgroup of H. For instance, if R is a space group and ’ is mapping any element ! W w W¼ 2R oT 1

Example 1.4.3.4.2. In Example 1.4.3.1.3, the group homomorphism Z ! pZ deﬁned by 1 7 !p is a group isomorphism (from the group Z onto its subgroup pZ). Example 1.4.3.4.3. For any group element g 2 G, conjugation by g deﬁnes an automorphism of G. In particular, if U is a subgroup of G, then U and its conjugate subgroup g U g 1 are isomorphic.

to its linear part W, then the kernel of ’ is the translation group T ðRÞ of R and the image is the point group R of R. The theorem says that the point group R is isomorphic to the factor group R=T ðRÞ.

Philosophy: If G and H are isomorphic groups, then all grouptheoretical properties of G and H are the same. The calculations in G can be translated by the isomorphism to calculations in H. Sometimes it is easier to calculate in one group than in the other and translate the result back via the inverse of the isomorphism. For example, the isomorphism between ðAn Þ and Vn in Section 1.4.2 is an isomorphism of groups. It even respects scalar multiplication with real numbers, so in fact it is an isomorphism of vector spaces. While the composition of translations is more concrete and easier to imagine, the calculation of the resulting vector is much easier in Vn . The concept of isomorphism says that you can translate to the more convenient group for your calculations and translate back afterwards without losing anything. Note that a homomorphism is injective, hence an isomorphism onto its image, if and only if its kernel is trivial ð¼ fegÞ.

Theorem 1.4.3.5.2. (Third isomorphism theorem.) Let N / G be a normal subgroup of the group G and U G be an arbitrary subgroup of G. Then U \ N / U is a normal subgroup of U and U=ðU \ N Þ ﬃ N U=N : (For the deﬁnition of the group N U see Proposition 1.4.3.2.11.) &

Deﬁnition 1.4.3.5.3. A subgroup U H is a characteristic sub& group U char H if ’ðUÞ ¼ U for all automorphisms ’ of H. Remarks (a) If H is a ﬁnite Abelian group and P is a Sylow p-subgroup of H, then P char H, because P is the only subgroup of H of order jPj. (b) If H is any group and U char H, then U / H is also a normal subgroup of H: for h 2 H deﬁne the mapping h : H ! H, x 7 !hxh 1 . Then h is an automorphism of H and h ðUÞ ¼ h U h1 ¼ U since U is characteristic in H.

Example 1.4.3.4.4 The mapping from the space ðAn Þ of translation vectors into the afﬁne group An deﬁned by

32

1.4. THE MATHEMATICAL BACKGROUND OF THE SUBGROUP TABLES (c) If U char N / H, then U / H, since the conjugation with any element of H induces an automorphism of N .

denote this group by D2d (another Schoenﬂies symbol for this group is S4v and its Hermann–Mauguin symbol is 42m). The group U above is contained in D2d . It is its own centralizer in Td : U ¼ CTd ðUÞ. Therefore, the factor group Td =U acts faithfully (and transitively) on the set fr 1 ; r 2 ; r 3 g. The stabilizer of r 1 is the subgroup D2d constructed above. Using this, one easily sees that Td =U ﬃ Sym3 . Another normal subgroup in Td is the set of all rotations in Td . This group contains the normal subgroup U above of index 3 and is of index 2 in Td (and hence has order 12). It is isomorphic to Alt4 , the alternating group of degree 4, and has Schoenﬂies symbol T and Hermann–Mauguin symbol 23.

1.4.3.6. An example Let us consider the tetrahedral group, Schoenﬂies symbol Td , which is deﬁned as the symmetry group of a tetrahedron. It permutes the four apices P1 ; P2 ; P3 ; P4 of the tetrahedron and hence every element of Td deﬁnes a bijection of V :¼ fP1 ; P2 ; P3 ; P4 g onto itself. The only element that ﬁxes all the apices is e. Therefore the set V is a faithful Td -set. Let us calculate the order of jTd j. Since there are elements in Td that map the ﬁrst apex P1 onto each one of the other apices, V is a transitive Td -set. Let S :¼ StabTd ðP1 Þ be the stabilizer of P1. By Theorem 1.4.3.2.8, jTd j ¼ jVjjSj ¼ 4jSj. The group S is generated by the threefold rotation r around the ‘diagonal’ of the tetrahedron through P1 and the reﬂection s at the symmetry plane of the tetrahedron which contains the edge ðP1 ; P2 Þ. In particular, S acts transitively on the set fP2 ; P3 ; P4 g. The stabilizer of P2 in S is the cyclic group hs i ﬃ Cyc2 generated by s . (The Schoenﬂies notation for hs i is Cs and the Hermann–Mauguin symbol is m.) Therefore jSj ¼ 3jhsij ¼ 6 and jTd j ¼ 24. In fact, we have seen that Td is isomorphic to the group of all bijections of V onto itself, which is the symmetric group Sym4 of degree 4 and the group S ﬃ Sym3 is the symmetric group on fP2 ; P3 ; P4 g. The Schoenﬂies notation for S is C3v and its Hermann–Mauguin symbol is 3m. In general, let n 2 N be a natural number. Then the group of all bijective mappings of the set f1; . . . ; ng onto itself is called the symmetric group of degree n and denoted by

1.4.4. Space groups 1.4.4.1. Deﬁnition of space groups In IT A (2005) Section 8.1.6 space groups are introduced as symmetry groups of crystal patterns. Deﬁnition 1.4.4.1.1 (a) Let Vn be the n-dimensional real vector space. A subset L Vn is called an (n-dimensional) lattice if there is a basis ðb1 ; . . . ; bn Þ of Vn such that L ¼ Zb1 þ . . . þ Zbn ¼ f

n P

ai bi j ai 2 Zg:

i¼1

(b) A crystal structure is a mapping f : En ! R of the Euclidean point space into the real numbers such that StabðAn Þ ðf Þ :¼ ft 2 ðAn Þ j f ðP þ tÞ ¼ f ðPÞ for all P 2 An g is an n-dimensional lattice in ðAn Þ. (c) The Euclidean group E n acts on the set of mappings En ! R via ðg f ÞðPÞ :¼ f ðg 1 PÞ for all P 2 En and for all g 2 E n and f : En ! R. A space group R is the stabilizer of a crystal structure f : En ! R; R ¼ StabE nðf Þ. (d) Let R E n be a space group. The translation subgroup T ðRÞ & of R is deﬁned as T ðRÞ :¼ R \ T n :

Symn :¼ ff : f1; . . . ; ng ! f1; . . . ; ng j f is bijectiveg: The alternating group is the normal subgroup Altn consisting of all even permutations of f1; . . . ; ng. Let us construct a normal subgroup of Td. The tetrahedral group contains three twofold rotations r 1 ; r 2 ; r 3 around the three axes of the tetrahedron through the midpoints of opposite edges. Since Td permutes these three axes and hence conjugates the three rotations into each other, the group

The deﬁnition introduced space groups in the way they occur in crystallography: The group of symmetries of an ideal crystal stabilizes the crystal structure. This deﬁnition is not very helpful in analysing the structure of space groups. If R is a space group, then the translation subgroup T :¼ T ðRÞ is a normal subgroup of R. It is even a characteristic subgroup of R, hence ﬁxed under every automorphism of R. By Deﬁnition 1.4.4.1.1, its image under the inverse 0 of the mapping in Example 1.4.3.4.4 deﬁned by ! v I 7!v 0 : T ! ðEn Þ; oT 1

U :¼ hr 1 ; r 2 ; r 3 i generated by these three rotations is a normal subgroup of Td. Since these three rotations commute with each other, the group U is Abelian. Now r 1 r 2 ¼ r 3 and hence U ¼ fe; r 1 ; r 2 ; r 3 g ﬃ D2 (in Schoenﬂies notation) ﬃ 222 (Hermann–Mauguin symbol) is of order 4. There are three normal subgroups of order 2 in U, namely hr i i for i ¼ 1; 2; 3. The factor group U=hr 1 i is again of order 2. Since all groups of order 2 are cyclic, hr 1 i ﬃ U=hr 1 i ﬃ Cyc2 . The set U is the set of all products of elements from the two normal subgroups hr 1 i and hr 2 i, hence U is isomorphic to the direct product Cyc2 Cyc2 in the sense of the following deﬁnition.

in ðAn Þ is an n-dimensional lattice LðRÞ. Since 0 is an isomorphism from T onto LðRÞ, the translation subgroup of R is isomorphic to the lattice LðRÞ. In particular, one has 0 ðt1 t2 Þ ¼ 0 ðt1 Þ þ 0 ðt2 Þ and the subgroup T p, formed by the pth powers of elements in T , is mapped onto pLðRÞ. Lattices are well understood. Although they are inﬁnite, they have a simple structure, so they can be examined algorithmically. Since they lie in a vector space, one can apply linear algebra to them. Now we want to look at how this lattice T ðRÞ ﬁts into the space group R. The afﬁne group An acts on T n by conjugation as well as on ðAn Þ via its linear part. Similarly the space group R acts on T ðRÞ by conjugation: For g 2 R and t 2 T , one gets

Deﬁnition 1.4.3.6.1. [cf. Ledermann (1976), Section 13, or Ledermann & Weir (1996), Section 2.7.] Let G and H be two groups. Then the direct product G H is the group G H ¼ fðg ; hÞ j g 2 G; h 2 Hg with multiplication ðg ; hÞðg 0 ; h0 Þ & :¼ ðgg 0 ; hh0 Þ. Let us return to the example above. The centralizer of one of the three rotations, say of r 1, is of index 3 in Td and hence a Sylow 2-subgroup of Td with order 8. Following Schoenﬂies, we will

33

1. SPACE GROUPS AND THEIR SUBGROUPS 0

1

0

ðgtg Þ ¼ g ðt Þ, where g is the linear part of g. Therefore, the kernel of this action is on the one hand the centralizer of T ðRÞ in R, on the other hand, since LðRÞ contains a basis of ðEn Þ, it is equal to the kernel of the mapping , which is R \ T n ¼ T ðRÞ, hence

The above picture looks as follows in the two cases:

CR ðT ðRÞÞ ¼ T ðRÞ: Let M be a t-subgroup of R. Then T ðRÞ M and M=T ðRÞ is a subgroup S of P ¼ R=T ðRÞ. On the other hand, any subgroup S of P deﬁnes a unique t-subgroup M of R with T ðRÞ M and M=T ðRÞ ¼ S, namely M ¼ fs 2 R j sT ðRÞ 2 Sg. Hence the t-subgroups of R are in bijection to the subgroups of P, which is a ﬁnite group according to the remarks below Deﬁnition 1.4.4.1.1. For future reference, we note this in the following corollary:

Hence only the linear part R ﬃ R=T ðRÞ of R acts faithfully on T ðRÞ by conjugation and linearly on LðRÞ. This factor group R=T ðRÞ is a ﬁnite group. Let us summarize this: Theorem 1.4.4.1.2. Let R be a space group. The translation subgroup T ðRÞ ¼ R \ T n is an Abelian normal subgroup of R which is its own centralizer, CR ðT ðRÞÞ ¼ T ðRÞ. The ﬁnite group R=T ðRÞ acts faithfully on T ðRÞ by conjugation. This action is similar to the action of the linear part R on the lattice & 0 ðT ðRÞÞ ¼ LðRÞ.

Corollary 1.4.4.2.4. The t-subgroups of the space group R are in & bijection with the subgroups of the ﬁnite group R=T ðRÞ. In the case n ¼ 3, which is the most important case in crystallography, the ﬁnite groups R=T ðRÞ are isomorphic to subgroups of either Cyc2 Sym4 (Hermann–Mauguin symbol m3m) or Cyc2 Cyc2 Sym3 (¼ 6=mmm). Here denotes the direct product (cf. Deﬁnition 1.4.3.6.1), Cyc2 the cyclic group of order 2, and Sym3 and Sym4 the symmetric groups of degree 3 or 4, respectively (cf. Section 1.4.3.6). Hence the maximal subgroups M of R that are t-subgroups can be read off from the subgroups of the two groups above. An algorithm for calculating the maximal t-subgroups of R which applies to all three-dimensional space groups is explained in Section 1.4.5. The more difﬁcult task is the determination of the maximal k-subgroups.

1.4.4.2. Maximal subgroups of space groups Deﬁnition 1.4.4.2.1. A subgroup M G of a group G is called maximal if M 6¼ G and for all subgroups U G with M U it & holds that either U ¼ M or U ¼ G. The translation subgroup T :¼ T ðRÞ of the space group R plays a very important role if one wants to analyse the space group R. Let U 6¼ R be a subgroup of R. Then U has either fewer translations (T ðUÞ < T ) or the order of the linear part of U, the index of T ðUÞ in U, gets smaller (jUj < jRj), or both happen. Deﬁnition 1.4.4.2.2. Let U be a subgroup of the space group R and T :¼ T ðRÞ.

Lemma 1.4.4.2.5. Let M be a maximal k-subgroup of the space group R. Then T ðMÞ ¼ T \ M / R is a normal subgroup of R. & Hence 0 ðT ðMÞÞ LðRÞ is an R-invariant lattice.

(t) U is called a translationengleiche or a t-subgroup if U \ T ¼T. (k) U is called a klassengleiche or a k-subgroup if U=U \ T & ﬃ R=T .

Proof. R ¼ T M, so every element g in R can be written as g ¼ tm where t 2 T and m 2 M. Therefore one obtains for t1 2 T \ M

Remark. The third isomorphism theorem, Theorem 1.4.3.5.2, implies that if U is a k-subgroup, then UT =T ﬃ U=U \ T ﬃ R=T . Hence U is a k-subgroup if and only if UT ¼ R.

g 1 t 1 g ¼ m1 t 1 t 1 tm ¼ m1 t 1 m;

Let M be a maximal subgroup of R. Then we have the following preliminary situation:

since T is Abelian. Since m 2 R and T is normal in R, one has m1 t 1 m 2 T . But m1 t 1 m is a product of elements in M and therefore lies in the subgroup M, hence m1 t 1 m 2 T \ M. QED The candidates for translation subgroups T ðMÞ of maximal k-subgroups M of R can be found by linear-algebra algorithms using the philosophy explained at the beginning of this section: R acts on T by conjugation and this action is isomorphic to the action of the linear part R ﬃ R=T of R on the lattice LðRÞ via the isomorphism 0 : T ! LðRÞ. Normal subgroups of R contained in T are mapped onto R-invariant sublattices of LðRÞ. An example for such a normal subgroup is the group T p formed by the pth powers of elements of T for any natural number, in particular for prime numbers p 2 N. One has 0 ðT p Þ ¼ pLðRÞ. If M is a maximal k-subgroup of R, then T ðMÞ is a normal subgroup of R that is maximal in T , which means that 0 ðT ðMÞÞ ¼ LðMÞ is a maximal R-invariant sublattice of LðRÞ. Hence it contains pLðRÞ for some prime number p. One may view T =T p ﬃ LðRÞ=pLðRÞ as a ﬁnite ðZ=pZÞR-module and ﬁnd all candidates for such normal subgroups as full pre-images of

Since T / R and M R, one has by Proposition 1.4.3.2.11 that MT R. Hence the maximality of M implies that MT ¼ M or MT ¼ R. If MT ¼ M then T M, hence M is a t-subgroup. If MT ¼ R, then by the third isomorphism theorem, Theorem 1.4.3.5.2, R=T ¼ MT =T ﬃ M=ðM \ T Þ ¼ M=T ðMÞ, hence M is a k-subgroup. This is given by the following theorem: Theorem 1.4.4.2.3. (Hermann) Let M R be a maximal subgroup of the space group R. Then M is either a k-subgroup or & a t-subgroup.

34

1.4. THE MATHEMATICAL BACKGROUND OF THE SUBGROUP TABLES M ! T ; m 7 !t m with this property deﬁnes some maximal subgroup M0 as above. Since M and T are ﬁnite, it is a ﬁnite problem to ﬁnd all such mappings. If there is no such complement M, this means that there is no (maximal) k-subgroup M of R with M \ T ¼ S.

maximal ðZ=pZÞR-submodules of LðRÞ=pLðRÞ. This gives an algorithm for calculating these normal subgroups, which is implemented in the package [CARAT]. The group G :¼ T =T p is an Abelian group, with the additional property that for all g 2 G one has g p ¼ e. Such a group is called an elementary Abelian p-group. From the reasoning above we ﬁnd the following lemma. Lemma 1.4.4.2.6. Let M be a maximal k-subgroup of the space group R. Then T =T ðMÞ is an elementary Abelian p-group for & some prime p. The order of T =T ðMÞ is pr with r n.

1.4.5. Maximal subgroups 1.4.5.1. Maximal subgroups and primitive G-sets To determine the maximal t-subgroups of a space group R, essentially one has to calculate the maximal subgroups of the ﬁnite group R=T ðRÞ. There are fast algorithms to calculate these maximal subgroups if this ﬁnite group is soluble (see Deﬁnition 1.4.5.2.1), which is the case for three-dimensional space groups. To explain this method and obtain theoretical consequences for the index of maximal subgroups in soluble space groups, we consider abstract groups again in this section. For an arbitrary group G, one has a fast method of checking whether a given subgroup U G of ﬁnite index ½G : U is maximal by inspection of the G-set G=U of left cosets of U in G. Assume that U M G and let M=U :¼ fm1 U; . . . ; mk Ug with mi 2 M, m1 ¼ e and G=M :¼ fg 1 M; . . . ; g l Mg with g i 2 G, g 1 ¼ e . Then the set G=U may be written as

Corollary 1.4.4.2.7. Maximal subgroups of space groups are again & space groups and of ﬁnite index in the supergroup. Hence the ﬁrst step is the determination of subgroups of R that are maximal in T and normal in R, and is solved by linearalgebra algorithms. These subgroups are the candidates for the translation subgroups T ðMÞ for maximal k-subgroups M. But even if one knows the isomorphism type of M=T ðMÞ, the group T ðMÞ does not in general determine M R. Given such a normal subgroup S / R that is contained in T , one now has to ﬁnd all maximal k-subgroups M R with S ¼ T \ M and T M ¼ R. It might happen that there is no such group M. This case does not occur if R is a symmorphic space group in the sense of the following deﬁnition:

G=U ¼

Deﬁnition 1.4.4.2.8. A space group R is called symmorphic if there is a subgroup P R such that P \ T ðRÞ ¼ I and PT ðRÞ ¼ R. The subgroup P is called a complement of the & translation subgroup T ðRÞ.

fg 1 m1 U; . . . ; g 2 m1 U; . . . ; .. .; ...; g l m1 U; . . . ;

g 1 mk U; g 2 mk U;

.. .;

:

g l mk Ug

Then G permutes the lines of the rectangle above: For all g 2 G and all j 2 f1; . . . ; lg, the left coset gg j M is equal to some g a M for an a 2 f1; . . . ; lg. Hence the jth line is mapped onto the set

Note that the group P in the deﬁnition is isomorphic to R=T ðRÞ and hence a ﬁnite group. If R is symmorphic and P R is a complement of T , then one may take M :¼ SP.

fgg j m1 U; . . . ; gg j mk Ug ¼ fg a m1 U; . . . ; g a mk Ug: Deﬁnition 1.4.5.1.1. Let G be a group and X a G-set. (i) A congruence fS1 ; . . . ; Sl g on X is a partition of X into l non-empty subsets X ¼ [_ i¼1 Si such that for all x1 ; x2 2 Si , g 2 G, gx1 2 Sj implies gx2 2 Sj . (ii) The congruences fXg and ffxg j x 2 Xg are called the trivial congruences. (iii) X is called a primitive G-set if G is transitive on X, jXj > 1 & and X has only the trivial congruences.

This shows the following: Lemma 1.4.4.2.9. Let R be a symmorphic space group with translation subgroup T and T 1 T an R-invariant subgroup of T (i.e. T 1 / R). Then there is at least one k-subgroup U R & with translation subgroup T 1.

Hence the considerations above have proven the following lemma. Lemma 1.4.5.1.2. Let M G be a subgroup of the group G. Then M is a maximal subgroup if and only if the G-set G=M is & primitive.

In any case, the maximal k-subgroups, M, of R satisfy MT ¼ R and M \ T ¼ S is a maximal R-invariant subgroup of T .

The advantage of this point of view is that the groups G having a faithful, primitive, ﬁnite G-set have a special structure. It will turn out that this structure is very similar to the structure of space groups. If X is a G-set and N / G is a normal subgroup of G, then G acts on the set of N -orbits on X, hence fN x j x 2 Xg is a congruence on X. If X is a primitive G-set, then this congruence is trivial, hence N x ¼ fxg or N x ¼ X for all x 2 X. This means that N either acts trivially or transitively on X.

To ﬁnd these maximal subgroups, M, one ﬁrst chooses such a subgroup S. It then sufﬁces to compute in the ﬁnite group R=S ¼: R. If there is a complement M of T ¼ T =S in R, then every element x 2 R may be written uniquely as x ¼ mt with 0 m 2 M, t 2 T . In particular, any other complement M of T in 0 R is of the form M ¼ fmt m j m 2 M; t m 2 T g. One computes 0 m1 t m1 m2 t m2 ¼ m1 m2 ðm1 2 t m1 m2 Þt m2 . Since M is a subgroup of R, it holds that t m1 m2 ¼ ðm1 2 t m1 m 2 Þt m2 . Moreover, every mapping

35

1. SPACE GROUPS AND THEIR SUBGROUPS One obtains the following:

1.4.5.2. Soluble groups Deﬁnition 1.4.5.2.1. Let G be a group. The derived series of G is the series ðG0 ; G1 ; . . .Þ deﬁned via G0 :¼ G, Gi :¼ hg 1 h1 gh j g ; h 2 Gi1 i. The group G1 is called the derived subgroup of G. The & group G is called soluble if Gn ¼ fe g for some n 2 N.

Theorem 1.4.5.1.3. [Theorem of Galois (ca 1830).] Let H be a ﬁnite group and let X be a faithful, primitive H-set. Assume that feg 6¼ N / H is an Abelian normal subgroup. Then (a) N is a minimal normal subgroup of H (i.e. for all N 1 / H, N 1 N , N 1 ¼ N or N 1 ¼ feg). (b) N is an elementary Abelian p-group for some prime p and jXj ¼ jN j is a prime power. (c) CH ðN Þ ¼ N and N is the unique minimal normal subgroup & of H.

Remarks (i) The Gi are characteristic subgroups of G. (ii) G is Abelian if and only if G1 ¼ feg. (iii) G1 is characterized as the smallest normal subgroup of G, such that G=G1 is Abelian, in the sense that every normal subgroup of G with an Abelian factor group contains G1. (iv) Subgroups and factor groups of soluble groups are soluble. (v) If N / G is a normal subgroup, then G is soluble if and only if G=N and N are both soluble.

Proof. Let feg 6¼ N / H be an Abelian normal subgroup. Then N acts faithfully and transitively on X. To establish a bijection between the sets N and X, choose x 2 X and deﬁne ’ : N ! X; n 7 !n x. Since N is transitive, ’ is surjective. To show the injectivity of ’, let n1 ; n2 2 N with ’ðn1 Þ ¼ ’ðn 2 Þ. Then 1 n 1 x ¼ n 2 x, hence n 1 1 n 2 x ¼ x. But then n 1 n 2 acts trivially on X, because if y 2 X then the transitivity of N implies that there 1 is an n 2 N with n x ¼ y. Then n1 1 n2 y ¼ n1 n2 n x ¼ 1 nn 1 n 2 x ¼ n x ¼ y, since N is Abelian. Since X is a faithful H-set, this implies n1 1 n 2 ¼ e and therefore n 1 ¼ n 2 . This proves jN j ¼ jXj. Since this equality holds for all nontrivial Abelian normal subgroups of H, statement (a) follows. If p is some prime dividing jN j, then the Sylow p-subgroup of N is normal in N , since N is Abelian. Therefore, it is also a characteristic subgroup of N and hence a normal subgroup in H (see the remarks below Deﬁnition 1.4.3.5.3). Since N is a minimal normal subgroup of H, this implies that N is equal to its Sylow p-subgroup. Therefore, the order of N is a prime power jN j ¼ pr for some prime p and p r 2 N. Similarly, the set N :¼ fnp j n 2 N g is a normal subgroup p of H properly contained in N . Therefore, N ¼ feg and N is elementary Abelian. This establishes (b). To see that (c) holds, let g 2 CH ðN Þ. Choose x 2 X. Then g x ¼ y 2 X. Since N acts transitively, there is an n 2 N such that n x ¼ y. Hence n1 g x ¼ x. As above, let z 2 X be any element of X. Then there is an element n1 2 N with z ¼ n1 x. Hence n1 g z ¼ n1 gn1 x ¼ n1 n1 g x ¼ n 1 x ¼ z. Since z was arbitrary and X is faithful, this implies that g ¼ n 2 N . Therefore, CH ðN Þ N . Since N is Abelian, one has N CH ðN Þ, hence N ¼ CH ðN Þ. To see that N is unique, let P 6¼ N be another normal subgroup of H. Since N is a minimal normal subgroup, one has N \ P ¼ feg, and, therefore, for p 2 P, n 2 N : n 1 p 1 np 2 N \ P ¼ fe g. Hence P centralizes N , P CH ðN Þ ¼ N , which is a contradiction. QED

Example 1.4.5.2.2 The derived series of Cyc2 Sym4 is Cyc2 Sym4 . Alt4 . Cyc2 Cyc2 . I (or in Hermann–Mauguin notation m3m . 23 . 222 . 1) and that of Cyc2 Cyc2 Sym3 is Cyc2 Cyc2 Sym3 . Cyc3 . I (Hermann–Mauguin notation: 6=mmm . 3 . 1). Hence these two groups are soluble. (For an explanation of the groups that occur here and later, see Section 1.4.3.6.) Now let R E 3 be a three-dimensional space group. Then T ðRÞ is an Abelian normal subgroup, hence T ðRÞ is soluble. The factor group R=T ðRÞ is isomorphic to a subgroup of either Cyc2 Sym4 or Cyc2 Cyc2 Sym3 and therefore also soluble. Using the remark above, one deduces that all three-dimensional space groups are soluble. Lemma 1.4.5.2.3. Let R be a three-dimensional space group. & Then R is soluble.

1.4.5.3. Maximal subgroups of soluble groups Now let G be a soluble group and M G a maximal subgroup of ﬁnite index in G. Then the set of left cosets X :¼ G=M is a primitive ﬁnite G-set. Let K ¼ core ðMÞ be the kernel of the action of G on X. Then the factor group H :¼ G=K acts faithfully on X. In particular, H is a ﬁnite group and X is a primitive, faithful H-set. Since G is soluble, the factor group H is also a soluble group. Let H . H1 . . . . . Hn1 . feg be the derived series of H with N :¼ Hn1 6¼ feg. Then N is an Abelian normal subgroup of H. The theorem of Galois (Theorem 1.4.5.1.3) states that N is an elementary Abelian p-group for some prime p and jXj ¼ jN j ¼ pr for some r 2 N. Since X ¼ G=M, the order of X is the index ½G : M of M in G. Therefore one gets the following theorem:

Hence the groups H that satisfy the hypotheses of the theorem of Galois are certain subgroups of an afﬁne group An ðZ=pZÞ over a ﬁnite ﬁeld Z=pZ. This afﬁne group is deﬁned in a way similar to the afﬁne group An over the real numbers where one has to replace the real numbers by this ﬁnite ﬁeld. Then N is the translation subgroup of An ðZ=pZÞ isomorphic to the n-dimensional vector space 0

1 x1 B C ðZ=pZÞ n ¼ fx ¼ @ ... A j x1 ; . . . ; xn 2 Z=pZg

Theorem 1.4.5.3.1. If M G is a maximal subgroup of ﬁnite index in the soluble group G, then its index jG=Mj is a prime & power.

xn over Z=pZ. The set X is the corresponding afﬁne space An ðZ=pZÞ. The factor group H ¼ H=N is isomorphic to a subgroup of the linear group of ðZ=pZÞn that does not leave invariant any nontrivial subspace of ðZ=pZÞn.

In the proof of Theorem 1.4.5.1.3, we have established a bijection between N and the H-set X, which is now X :¼ G=M. Taking the full pre-image

36

1.4. THE MATHEMATICAL BACKGROUND OF THE SUBGROUP TABLES ality of M implies that either N G ðMÞ ¼ G and M is a normal subgroup of G or N G ðMÞ ¼ M and G has i maximal subgroups that are conjugate to M.

0

N :¼ N core ðMÞ 0

0

of N in G, then one has G ¼ N M and M \ N ¼ core ðMÞ. Hence we have seen the ﬁrst part of the following theorem:

The smallest possible index of a proper subgroup is 2. It is well known and easy to see that subgroups of index 2 are normal subgroups:

Theorem 1.4.5.3.2. Let M G be a maximal subgroup of the soluble group G. Then the factor group H :¼ G=coreðMÞ acts primitively and faithfully on X :¼ G=M, and there is a normal 0 0 0 subgroup N / G with MN ¼ G and M \ N ¼ coreðMÞ. 0 0 Moreover, if M is another subgroup of G, with M0 N ¼ G and 0 0 0 M \ N ¼ coreðMÞ, then M is conjugate to M.

Proposition 1.4.6.1.1. Let G be a group and M G a subgroup of & index 2 ¼ jG=Mj. Then M is a normal subgroup of G. Proof. Choose an element g 2 G, g 62 M. Then G ¼ M [ g M ¼ M [ Mg . Hence g M ¼ Mg and therefore g Mg 1 ¼ M. Since this is also true if g 2 M, the proposition follows. QED Let M be a subgroup of a group G of index 2. Then M / G is a normal subgroup and the factor group G=M is a group of order 2. Since groups of order 2 are Abelian, it follows that the derived subgroup G1 of G (cf. Deﬁnition 1.4.5.2.1) (which is the smallest normal subgroup of G such that the factor group is Abelian) is contained in M. Hence all maximal subgroups of index 2 in G contain G1 . If one deﬁnes N :¼ \fM G j ½G : M ¼ 2g, then G=N is an elementary Abelian 2-group and hence a vector space over the ﬁeld with two elements. The maximal subgroups of G=N are the maximal subspaces of this vector space, hence their number is 2a 1, where a :¼ dimZ=2Z ðG=N Þ. This shows the following:

&

Example 1.4.5.3.3 G ¼ Sym4 ﬃ Td is the tetrahedral group from Section 1.4.3.2 and Sym3 ﬃ M ¼ C3v G is the stabilizer of one of the four apices in the tetrahedron. Then coreðMÞ ¼ feg and G=M is a faithful G-set which can be identiﬁed with the set of apices of 0 the tetrahedron. The normal subgroup N ¼ N is the normal subgroup U of Section 1.4.3.2.

Corollary 1.4.6.1.2. The number of subgroups of G of index 2 is of & the form 2a 1 for some a 0. Dealing with subgroups of index 3, one has the following: Proposition 1.4.6.1.3. Let U be a subgroup of the group G with ½G : U ¼ 3. Then U is either a normal subgroup of G or G=coreðUÞ ﬃ S 3 and there are three subgroups of G conjugate & to U.

Now let G ¼ Sym4 ﬃ Td be as above, and take D2d ﬃ M G a Sylow 2-subgroup of G. Then coreðMÞ ¼ D2 ﬃ Cyc2 Cyc2 is the normal subgroup U from Section 1.4.3.2 and H ¼ G=coreðMÞ ﬃ Sym3 .

Proof. G=core ðUÞ is isomorphic to a subgroup of Sym3 that acts primitively on f1; 2; 3g. Hence either G=core ðUÞ ﬃ Cyc3 and U ¼ core ðUÞ is a normal subgroup of G or G=core ðUÞ ﬃ Sym3 , U=core ðUÞ ﬃ Cyc2 and there are three subgroups of G conjugate to U. QED 1.4.6.2. Three-dimensional space groups We now come to space groups. By Lemma l.4.5.2.3, all threedimensional space groups are soluble. Theorem 1.4.5.3.1 says that the index of a maximal subgroup of a soluble group is a prime power (or inﬁnite). Since the index of a maximal subgroup of a space group is always ﬁnite (see Corollary 1.4.4.2.7), we get:

These observations result in an algorithm for computing maximal subgroups of soluble groups G: (1) compute normal subgroups C [candidates for coreðMÞ]; (2) compute a minimal normal subgroup N =C of G=C; (3) ﬁnd M=C as a complement of N =C in G=C.

Corollary 1.4.6.2.1. Let G be a three-dimensional space group and M G a maximal subgroup. Then ½G : M is a prime power. & Let R be a three-dimensional space group and P ¼ R=T ðRÞ its point group. It is well known that the order of P is of the form 2a 3b with a ¼ 0; 1; 2; 3 or 4 and b ¼ 0; 1. By Corollary 1.4.4.2.4, the t-subgroups of R are in one-to-one correspondence with the subgroups of P. Let us look at the t-subgroups of R of index 3. It is clear that P has no subgroup of index 3 if b ¼ 0, since the index of a subgroup divides the order of the ﬁnite group P by the theorem of Lagrange. If b ¼ 1, then any subgroup S of P of index 3 has order jPj=3 ¼ 2a and hence is a Sylow 2-subgroup of P. Therefore there is such a subgroup S of index 3 in P by the ﬁrst theorem of Sylow, Theorem 1.4.3.3.1. By the second theorem of Sylow, Theorem 1.4.3.3.2, all these Sylow 2-subgroups of P are

1.4.6. Quantitative results This section gives estimates for the number of maximal subgroups of a given index in space groups. 1.4.6.1. General results The ﬁrst very easy but useful remark applies to general groups G: Remark. Let M G be a maximal subgroup of G of ﬁnite index i :¼ ½G : M < 1. Then M N G ðMÞ G. Hence the maxim-

37

1. SPACE GROUPS AND THEIR SUBGROUPS (i) There are normal subgroups S i / Ri with R1 =S 1 ﬃ R2 =S 2 and with S i d2i T ðRi Þ if di 6¼ 2 and S i 16T ðRi Þ if di ¼ 2 (i ¼ 1; 2). (ii) ðR1 Þðd1 Þ ﬃ ðR2 Þðd2 Þ . (iii) There is a bijection : NðR1 Þ ! NðR2 Þ such that R1 =N & ﬃ R2 =ðN Þ for all N 2 NðR1 Þ.

conjugate in P. Therefore, by Proposition 1.4.6.1.3, the number of these groups is either 1 or 3: Corollary 1.4.6.2.2. Let R be a three-dimensional space group. If the order of the point group of R is not divisible by 3 then R has no t-subgroups of index 3. If 3 is a factor of the order of the point group of R, then R has either one t-subgroup of index 3 (which is then normal in R) or & three conjugate t-subgroups of index 3.

For a proof of this theorem, see Finken et al. (1980). Remark. If Ri are three- or four-dimensional space groups, the isomorphism in (ii) already implies the isomorphism of R1 and R2 , but there are counterexamples for dimension 5.

1.4.7. Qualitative results 1.4.7.1. General theory

1.4.7.2. Three-dimensional space groups Corollary 1.4.7.2.1. Let R be a three-dimensional space group with translation subgroup T and p be a prime not dividing the order of the point group R=T . Let U be a subgroup of R of index p for some 2 Z>0. Then (a) U is a k-subgroup. & (b) U is isomorphic to R.

In this section, we want to comment on the very subtle question of deciding whether two space groups R1 and R2 are isomorphic. This problem can be treated in several stages: Let R1 and R2 be space groups. Since the translation subgroups T ðRi Þ are characteristic subgroups of Ri (the maximal Abelian normal subgroup of ﬁnite index), each isomorphism ’ : R1 ! R2 induces isomorphisms of the corresponding translation subgroups

Proof: (a) U UT R implies that ½R : UT divides ½R : U ¼ p. Since T UT R, one obtains ½R : UT as a factor of ½R : T . But p is not a factor of ½R : T , hence ½R : UT ¼ 1 and R ¼ UT . According to the remark following Deﬁnition 1.4.4.2.2, U is a k-subgroup. (b) Let d1 :¼ jR=T j ¼ jU=T ðUÞj. Let d :¼ d21 if d1 6¼ 2 and d :¼ 16 otherwise, and let T 0 :¼ dT . Since gcdð½R : U; dÞ ¼ 1, one has UT 0 ¼ R and T 0 \ U ¼ dT ðUÞ. By the third isomorphism theorem, Theorem 1.4.3.5.2, it follows that

’0 : T ðR1 Þ ! T ðR2 Þ (by restriction) as well as of the point groups ’ : P 1 :¼ R1 =T ðR1 Þ ! R2 =T ðR2 Þ ¼: P 2 : It is convenient to view T ðRi Þ as a lattice on which the point group P i acts as group of linear mappings (cf. the start of Section 1.4.4). Then the isomorphism ’0 is an isomorphism of P 1 -sets, where P 1 acts on T ðR1 Þ via conjugation and on T ðR2 Þ via 1

g T ðR1 Þ t :¼ ’ðg Þt ’ðg Þ

R=T 0 ¼ UT 0 =T 0 ﬃ U=T 0 \ U ¼ U=dT ðUÞ: By Theorem 1.4.7.1.2 (i) ) (ii), one has Rðd1 Þ ﬃ U ðd1 Þ. By the remark above, this already implies that R and U are isomorphic. QED

for all g T ðR1 Þ 2 P 1 ; t 2 T ðR2 Þ:

Since ’ðT ðR1 ÞÞ ¼ T ðR2 Þ and T ðR2 Þ centralizes itself, this action is well deﬁned, i.e. independent of the choice of the coset representative g. The following theorem will show that the isomorphism of sufﬁciently large factor groups of R1 and R2 implies a ‘near’ isomorphism of the space groups themselves. To give a precise formulation we need one further deﬁnition.

Theorem 1.4.7.2.2. Let R be a three-dimensional space group and U be a maximal subgroup of R of index > 4. Then (a) U is a k-subgroup. & (b) U is isomorphic to R. Proof. Since R is soluble, the index ½R : U ¼ p is a prime power (see Theorem 1.4.5.3.1). If p is not a factor of jR=T ðRÞj, the statement follows from Corollary 1.4.7.2.1. Hence we only have to consider the cases p ¼ 2, > 2 and p ¼ 3, > 1. Since 9 is not a factor of the order of any crystallographic point group in dimension 3, assertion (a) follows if the index of U is divisible by 9. If U is a maximal t-subgroup, then R=U is a primitive P-set for the point group P of R. Since the point groups P of dimension 3 have no primitive P-sets of order divisible by 8, assertion (a) also follows if the index of U is divisible by 8. For all three-dimensional space groups R, the module LðRÞ=2LðRÞ [where T ðRÞ is identiﬁed with the corresponding lattice LðRÞ in ðE3 Þ as in Section 1.4.4] is not simple as a module for the point group P ¼ R=T ðRÞ. [It sufﬁces to check this property for the two maximal point groups Cyc2 Sym4 (¼ m3m) and Cyc2 Cyc2 Sym3 ð¼ 6=mmmÞ.] This means that 2LðRÞ is not a maximal R-invariant sublattice of LðRÞ. Since the translation subgroup T ðUÞ of a maximal k-subgroup U of index equal to a power of 2 in R is a maximal R-invariant subgroup of T ðRÞ that contains 2T ðRÞ, one now ﬁnds that R has no maximal k-subgroup of index 8.

Deﬁnition 1.4.7.1.1. For d 2 N deﬁne a Od :¼ f j a; b 2 Z; b 6¼ 0; gcdðb; dÞ ¼ 1g Q; b which is the set of all rational numbers for which the denominator is prime to d. For the space group R E n let R RðdÞ E n be the group RðdÞ :¼ hT ðRÞðdÞ ; Ri, where T ðRÞðdÞ ¼ fat j a 2 OðdÞ ; t 2 T ðRÞg T n ; i.e. one allows denominators that are prime to d in the translation & subgroup. One has the following: Theorem 1.4.7.1.2. Let R1 and R2 be two space groups with point groups of order di :¼ jRi =T ðRi Þj. Let NðRi Þ denote the set of normal subgroups of Ri having ﬁnite index in Ri . Then the following three conditions are equivalent:

38

1.4. THE MATHEMATICAL BACKGROUND OF THE SUBGROUP TABLES Now assume that ½R : U ¼ 9. By Corollary 1.4.7.2.1, one only needs to deal with groups R such that the order of the point group P :¼ R=T ðRÞ is divisible by 3. P is isomorphic to a subgroup of Cyc2 Sym4 or Cyc2 Cyc2 Sym3 . If Alt4 P is a subgroup of P, then LðRÞ=3LðRÞ is simple and U is of index 27 in R [with LðUÞ ¼ 3LðRÞ]. It turns out that U is isomorphic to R in these cases. If P does not contain a subgroup isomorphic to Alt4, then the maximality of U implies that T ðUÞ T ðRÞ is of index 3 in T ðRÞ. Hence ½R : U ¼ 3 in this case. QED

The determination of the k-supergroups of a given space group R is the easier task. For instance, if R is a symmorphic space group then all its k-supergroups are also symmorphic. This is not true for k-subgroups of R. Theorem 1.4.8.3. Let S be a k-supergroup of the space group R. Then S ¼ RT ðSÞ. If S is a minimal k-supergroup of R then the translation lattice LðSÞ of S is an R-invariant lattice that contains & LðRÞ as a maximal sublattice. Proof. Let T :¼ T ðSÞ be the translation subgroup of S. Then T \ R ¼ T ðRÞ and R=ðT \ RÞ is isomorphic to the point group R, which is a ﬁnite group. By the isomorphism theorem

Corollary 1.4.7.2.3. Let M be a maximal subgroup of the threedimensional space group R. (a) If the index of M is a power of 2, then ½R : M ¼ 2 or 4. (b) If 3 is a factor of the order of the point group ½R : T ðRÞ and the index of M is a power of 3, then ½R : M ¼ 3 or 27. For ½R : M ¼ 27, M is necessarily isomorphic to R (by & Theorem 1.4.7.2.2).

R=ðT \ RÞ ﬃ RT =T : Therefore, group RT generated by T and R is a subgroup of S containing T with the same index and, therefore, RT ¼ S. Moreover, T contains T ðRÞ and hence LðSÞ contains LðRÞ. Since T is a normal subgroup of S, the space group R acts on T by conjugation and therefore LðSÞ is R invariant. If there is an R invariant lattice L such that LðRÞ L LðSÞ, then, applying the isomorphism from Example 1.4.3.4.4, the group G :¼ R1 ðLÞ is a space group with R G S. Hence the minimality of the supergroup S implies that LðSÞ is an R-invariant lattice that contains LðRÞ as a maximal sublattice. QED

This interesting fact explains why there are no maximal subgroups of index 8 in a three-dimensional space group. If there is a maximal subgroup M of a three-dimensional space group R of index 9, then the order of the point group of R is not divisible by three and the subgroup M is a k-subgroup and isomorphic to R. In particular, there are no maximal subgroups of index 9 for trigonal, hexagonal or cubic space groups, whereas there are such subgroups of tetragonal space groups.

As for maximal k-subgroups, the index ½S : R of a minimal k-supergroup S of R is a prime power and for each prime p there is some a 2 N such that R has a minimal k-supergroup S of index ½S : R ¼ pa . Because of the inﬁnite number of prime numbers, a space group R has inﬁnitely many minimal k-supergroups, but there are only ﬁnitely many minimal k-supergroups containing a given space group R of given index. This is different for t-supergroups, as the following example shows.

1.4.8. Minimal supergroups For several problems, for example for the prediction of a phase transition or in the search for overlooked symmetry in crystalstructure determinations etc., it is helpful to know all space groups S containing a given space group R, which means that R S. Then S is called a supergroup of R. Note that – in contrast to subgroups – the supergroups G R containing a space group R of ﬁnite index need not be space groups. For instance, the one-dimensional translation group 1 R¼h 0

Example 1.4.8.4 Let 1 B0 R ¼ [email protected]

0 1

1 0 1 1 B0 0C A; @

0 1

1 0 2 1C Ai ﬃ Z

0

0

1

0

1

0

! 1 iﬃZ 1

0

be the two-dimensional translation group p1. Then for all x 2 R the group

has a supergroup G of index 2 isomorphic to Cyc2 Z which is not a subgroup of the one-dimensional afﬁne group. G is Abelian but has an element of ﬁnite order, so G cannot be a space group. For the applications in crystallography, we are restricted to those supergroups S of R that are again space groups.

0

Deﬁnition 1.4.8.1. Let S be a space group that is a supergroup of the space group R and T :¼ T ðSÞ.

1 B0 S x :¼ hR; @

0 1

0

0

1 0 xC Ai 1

is a minimal t-supergroup of R containing R of index 2. These groups are conjugate under the normalizer of R in the afﬁne group A2 [see Example 3.47 in Heidbu¨chel (2003)],

(i) S is called a translationengleiche or a t-supergroup if R\T ¼T. (ii) S is called a klassengleiche or a k-supergroup if & R=R \ T ﬃ S=T .

0

Clearly S is a t-supergroup (or k-supergroup, respectively) of R if and only if R is a t-subgroup (or k-subgroup) of S and the theorem of Hermann implies the following:

1 B0 @

0 1

0

0

1 0 0 1 B0 x=2 C AS x @ 1

0

0 1 0

1 0 x=2 C A ¼ S0 : 1

Visually this means that the discrete sets of symmetry lines of the different plane groups may be shifted by any real distance against each other or relative to an arbitrarily chosen origin. This yields uncountably many different t-supergroups of p1 which are all of the same type.

Theorem 1.4.8.2. (Theorem of Hermann.) Let S and R be space groups such that S R is a minimal supergroup of R. Then S is & either a k-supergroup or a t-supergroup.

39

1. SPACE GROUPS AND THEIR SUBGROUPS The use of the afﬁne and Euclidean normalizers of a space group R is described in Part 15 of IT A. The afﬁne normalizer

This proof also provides an algorithm to determine representatives of the N -orbits of minimal t-supergroups of a given space group R, provided that one knows representatives of all afﬁne classes of space groups and their maximal t-subgroups. For dimensions 2 and 3 these are given in this volume. Since maximal t-subgroups of three-dimensional space groups have index 2, 3 or 4, this also holds for the minimal t-supergroups of these groups. Up to dimension n 4, the minimal t-supergroups and the minimal k-supergroups of a given space group R An can be obtained with the commands TSupergroups and KSupergroups in CARAT [see also Heidbu¨chel (2003)].

N :¼ N An ðRÞ ¼ fg 2 An j gRg1 ¼ Rg of an n-dimensional space group R An acts on the set of all minimal t-supergroups S of R by conjugation. Theorem 1.4.8.5. N has ﬁnitely many orbits on the set of & (minimal) t-supergroups S of R. Proof. Let S 1 ; . . . ; S m be representatives of the An -orbits on the set of n-dimensional space groups, i.e. of the types of n-dimensional space groups. For 1 i m let

References Finken, H., Neubu¨ser, J. & Plesken, W. (1980). Space groups and groups of prime power order II. Arch. Math. 35, 203–209. Hahn, Th. & Wondratschek, H. (1994). Symmetry of Crystals. Introduction to International Tables for Crystallography, Vol. A. Soﬁa: Heron Press. Heidbu¨chel, O. (2003). Beitra¨ge zur Theorie der kristallographischen Raumgruppen. Aachener Beitra¨ge zur Mathematik, 29. ISBN 3-86073649-3. International Tables for Crystallography (2005). Vol. A, Space-Group Symmetry, edited by Th. Hahn, 5th ed. Heidelberg: Springer. Ledermann, W. (1976). Introduction to Group Theory. London: Longman. (German: Einfu¨hrung in die Gruppentheorie, Braunschweig: Vieweg, 1977.) Ledermann, W. & Weir, A. J. (1996). Introduction to Group Theory. 2nd ed. Harlow: Addison Wesley Longman. Opgenorth, J., Plesken, W. & Schulz, T. (1998). Crystallographic algorithms and tables. Acta Cryst. A54, 517–531.

Mi :¼ fRij j 1 j ai g denote the set of all (maximal) t-subgroups of S i that are isomorphic to R. If S is a (minimal) t-supergroup of the space group R, then there is some g 2 An and i 2 f1; . . . ; mg such that gSg1 ¼ S i and gRg1 ¼ Rij 2 Mi , hence the pair of space groups ðR; SÞ ¼ ðg1 Rij g; g1 S i gÞ for some i 2 f1; . . . ; mg, Rij 2 Mi and g 2 An . If S 0 is a second supergroup of R and h 2 An such that 1 ðR; S 0 Þ ¼ ðh1 Rij h; h1 S i hÞ for the same Pm i; j, then h g 2 N normalizes R. Hence there are at most i¼1 ai orbits of N on the set of (minimal) t-supergroups of R. QED

40

references

International Tables for Crystallography (2011). Vol. A1, Chapter 1.5, pp. 41–43.

1.5. Remarks on Wyckoff positions By Ulrich Mu¨ller

1.5.1. Crystallographic orbits and Wyckoff positions

equivalent under the Euclidean normalizer may have to be interchanged. The permutations they undergo when the origin is shifted have been listed by Boyle & Lawrenson (1973). An origin shift of 0; 0; 12 will interchange the Wyckoff positions 4e and 4f as well as 4g and 4h of I222.

The set of symmetry-equivalent sites in a space group is referred to as a (crystallographic point) orbit (Koch & Fischer, 1985; Wondratschek, 1976, 1980, 2005; also called point conﬁguration). If the coordinates of a site are completely ﬁxed by symmetry (e.g. 1 1 1 4 ; 4 ; 4), then the orbit is identical with the corresponding Wyckoff position of that space group (in German Punktlage). However, if there are one or more freely variable coordinates (e.g. z in 0; 12 ; z), the Wyckoff position comprises an inﬁnity of possible orbits; they differ in the values of the variable coordinate(s). The set of sites that are symmetry equivalent to, say, 0; 12 ; 0:391 makes up one orbit. The set corresponding to 0; 12 ; 0:468 belongs to the same Wyckoff position, but to another orbit (its variable coordinate z is different). The Wyckoff positions of the space groups are listed in Volume A of International Tables for Crystallography (2005). They are labelled with letters a, b, . . ., beginning from the position having the highest site symmetry. A Wyckoff position is usually given together with the number of points belonging to one of its orbits within a unit cell. This number is the multiplicity listed in Volume A, and commonly is set in front of the Wyckoff letter. For example, the denomination 4c designates the four symmetryequivalent points belonging to an orbit c within the unit cell. In many space groups, for some Wyckoff positions there exist several Wyckoff positions of the same kind that can be combined to form a Wyckoff set [called a Konﬁgurationslage by Koch & Fischer (1975)]. They have the same site symmetries and they are mapped onto one another by the afﬁne normalizer of the space group (Koch & Fischer, 1975; Wondratschek, 2005).

Example 1.5.1.2 In the space group Fm3m, No. 225, the orbits of the Wyckoff positions 4a ð0; 0; 0Þ and 4b ð12 ; 12 ; 12Þ are equivalent under the Euclidean normalizer. The copper structure can be described equivalently either by having the Cu atoms occupy the position 4a or the position 4b. If we take Cu atoms in the position 4a and shift the origin from ð0; 0; 0Þ to ð12 ; 12 ; 12Þ, then they result in the position 4b. Unique relations exist between the Wyckoff positions of a space group and the Wyckoff positions of any of its subgroups (Billiet et al., 1978; Wondratschek, 1993; Wondratschek et al., 1995). Given the relative positions of their unit cells (axes transformations and relative origin positions), the labels of these Wyckoff positions are unique. Example 1.5.1.3 In diamond, the carbon atoms occupy the orbit belonging to the Wyckoff position 8a of the space group Fd3m, No. 227. Zinc blende (sphalerite) crystallizes in the maximal subgroup F43m, No. 216, of Fd3m. With the transition Fd3m ! F43m the Wyckoff position 8a splits into the positions 4a and 4c of F43m. These are two symmetry-independent positions that allow an occupation by atoms of two different elements (zinc and sulfur). In this example, all of the positions retain the site symmetry 43m and each Wyckoff position comprises only one orbit.

Example 1.5.1.1 In space group I222, No. 23, there are six Wyckoff positions with the site symmetry 2: 4e ðx; 0; 0Þ, 4f ðx; 0; 12Þ on twofold rotation axes parallel to a, 4g ð0; y; 0Þ, 4h ð12 ; y; 0Þ on twofold rotation axes parallel to b, 4i ð0; 0; zÞ, 4j ð0; 12 ; zÞ on twofold rotation axes parallel to c.

1.5.2. Derivative structures and phase transitions In crystal chemistry, structural relations such as the relation diamond–sphalerite are of fundamental interest. Structures that result from a basic structure by the substitution of atoms of one kind for atoms of different elements, the topology being retained, are called derivative structures after Buerger (1947, 1951). For the basic structure the term aristotype has also been coined, while its derivative structures are called hettotypes (Megaw, 1973). For more details, see Chapter 1.6. When searching for derivative structures, one must look for space groups that are subgroups of the space group of the aristotype and in which the orbit of the atom(s) to be substituted splits into different orbits. Similar relations also apply to many phase transitions (cf. Section 1.6.6). Very often the space group of one of the phases is a subgroup of the space group of the other. For second-order phase transitions this is even mandatory (cf. Section 1.2.7). The positions of the atoms in one phase are related to those in the other one.

They are mapped onto one another by the afﬁne normalizer of No. 221. These six Wyckoff I222, which is isomorphic to Pm3m, positions make up one Wyckoff set. However, in this example the positions 4e, 4f vs. 4g, 4h vs. 4i, 4j, being on differently oriented axes, cannot be considered to be equivalent if the lattice parameters are a 6¼ b 6¼ c. The subdivision of the positions of the Wyckoff set into these three sets is accomplished with the aid of the Euclidean normalizer of the space group I222. The Euclidean normalizer is that supergroup of a space group that maps all equivalent symmetry elements onto one another without distortions of the lattice. It is a subgroup of the afﬁne normalizer (Fischer & Koch, 1983; Koch et al., 2005). In Example 1.5.1.1 (space group I222), the positions 4e and 4f are equivalent under the Euclidean normalizer (and so are 4g, 4h and also 4i, 4j). The Euclidean normalizer of the space group I222 is Pmmm, No. 47, with the lattice parameters 12 a; 12 b; 12 c (if a 6¼ b 6¼ c). If the origin of a space group is shifted, Wyckoff positions that are Copyright © 2011 International Union of Crystallography

Example 1.5.2.1 The disorder–order transition of -brass (CuZn) taking place at 741 K involves a space-group change from the space group

41

1. SPACE GROUPS AND THEIR SUBGROUPS Im3m, No. 229, to its subgroup Pm3m, No. 221. In the hightemperature form, Cu and Zn atoms randomly take the orbit of the Wyckoff position 2a of Im3m. Upon transition to the ordered form, this position splits into the independent positions 1a and 1b of the subgroup Pm3m. These positions are occupied by the Cu and Zn atoms, respectively. See also Example 1.2.7.3.3.

Rj ¼ jSG ðXj Þj=jSH ðXj Þj: When the space-group symmetry is reduced from G to H and the orbit of the point Xj splits into n orbits, the following relation holds (Wondratschek, 1993): i¼

n P

Rj :

j¼1

Phase transitions in which a paraelectric crystal becomes ferroelectric occur when atoms that randomly occupy several symmetry-equivalent positions become ordered in a space group with lower symmetry, or when a key atom is displaced to a position with reduced site symmetry, thus allowing a distortion of the structure. In both cases, the space group of the ferroelectric phase is a subgroup of the space group of the paraelectric phase. In the case of ordering, the orbits of the atoms concerned split; in the case of displacement this is not necessary.

i ¼ jG : Hj is the index of H in G (cf. Section 1.2.4.2). Example 1.5.3.1 The orbit of the Wyckoff position 24d of space group Fm3m, No. 225, has the site symmetry mmm with the order jmmmj ¼ 8. Upon symmetry reduction to the space group I4=mmm, No. 139, this orbit splits into the two orbits 4c and 8f of I4=mmm with the site symmetries mmm and 2=m, respectively. j2=mj ¼ 4. The reduction factors of the site symmetries are

Example 1.5.2.2 In paraelectric NaNO2, space group Immm, No. 71, Na+ ions randomly occupy two sites close to each other around an inversion centre ð0; 0; 12Þ with half occupation (position 4i at 0; 0; 0:540Þ. The same applies to the nitrite ions, which are disordered in two opposite orientations around the inversion centre at 0; 0; 0, with the N atoms at 4i (0; 0; 0:072). At the transition to the ferroelectric phase at 438 K, the space-group symmetry decreases to the subgroup Imm2, No. 44, and the ions become ordered in one orientation. Each of the 4i orbits splits into two 2a orbits, but for every ion only one of the resulting orbits is now fully occupied: Na+ at 2a ð0; 0; 0:540Þ and N at 2a ð0; 0; 0:074Þ.

jmmmj=jmmmj ¼ 8=8 ¼ 1 and jmmmj=j2=mj ¼ 8=4 ¼ 2: They add up to 1 þ 2 ¼ 3, which is the index of I4=mmm in Fm3m. The multiplicities commonly used together with the Wyckoff labels depend on the size of the chosen unit cell. As a consequence, a change of the size of the unit cell also changes the multiplicities. For example, the multiplicities of the Wyckoff positions listed in Volume A are larger by a factor of three for rhombohedral space groups when the unit cell is referred not to rhombohedral, but to hexagonal axes. The multiplicity of a Wyckoff position shows up in the sum of the multiplicities of the corresponding positions of the subgroup. If the unit cell selected to describe the subgroup does not change in size, then the sum of the multiplicities of the positions of the subgroup must be equal to the multiplicity of the position of the starting group. For example, from a position with a multiplicity of 6, a position with multiplicity of 6 can result, or it can split into two positions of multiplicity of 3, or into two with multiplicities of 2 and 4, or into three with multiplicity of 2 etc. If the unit cell of the subgroup is enlarged or reduced by a factor f, then the sum of the multiplicities must also be multiplied or divided by this factor f. Relations between the Wyckoff positions of space groups and the Wyckoff positions of their maximal subgroups were listed by Lawrenson (1972). However, his tables are not complete, and they were never published. In addition, they lack information about the transformations of axes and coordinates when these differ in the subgroup. More recently, a computer program to calculate these relations has been developed (Kroumova et al., 1998; cf. Section 1.7.4). To be used, the program requires knowledge of the subgroups (maximal or non-maximal) and of the necessary axes transformations and origin shifts. The Wyckoff position(s) to be considered can be marked or speciﬁc coordinates of a position must be given. The output is a listing of the Wyckoff position(s) of the speciﬁed subgroup and optionally all corresponding site coordinates. Depending on the relations and positions considered, the listings of coordinates can be rather long. The program has not been designed to give a fast overview of the relations. If one is looking for those subgroups that will exhibit a splitting of a certain position, all subgroups have to be tried one by one. For these reasons, the program cannot substitute the present tables;

Example 1.5.2.3 Paraelectric BaTiO3 crystallizes in the space group Pm3m, No. 221, and the position 1a of a Ti atom (0; 0; 0 with site symmetry m3m) is in the centre of an octahedron of oxygen atoms. At 393 K, a phase transition to a ferroelectric phase takes place. It has space group P4mm, No. 99, which is a subgroup of Pm3m; the Ti atom is now at 0; 0; z (1a, site symmetry 4mm) and is displaced from the octahedron centre. The orbit does not split, but the site symmetry is reduced.

1.5.3. Relations between the positions in group–subgroup relations The following statements are universally valid: (1) Between the points of an orbit and the corresponding points in a subgroup there exists a one-to-one relation; both sets of points have the same magnitude. (2) Between the Wyckoff positions of a space group and those of its subgroups there exist unique relations. These may involve different Wyckoff labels for different relative positions of the origins. (3) With the symmetry reduction from a group to a subgroup, an orbit either splits into different orbits, or its site symmetry is reduced, or both happen. In addition, coordinates ﬁxed or coupled by symmetry may become independent. Let G be a space group and H a subgroup of G. Let the sitesymmetry groups of a point Xj under the space groups G and H be SG ðXj Þ and SH ðXj Þ, respectively. The reduction factor of the site symmetries is then

42

1.5. REMARKS ON WYCKOFF POSITIONS for practical work, the program and the tables listed in Part 3 complement each other. The tables in Part 3 are a complete compilation for all space groups and all of their maximal subgroups. For all Wyckoff positions of a space group, all relations to the Wyckoff positions of its subgroups are listed. This also applies to the inﬁnite number of maximal isomorphic subgroups; for these, a parameterized form has been developed that allows the listing of all maximal subgroups and all of their resulting Wyckoff positions completely for every allowed index of symmetry reduction.

International Tables for Crystallography (2005). Vol. A, Space-Group Symmetry, edited by Th. Hahn, 5th ed. Heidelberg: Springer. Koch, E. & Fischer, W. (1975). Automorphismengruppen von Raumgruppen und die Zuordnung von Punktlagen zu Konﬁgurationslagen. Acta Cryst. A31, 88–95. Koch, E. & Fischer, W. (1985). Lattice complexes and limiting complexes versus orbit types and non-characteristic orbits: a comparative discussion. Acta Cryst. A41, 421–426. Koch, E., Fischer, W. & Mu¨ller, U. (2005). Normalizers of space groups and their use in crystallography. International Tables for Crystallography, Vol. A, Space-Group Symmetry, edited by Th. Hahn, Part 15. Heidelberg: Springer. Kroumova, E., Perez-Mato, J. M. & Aroyo, M. I. (1998). WYCKSPLIT. Computer program for the determination of the relations of Wyckoff positions for a group–subgroup pair. J. Appl. Cryst. 31, 646. Accessible at: http://www.cryst.ehu.es/cryst/wpsplit.html. Lawrenson, J. E. (1972). Theoretical Studies in Crystallography. Dissertation, University of Kent at Canterbury. Megaw, H. D. (1973). Crystal Structures: A Working Approach. Philadelphia: Saunders. Wondratschek, H. (1976). Extraordinary orbits of space groups. Theoretical considerations. Z. Kristallogr. 143, 460–470. Wondratschek, H. (1980). Crystallographic orbits, lattice complexes, and orbit types. MATCH Commun. Math. Chem. 9, 121–125. Wondratschek, H. (1993). Splitting of Wyckoff positions (orbits). Mineral. Petrol. 48, 87–96. Wondratschek, H. (2005). Special topics on space groups. International Tables for Crystallography, Vol. A, Space-Group Symmetry, edited by Th. Hahn, Part 8. Heidelberg: Springer. Wondratschek, H., Mu¨ller, U., Aroyo, M. I. & Sens, I. (1995). Splitting of Wyckoff positions (orbits). II. Group–subgroup chains of index 6. Z. Kristallogr. 210, 567–573.

References Billiet, Y., Sayari, A. & Zarrouk, H. (1978). L’association des familles de Wyckoff dans les changements de repe`re conventionnel des groupes spatiaux et les passages aux sous-groupes spatiaux. Acta Cryst. A34, 811–819. Boyle, L. L. & Lawrenson, J. E. (1973). The origin dependence of the Wyckoff site description of a crystal structure. Acta Cryst. A29, 353– 357. Buerger, M. J. (1947). Derivative crystal structures. J. Chem. Phys. 15, 1– 16. Buerger, M. J. (1951). Phase Transformations in Solids, ch. 6. New York: Wiley. Fischer, W. & Koch, E. (1983). On the equivalence of point conﬁgurations due to Euclidean normalizers (Cheshire groups) of space groups. Acta Cryst. A39, 907–915.

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references

International Tables for Crystallography (2011). Vol. A1, Chapter 1.6, pp. 44–56.

1.6. Relating crystal structures by group–subgroup relations By Ulrich Mu¨ller

1.6.1. Introduction

(1) In crystal structures the arrangement of atoms reveals a pronounced tendency towards the highest possible symmetry. (2) Counteracting factors due to special properties of the atoms or atom aggregates may prevent the attainment of the highest possible symmetry. However, in most cases the deviations from the ideal symmetry are only small (key word: pseudosymmetry). (3) During phase transitions and solid-state reactions which result in products of lower symmetry, the higher symmetry of the starting material is often indirectly preserved by the formation of oriented domains.

Symmetry relations using crystallographic group–subgroup relations have proved to be a valuable tool in crystal chemistry and crystal physics. Some important applications include: (1) Structural relations between crystal-structure types can be worked out in a clear and concise manner by setting up family trees of group–subgroup relations (see following sections). (2) Elucidation of problems concerning twinned crystals and antiphase domains (see Section 1.6.6). (3) Changes of structures and physical properties taking place during phase transitions; applications of Landau theory (Aizu, 1970; Aroyo & Perez-Mato, 1998; Birman, 1966a,b, 1978; Cracknell, 1975; Howard & Stokes, 2005; Igartua et al., 1996; Izyumov & Syromyatnikov, 1990; Landau & Lifshitz, 1980; Lyubarskii, 1960; Salje, 1990; Stokes & Hatch, 1988; Tole´dano & Tole´dano, 1987). (4) Prediction of crystal-structure types and calculation of the numbers of possible structure types (see Section 1.6.4.7). (5) Solution of the phase problem in the crystal structure analysis of proteins (Di Costanzo et al., 2003).

Aspect (1) is due to the tendency of atoms of the same kind to occupy equivalent positions, as stated by Brunner (1971). This has physical reasons: depending on chemical composition, the kind of chemical bonding, electron conﬁguration of the atoms, relative sizes of the atoms, pressure, temperature etc., there exists one energetically most favourable surrounding for atoms of a given species which all of these atoms strive to attain. Aspect (2) of the symmetry principle is exploited in the following sections. Factors that counteract the attainment of the highest symmetry include: (1) stereochemically active lone electron pairs; (2) Jahn–Teller distortions; (3) covalent bonds; (4) Peierls distortions; (5) ordered occupation of originally equivalent sites by different atomic species (substitution derivatives); (6) partial occupation of voids in a packing of atoms; (7) partial vacation of atomic positions; (8) freezing (condensation) of lattice vibrations (soft modes) giving rise to phase transitions; and (9) ordering of atoms in a disordered structure. Aspect (3) of the symmetry principle has its origin in an observation by Bernal (1938). He noted that in the solid state reaction MnðOHÞ2 ! MnOOH ! MnO2 the starting and the product crystal had the same orientation. Such reactions are called topotactic reactions after Lotgering (1959) (for a more exact deﬁnition see Giovanoli & Leuenberger, 1969). In a paper by Bernal & Mackay (1965) we ﬁnd the sentence: ‘One of the controlling factors of topotactic reactions is, of course, symmetry. This can be treated at various levels of sophistication, ranging from Lyubarskii’s to ours, where we ﬁnd that the simple concept of Buridan’s ass illumines most cases.’ According to the metaphor of Buridan (French philosopher, died circa 1358), the ass starves to death between two equal and equidistant bundles of hay because it cannot decide between them. Referred to crystals, such an asinine behaviour would correspond to an absence of phase transitions or solid-state reactions if there are two or more energetically equivalent orientations of the domains of the product. Crystals, of course, do not behave like the ass; they take both.

Ba¨rnighausen (1975, 1980) presented a standardized procedure to set forth structural relations between crystal structures with the aid of symmetry relations between their space groups. For a review on this subject see Mu¨ller (2004). Short descriptions are given by Chapuis (1992) and Mu¨ller (2006). The main concept is to start from a simple, highly symmetrical crystal structure and to derive more and more complicated structures by distortions and/or substitutions of atoms. Similar to the ‘diagrams of lattices of subgroups’ used in mathematics, a tree of group–subgroup relations between the space groups involved, now called a Ba¨rnighausen tree, serves as the main guideline. The highly symmetrical starting structure is called the aristotype after Megaw (1973) or basic structure after Buerger (1947, 1951); other terms used in the literature on phase transitions in physics are prototype or parent structure. The derived structures are the hettotypes or derivative structures. In Megaw’s (1973) terminology, the structures mentioned in the tree form a family of structures. The structure type to be chosen as the aristotype depends on the speciﬁc problem and, therefore, the term aristotype cannot be deﬁned in a strict manner. For example, a body-centred packing of spheres (space group Im3m) can be chosen as the aristotype for certain intermetallic structures. By symmetry reduction due to a loss of the centring, the CsCl type (space group Pm3m) can be derived. However, if all the structures considered are ionic, there is no point in starting from the body-centred packing of spheres and one can choose the CsCl type as the aristotype.

1.6.2. The symmetry principle in crystal chemistry 1.6.3. Ba¨rnighausen trees

The usefulness of symmetry relations is intimately related to the symmetry principle in crystal chemistry. This is an old principle based on experience which has been worded during its long history in rather different ways. Ba¨rnighausen (1980) summarized it in the following way: Copyright © 2011 International Union of Crystallography

To represent symmetry relations between different crystal structures in a concise manner, we construct a tree of group– subgroup relations in a modular design, beginning with the space group of the aristotype at its top. Each module represents one

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1.6. RELATING CRYSTAL STRUCTURES BY GROUP–SUBGROUP RELATIONS

Fig. 1.6.3.1. Scheme of the formulation of the smallest step of symmetry reduction connecting two related crystal structures.

step of symmetry reduction to a maximal subgroup. Therefore, we have to discuss only one of these modules in detail. For two structures we want to interrelate, we place their spacegroup symbols one below the other and indicate the direction of the symmetry reduction by an arrow pointing downwards (Fig. 1.6.3.1). Since they are more informative, it is advisable to use only the full Hermann–Mauguin symbols. In the middle of the arrow we insert the kind of maximal subgroup and the index of symmetry reduction, using the abbreviations t for translationengleiche, k for klassengleiche and i for isomorphic. If the unit cell changes, we also insert the new basis vectors expressed as vector sums of the basis vectors of the higher-symmetry cell. If there is an origin shift, we enter this as a triplet of numbers which express the coordinates of the new origin referred to the coordinate system of the higher-symmetry cell. This is a shorthand notation for the transformation matrices. Any change of the basis vectors and the origin is essential information that should never be omitted. If the atomic coordinates of two related crystal structures differ because of different settings of their unit cells, the similarities of the structures become less clear and may even be obscured. Therefore, it is recommended to avoid cell transformations whenever possible. If necessary, it is much better to fully

exploit the possibilities offered by the Hermann–Mauguin symbolism and to choose nonconventional space-group settings [see Chapter 4.3 of International Tables for Crystallography Volume A (2005) and Section 3.1.4 of this volume]. Origin shifts also tend to obscure relations. However, they often cannot be avoided. There is no point in deviating from the standard origin settings, because otherwise much additional information would be required for an unequivocal description. Note: The coordinate triplet specifying the origin shift in the group–subgroup arrow refers to the coordinate system of the higher-symmetry space group, whilst the corresponding changes of the atomic coordinates refer to the coordinate system of the subgroup and therefore are different. Details are given in Section 3.1.3. Also note that in the tables of Parts 2 and 3 of this volume the origin shifts are given in different ways. In Part 2 they refer to the higher-symmetry space group. In Part 3 (relations of the Wyckoff positions) they are given only as parts of the coordinate transformations, i.e. in the coordinate systems of the subgroups. As explained in the Appendix, the chosen origin shifts themselves also differ in Parts 2 and 3; an origin transformation taken from Part 3 may be different from the one given in Part 2 for the same group–subgroup relation. If needed, one has to calculate the corresponding values with the formulae given in Section 3.1.3.

45

1. SPACE GROUPS AND THEIR SUBGROUPS The calculation of coordinate changes due to cell transformations and origin shifts is prone to errors. Some useful hints are given in Section 1.6.5. For space groups with two possible choices of origin (‘origin choice 1’ and ‘origin choice 2’), the choice is speciﬁed by a superscript (1) or (2) after the space-group symbol, for example P4=nð2Þ . The setting of rhombohedral space groups is speciﬁed, if necessary, by superscript (rh) or (hex). Occasionally it may be useful to use a nonconventional rhombohedral ‘reverse’ setting, i.e. with the centring vectors ð 13 ; 23 ; 13 Þ instead of ‘obverse’ with ð 23 ; 13 ; 13 Þ; this is speciﬁed by superscript (rev), for example R 3 ðrevÞ . In a Ba¨rnighausen tree containing several group–subgroup relations, it is recommended that the vertical distances between the space-group symbols are kept proportional to the logarithms of the corresponding indices. This way all subgroups that are at the same hierarchical distance, i.e. at the same index, from the aristotype appear on the same line. If several paths can be constructed from one space group to a general subgroup, via different intermediate groups, usually there is no point in depicting all of them. There is no general recipe indicating which of several possible paths should be preferred. However, crystal-chemical and physical aspects should be used as a guide. First of all, the chosen intermediate groups should be:

Group–subgroup relations are of little value if the usual crystallographic data are not given for every structure. The mere mention of the space groups is insufﬁcient. The atomic coordinates are of special importance. It is also important to present all structures in such a way that their relations become clearly visible. In particular, all atoms of the asymmetric units should exhibit strict correspondence, so that their positional parameters can immediately be compared. For all structures, the same coordinate setting and among several symmetry-equivalent positions for an atom the same location in the unit cell should be chosen, if possible. For all space groups, except Im3m and Ia3d, one can choose several different equivalent sets of coordinates describing one and the same structure in the same space-group setting. It is by no means a simple matter to recognize whether two differently documented structures are alike or not (the literature abounds with examples of ‘new’ structures that really had been well known). One is often forced to transform coordinates from one set to another to attain the necessary correspondence. In Section 15.3 of Volume A (editions 1987–2005) and in a paper by Koch & Fischer (2006) one can ﬁnd a procedure for and examples of how to interconvert equivalent coordinate sets with the aid of the Euclidean normalizers of the space groups. Note: For enantiomorphic (chiral) space groups like P31 this procedure will yield equivalent sets of coordinates without a change of chirality; for chiral structures in non-enantiomorphic (non-chiral) space groups like P21 21 21 the sets of coordinates include the enantiomorphic pairs [for the distinction between chiral and non-chiral space groups see Flack (2003)]. If space permits, it is useful to list the site symmetries and the coordinates of the atoms next to the space-group symbols in the Ba¨rnighausen tree, as shown in Fig. 1.6.3.1 and in the following examples. If there is not enough space, this information must be provided in a separate table.

(1) Space groups having actually known representatives. (2) Space groups that disclose a physically realizable path for the symmetry reduction. Observed phase transitions should be given high priority. For phase transitions that are driven by certain lattice vibrations, those intermediate space groups should be considered that are compatible with these lattice modes (i.e. irreducible representations; Stokes & Hatch, 1988). (3) In the case of substitution derivatives: Space groups showing a splitting of the relevant Wyckoff position(s). These intermediate groups allow for substitution derivatives, even if no representative is yet known.

1.6.4. The different kinds of symmetry relations among related crystal structures

Ba¨rnighausen trees sometimes contain intermediate space groups which, in Howard & Stokes’ (2005) opinion, ‘have no physical signiﬁcance’. As an example, they cite the phase transition induced by the displacement of the octahedrally coordinated cations in a cubic perovskite along the +z axis. This lowers the symmetry directly from Pm3m to the non-maximal subgroup P4mm, skipping the intermediate space group P4=mmm. In this particular case, P4=mmm ‘has no physical signiﬁcance’ in the sense that it cannot actually occur in this kind of phase transition. However, the intermediate space group P4=mmm can occur in other instances (e.g. in order–disorder transitions) and it has been found among several perovskites. In addition, we are not dealing merely with phase transitions. Intermediate space groups do have signiﬁcance for several reasons: (1) Any step of symmetry reduction reduces the restrictions for every Wyckoff position (either the site symmetry is reduced or the position splits into independent positions, or both happen); every intermediate space group offers new scope for effects with physical signiﬁcance, even if none have yet been observed. (2) Skipping intermediate space groups in the tree of group–subgroup relations reduces the informative value of the symmetry relations. For example, it is no longer directly evident how many translationengleiche and klassengleiche steps are involved; this is useful to decide how many and what kind of twin domains may appear in a phase transition or topotactic reaction (see Sections 1.2.7 and 1.6.6).

In this section, using a few simple examples, we point out the different kinds of group–subgroup relations that are important among related (homeotypic) crystal structures. The crystal structures of two compounds are isotypic if their atoms are arranged in the same way and if they have the same or the enantiomorphic space group. The absolute values of the lattice dimensions and interatomic distances may differ, and small deviations are permitted for non-ﬁxed coordinates of corresponding atoms. The axial ratios and interaxial angles must be similar. Two structures are homeotypic if the conditions for isotypism are relaxed because: (1) their space groups differ, allowing for a group–subgroup relation; (2) the geometric conditions differ (axial ratios, interaxial angles, atomic coordinates); or (3) corresponding atomic positions are occupied by various atomic species (substitution derivatives). (Lima-de-Faria et al., 1990.) 1.6.4.1. Translationengleiche maximal subgroups The space group Pbca of PdS2 is a translationengleiche maximal subgroup of Pa3, the space group of pyrite (FeS2; Ba¨rnighausen, 1980). The threefold axes of the cubic space group have been lost, the index of the symmetry reduction is 3. As shown in Fig. 1.6.4.1, the atomic coordinates have not changed

46

1.6. RELATING CRYSTAL STRUCTURES BY GROUP–SUBGROUP RELATIONS much. However, the two structures differ widely, the c axis of PdS2 being strongly stretched. This is due to the tendency of bivalent palladium towards square-planar coordination (electron conﬁguration d8), whereas the iron atoms in pyrite have octahedral coordination. Strictly speaking, the space groups of FeS2 and PdS2 are not really translationengleiche because of the different lattice parameters. In the strict sense, however, FeS2 at 293.0 and at 293.1 K would not have the same space group either, due to thermal expansion. Such a strict treatment would render it impossible to apply group-theoretical methods in crystal chemistry and physics. Instead, we use the concept of the parent-clamping approximation, i.e. we act as if the translations of the two homeotypic structures were the same (see Section 1.2.7). With the parentclamping approximation we also treat isotypic structures with different lattice parameters (like NaCl and MgO) as if they had the same space group with the same translational lattice. Upon transition from Pa3 to Pbca none of the occupied Wyckoff positions split, but their site symmetries are reduced. Without the symmetry reduction from 3 to 1 the square coordination of the Pd atoms would not be possible. If the positions of the sulfur atoms of pyrite and PdS2 are substituted by two different kinds of atoms in an ordered 1:1 ratio, this enforces further symmetry reductions. The corresponding subgroups may only be subgroups in which the sulfur positions split into symmetry-independent positions. In the chosen examples NiAsS and PtGeSe the symmetry reductions consist of the loss of the inversion centres of Pa3 and Pbca (Fig. 1.6.4.1). Coordinate changes are not necessary, but may occur depending on the site symmetries. In our examples there are minor coordinate changes. To avoid a basis transformation, the nonconventional spacegroup setting Pbc21 has been chosen for PtGeSe; it corresponds to Pca21 after interchange of the axes a and b. Mind the origin shift from PdS2 to PtGeSe; in the conventional description of Pca21 , and therefore also of Pbc21, the origin is situated on one of the 21 axes and thus differs from that of Pbca. The origin shift of 14 ; 0; 0 in the coordinate system of Pbca involves a change of the atomic coordinates by þ 14 ; 0; 0, i.e. with opposite sign. The substitution derivatives NiAsS and PtGeSe can only be connected by the common supergroup P21 =a3. A direct group– subgroup relation from P21 3 to Pbc21 does not exist.

Fig. 1.6.4.1. Ba¨rnighausen tree for the family of structures of pyrite. Coordinates in brackets (not stated normally) refer to symmetryequivalent positions. Lattice parameters are taken from Ramsdell (1925) and Brostigen & Kjeskhus (1969) for pyrite; from Ramsdell (1925), Steger et al. (1974) and Foecker & Jeitschko (2001) for NiAsS; from Grønvold & Røst (1957) for PdS2; and from Entner & Parthe´ (1973) for PtGeSe.

left), rendering possible occupation by atoms of two different species. Figs. 1.6.4.2 and 1.6.4.3 show another peculiarity. P6=mmm has two different klassengleiche maximal subgroups of the same type P63 =mmc with doubled basis vector c. The second one corresponds to CaIn2 (Iandelli, 1964; Wendorff & Roehr, 2005). Here the graphite-like sheets of the AlB2 type have become puckered layers of indium atoms; the indium atoms of adjacent layers have shifted parallel to c and have come close to each other in pairs, so that the result is a network as in lonsdaleite (hexagonal diamond). The alternating shift of the atoms no longer permits the existence of mirror planes in the layers; however, neighbouring layers are mutually mirror-symmetrical. The calcium atoms are on the mirror planes, but no longer on inversion centres. The difference between the two subgroups P63 =mmc consists of the selection of the symmetry operations that are being lost with the doubling of c. The conventional description of the space groups requires an inversion centre to be at the origin of space group P63 =mmc. The position of the origin at an Al atom of the AlB2 type can be kept when the symmetry is reduced to that of ZrBeSi (i.e. Zr at the origin). The symmetry reduction to CaIn2, however, requires an origin shift to the centre of one of the six-membered rings. In the coordinate system of the aristotype that is a shift by 0; 0; 12, as marked in the middle of the group–subgroup arrow in Fig. 1.6.4.3. For the new atomic coordinates (in the coordinate system of the subgroup), the origin shift results in the addition of þ 14 to the z coordinates; in addition, due to the doubling of c, the z coordinates of the aristotype have to be halved. Therefore, the new z coordinate of the In atom is approximately z0 ’ 12 z þ 14 = 1 1 1 2 2 þ 4 ¼ 0:5. It cannot be exactly this value, because then there would have been no symmetry reduction and the space group would still be P6=mmm. The symmetry reduction requires the atom shift to z0 ¼ 0:452. In the relation AlB2 ! ZrBeSi, the site symmetry 6m2 of the boron atoms is retained and the Wyckoff position splits. In the relation AlB2 ! CaIn2 it is the other way; the position does

1.6.4.2. Klassengleiche maximal subgroups Consider two derivatives of the AlB2 type as an example of klassengleiche subgroups (Po¨ttgen & Hoffmann, 2001). AlB2 has a simple hexagonal structure in the space group P6=mmm. In the c direction, aluminium atoms and sheets of boron atoms alternate; the boron-atom sheets are planar, like in graphite (Fig. 1.6.4.2; Hoffmann & Ja¨niche, 1935). The ZrBeSi type has a similar structure (Nielsen & Baenziger, 1953, 1954), but the sheets consist of Be and Si atoms. As a consequence, the inversion centres in the middles of the six-membered rings cannot be retained. This enforces a symmetry reduction to the klassengleiche subgroup P63 =mmc with doubled c vector. The doubling of c is the essential aspect in the symmetry reduction from the AlB2 to the ZrBeSi type. The index is 2: half of all translations are lost, together with half of the inversion centres, half of the symmetry axes perpendicular to c and half of the mirror planes perpendicular to c. The Wyckoff position 2d of the boron atoms of AlB2 splits into the two symmetryindependent positions 2c and 2d of the subgroup (Fig. 1.6.4.3,

47

1. SPACE GROUPS AND THEIR SUBGROUPS

Fig. 1.6.4.2. The structures of AlB2, ZrBeSi and CaIn2. The mirror planes of P63 =mmc perpendicular to c are at z ¼ 14 and z ¼ 34.

Fig. 1.6.4.3. Both hettotypes of the AlB2 type have the same space-group type and a doubled c axis, but the space groups are different due to different origin positions relative to the origin of the aristotype.

not split, the atoms remain symmetry-equivalent, but their site symmetry is reduced to 3m1 and the z coordinate becomes variable. Among klassengleiche subgroups there often exist two and sometimes four or even eight nonconjugate subgroups of the same space-group type with different origin positions. It is important to choose the correct one, with the correct origin shift. All of these subgroups are listed in this volume if they are maximal. Compared to AlB2, ZrBeSi and CaIn2 have so-called ‘superstructures’ (they have additional reﬂections in the X-ray diffraction patterns). Whereas the term superstructure gives only a qualitative, informal outline of the facts, the group-theoretical approach permits a precise treatment.

tetragonal, hexagonal and trigonal space groups, and for cubic space groups only cubes of prime numbers ( 33 ) are possible. For many space groups, not all prime numbers are permitted. The prime number 2 is often excluded, and additional restrictions may apply. In the tables in Parts 2 and 3 of this volume all permitted isomorphic maximal subgroups are listed. Usually, in accordance with the symmetry principle, only small index values are observed (mostly 2 and 3, sometimes 4, less frequently 5, 7 or 9). However, seemingly curious values like 13, 19, 31 or 37 do occur [for examples with indices of 13 and 37, discovered by Ba¨rnighausen, see Mu¨ller (2004)]. A classic example of a relation between isomorphic space groups concerns trirutile (Billiet, 1973). The space group of rutile, P42 =mnm, has an isomorphic subgroup of index 3, but none of index 2. By triplication of c it becomes possible to substitute the titanium-atom positions of rutile by two different kinds of atoms in a ratio of 1:2, as for example in ZnSb2O6 (Fig. 1.6.4.4; Bystro¨m et al., 1941; Ercit et al., 2001). Since the space group P42 =mnm has no isomorphic subgroup of index 2, a ‘dirutile’ with this spacegroup type cannot exist. Note that rutile and trirutile have different space groups of the same space-group type.

1.6.4.3. Isomorphic maximal subgroups Isomorphic subgroups comprise a special category of klassengleiche subgroups. Every space group has an inﬁnity of isomorphic maximal subgroups. The index agrees with the factor by which the unit cell has been enlarged. The indices are prime numbers; squares of prime numbers may occur in the case of

48

1.6. RELATING CRYSTAL STRUCTURES BY GROUP–SUBGROUP RELATIONS

Fig. 1.6.4.4. The group–subgroup relation rutile–trirutile. The twofold rotation axes have been included in the plots of the unit cells, showing that only one third of them are retained upon the symmetry reduction.

1.6.4.4. The space groups of two structures having a common supergroup

same. In the listings of the supergroups the origin shifts are not mentioned either in Volume A or in this volume. Therefore, one has to look up the subgroups of Cmcm and Cmce in this volume and check in which cases the origin shifts coincide. One ﬁnds that the relation Cmce ! C12=c1 requires an origin shift of 14 ; 14 ; 0 (or 14 ; 14 ; 0), while all other relations (Cmcm ! C12=c1, Cmcm ! C2=m11, Cmce ! C2=m11) require no origin shifts. As a consequence, only Cmcm and not Cmce can be the common supergroup. No structure is known that has the space group Cmcm and that can be related to these carbonate structures. Could there be any other structure with even higher symmetry? A supergroup of Cmcm is P 63 =mmc and, in fact, -K2CO3 and -Na2CO3 are high-temperature modiﬁcations that crystallize in this space group. They have the carbonate groups perpendicular to c. The group–subgroup relations are depicted in Fig. 1.6.4.6. In this case there exists a higher-symmetry structure that can be chosen as the common aristotype. In other cases, however, the common supergroup refers to a hypothetical structure; one can speculate why it does not exist or one can try to prepare a compound having this structure. Among the alkali metal carbonates several other modiﬁcations are known which we do not discuss here.

Two crystal structures can be intimately related even when there is no direct group–subgroup relation between their space groups. Instead, there may exist a common supergroup. The structures of NiAsS and PtGeSe, presented in Section 1.6.4.1, offer an example. In this case, the pyrite type corresponds to the common supergroup. Even if there is no known representative, it can be useful to look for a common supergroup. -K2CO3 and -Na2CO3 have similar structures and unit cells (Fig. 1.6.4.5; Jansen & Feldmann, 2000). The planes of the carbonate ions are not aligned exactly perpendicular to c; compared to the perpendicular orientation, in the case of -K2CO3, they are rotated about b by 22.8 and those of -Na2CO3 are rotated about a by 27.3 . There is no group–subgroup relation between the space groups C12=c1 and C2=m11 of the two structures (the nonconventional setting C2=m11 has been chosen for -Na2CO3 to ensure a correspondence between the cells of both structures). Looking for common minimal supergroups of C12=c1 and C2=m11 one can ﬁnd two candidates: Cmcm, No. 63, and Cmce, No. 64. Since the atomic coordinates of -K2CO3 and -Na2CO3 are very similar, any origin shifts in the relations from the common supergroup to C12=c1 as well as C2=m11 must be the

1.6.4.5. Can a structure be related to two aristotypes? Occasionally a crystal structure shows a pronounced distortion compared to a chosen aristotype and another aristotype can be chosen just as well with a comparable distortion. When pressure is exerted upon silicon, it ﬁrst transforms to a modiﬁcation with the -tin structure (Si-II, I41 =amd). Then it is transformed to silicon-XI (McMahon et al., 1994). At even higher pressures it is converted to silicon-V forming a simple hexagonal structure (P6=mmm). The space group of Si-XI, Imma, is a subgroup of both I41 =amd and P6=mmm, and the structure of SiXI can be related to either Si-II or Si-V (Fig. 1.6.4.7). Assuming no distortions, the calculated coordinates of a silicon atom of SiXI would be 0; 14; 0:125 when derived from Si-II, and 0; 14; 0:0 when derived from Si-V. The actual coordinates are halfway between. The metric deviations of the lattices are small; taking into account the basis transformations given in Fig. 1.6.4.7, the expected lattice parameters for Si-XI, calculated from those of Sipﬃﬃﬃ V, would be aXI = aV 3 = 441.5 pm, bXI = 2cV = 476.6 pm and cXI = aV = 254.9 pm.

Fig. 1.6.4.5. The unit cells of -K2CO3 and -Na2CO3. The angles of tilt of the CO2 3 ions are referred relative to a plane perpendicular to c.

49

1. SPACE GROUPS AND THEIR SUBGROUPS

Fig. 1.6.4.6. Group–subgroup relations among some modiﬁcations of the alkali metal carbonates. Lattice parameters for - and -Na2CO3 are taken from Swainson et al. (1995) and those for - and -K2CO3 are taken from Becht & Struikmans (1976).

physical or chemical evidence. See also the statements on this matter in Section 1.6.4.6, at the end of Section 1.6.6 and in Section 1.6.7.

These phase transitions of silicon involve small atomic displacements and small volume changes. The lattice parameter c of the hexagonal structure is approximately half the value of a of tetragonal Si-II. There are two separate experimentally observable phase transitions. In a certain pressure range, the whole crystal actually consists of stable Si-XI; it is not just a hypothetical intermediate. Taking all these facts together, a group-theoretical relation between Si-II and Si-V exists via the common subgroup of Si-XI. Usually, however, there is no point in relating two structures via a common subgroup. Two space groups always have an inﬁnity of common subgroups, and it may be easy to set up relations via common subgroups in a purely formal manner; this can be quite meaningless unless it is based on well founded

1.6.4.6. Treating voids like atoms To comprehend the huge amount of known crystal-structure types, chemists have very successfully developed quite a few concepts. One of them is the widespread description of structures as packings of spheres with occupied interstices. Group–subgroup relations can help to rationalize this. This requires that unoccupied interstices be treated like atoms, i.e. that the occupation of voids is treated like a substitution of ‘zero atoms’ by real atoms. The crystal structure of FeF3 (VF3 type) can be derived from the ReO3 type by a mutual rotation of the coordination octahedra about the threefold rotation axes parallel to one of the four diagonals of the cubic unit cell of the space group Pm3m (Fig. 1.6.4.8). This involves a symmetry reduction by two steps to a rhombohedral hettotype with the space group R3c (Fig. 1.6.4.9, left). FeF3 can also be described as a hexagonal closest packing of ﬂuorine atoms in which one third of the octahedral voids have been occupied by Fe atoms (Fig. 1.6.4.9, right). This is a more

Fig. 1.6.4.8. The connected coordination octahedra in ReO3 and FeF3 (VF3 type). Light, medium and dark grey refer to octahedron centres at z ¼ 0, z ¼ 13 and z ¼ 23 , respectively (hexagonal setting).

Fig. 1.6.4.7. The structure of the high-pressure modiﬁcation Si-XI is related to the structures of both Si-II and Si-V.

50

1.6. RELATING CRYSTAL STRUCTURES BY GROUP–SUBGROUP RELATIONS descriptive and more formal point of view, because the number of atoms and the kind of linkage between them are altered. The calculated ideal x coordinates of the ﬂuorine atoms of FeF3, assuming no distortions, are x = 0.5 when derived from ReO3 and x = 0.333 when derived from the packing of spheres. The actual value at ambient pressure is x = 0.412, i.e. halfway in between. When FeF3 is subjected to high pressures, the coordination octahedra experience a mutual rotation which causes the structure to come closer to a hexagonal closest packing of ﬂuorine atoms; at 9 GPa this is a nearlypﬃﬃundistorted closest ﬃ packing with the x parameter and the c=ða 3Þ ratio close to the ideal values of 0.333 and 1.633 (Table 1.6.4.1). Therefore, at high pressures the hexagonal closest packing of spheres could be regarded as the more appropriate aristotype. This, however, is only true from the descriptive point of view, i.e. if one accepts that the positions of voids can be treated like atoms. If one studies the mutual rotation of the octahedra in FeF3 or if phase transitions from the ReO3 type to the VF3 type are of interest, there is no point in allowing a change of the occupation of the octahedra; in this case only the group–subgroup relations given in the left part of Fig. 1.6.4.9 should be considered. However, if one wants to derive structures from a hexagonal closest packing of spheres and point out similarities among them, the group–subgroup relations given in the right part of Fig. 1.6.4.9 are useful. For example, the occupation of the octahedral voids in the Wyckoff position 12c instead of 6b of R3c (Fig. 1.6.4.9, lower right) results in the structure of corundum (-Al2O3); this shows that both the VF3 type and corundum have the same kind of packing of their anions, even though the linkage of their coordination octahedra is quite different and the number of occupied positions does not coincide. It should also be kept in mind that the occupation of voids in a packing of atoms can actually be achieved in certain cases. Examples are the intercalation compounds and the large number

Table 1.6.4.1. Crystal data for FeF3 at different pressures The observed lattice parameters a and c, the x coordinates of the F atoms and the angles of rotation of the coordination octahedra (0 = ReO3 type, 30 = hexagonal closest packing) are given (Sowa & Ahsbahs, 1998; Jørgenson & Smith, 2006). pﬃﬃﬃ Pressure /GPa a /pm c /pm c=ða 3Þ x Angle / 104 1.54 4.01 6.42 9.0

520.5 503.6 484.7 476 469.5

1332.1 1340.7 1348.3 1348 1349

1.48 1.54 1.61 1.64 1.66

0.412 0.385 0.357 0.345 0.335

17.0 21.7 26.4 28.2 29.8

of metal hydrides MHx that can be prepared by diffusion of hydrogen into the metals. The hydrides often keep the closest packed structures of the metals while the hydrogen atoms occupy the tetrahedral or octahedral voids. It is not recommended to plot group–subgroup relations in which FeF3 is depicted as a common subgroup of both the ReO3 type and the hexagonal closest packing of spheres. This would mix up two quite different points of view. 1.6.4.7. Large families of structures. Prediction of crystal-structure types Large trees can be constructed using the modular way to put together Ba¨rnighausen trees, as set forth in Fig. 1.6.3.1 and in the preceding sections. Headed by an aristotype, they show structural relations among many different crystal structures belonging to a family of structures. As an example, Fig. 1.6.4.10 shows structures that can be derived from the ReO3 type (Bock & Mu¨ller, 2002b). Many other trees of this kind have been set up, for example hettotypes of perovskite (Ba¨rnighausen, 1975, 1980; Bock & Mu¨ller, 2002a), rutile (Baur, 1994, 2007; Meyer, 1981), CaF2 (Meyer, 1981), hexagonal closest packed structures (Mu¨ller, 1998), AlB2 (Po¨ttgen & Hoffmann, 2001), zeolites (Baur & Fischer, 2000, 2002, 2006) and tetraphenylphosphonium salts (Mu¨ller, 1980, 2004). In the left branch of Fig. 1.6.4.10 only one compound is mentioned, WO3. This is an example showing the symmetry relations among different polymorphic forms of a compound. The right branch of the tree shows the relations for substitution derivatives, including distortions caused by the Jahn–Teller effect, hydrogen bonds and different relative sizes of the atoms. In addition to showing relations between known structure types, one can also ﬁnd subgroups of an aristotype for which no structures are known. This can be exploited in a systematic manner to search for new structural possibilities, i.e. one can predict crystal-structure types. For this purpose, one starts from an aristotype in conjunction with a structural principle and certain additional restrictions. For example, the aristotype can be a hexagonal closest packing of spheres and the structural principle can be the partial occupation of octahedral voids in this packing. Additional restrictions could be things like the chemical composition, a given molecular conﬁguration or a maximal size of the unit cell. Of course, one can only ﬁnd such structure types that meet these starting conditions. For every space group appearing in the Ba¨rnighausen tree, one can calculate how many different structure types are possible for a given chemical composition (McLarnan, 1981a,b,c; Mu¨ller, 1992). Examples of studies of structural possibilities include compounds AX3, AX6, AaBbX6 (with the X atoms forming the packing of spheres and atoms A and B occupying the octahedral voids; a þ b < 6; Mu¨ller, 1998), molecular compounds (MX5)2 (Mu¨ller, 1978), chain structures MX4 (Mu¨ller, 1981) and MX5

Fig. 1.6.4.9. Derivation of the FeF3 structure either from the ReO3 type or from the hexagonal closest packing of spheres. The coordinates for FeF3 given in the boxes are ideal values calculated from the aristotypes assuming no distortions. A y coordinate given as x means y = x. The Schottky symbol & designates an unoccupied octahedral void.

51

1. SPACE GROUPS AND THEIR SUBGROUPS

Fig. 1.6.4.10. Ba¨rnighausen tree of hettotypes of the ReO3 type. For the atomic parameters and other crystallographic data see Bock & Mu¨ller (2002b). F32=c and F3 are nonconventional face-centred settings of R32=c and R 3, respectively, with rhombohedral-axes setting and nearly cubic metric, F1 is a nearly cubic setting of P1. Every space-group symbol corresponds to one space group (not space-group type) belonging to a speciﬁc crystal structure. Note that the vertical distances between space-group symbols are proportional to the logarithms of the indices.

(Mu¨ller, 1986), wurtzite derivatives (Baur & McLarnan, 1982) and NaCl derivatives with doubled cell (Sens & Mu¨ller, 2003).

of a maximal subgroup from those of the preceding space group. If there is a sequence of several transformations, the overall changes can be calculated by multiplication of the 4 4 matrices. Let P1 ; P2 ; . . . be the 4 4 matrices expressing the basis vector and origin changes of several consecutive cell transformations. 1 P1 1 ; P2 ; . . . are the corresponding inverse matrices. Let a, b, c be the starting (old) basis vectors and a0 , b0 , c0 be the (new) basis vectors after the consecutive transformations. Let x and x0 be the augmented columns of the atomic coordinates before and after the transformations. Then the following relations hold:

1.6.5. Handling cell transformations It is important to keep track of the coordinate transformations in a sequence of group–subgroup relations. A Ba¨rnighausen tree can only be correct if every atomic position of every hettotype can be derived from the corresponding positions of the aristotype. The mathematical tools are described in Sections 1.2.2.3, 1.2.2.4 and 1.2.2.7 of this volume and in Part 5 of Volume A. A basis transformation and an origin shift are mentioned in the middle of a group–subgroup arrow. This is a shorthand notation for the matrix–column pair (Seitz symbol; Section 1.2.2.3) or the 4 4 matrix (augmented matrix, Section 1.2.2.4) to be used to calculate the basis vectors, origin shift and coordinates

ða0 ; b0 ; c0 ; pÞ ¼ ða; b; c; oÞ P1 P2 ; . . .

and

1 x0 ¼ . . . P1 2 P1 x :

p is the vector of the origin shift, i.e. its components are the coordinates of the origin of the new cell expressed in the coor-

52

1.6. RELATING CRYSTAL STRUCTURES BY GROUP–SUBGROUP RELATIONS dinate system of the old cell; o is a zero vector. Note that the inverse matrices have to be multiplied in the reverse order. Example 1.6.5.1 Take a transformation from a cubic to a rhombohedral unit cell (hexagonal setting) combined with an origin shift of p1 ¼ ð 14; 14; 14 Þ (in the cubic coordinate system), followed by transformation to a monoclinic cell with a second origin shift of p2 ¼ ð12; 12; 0Þ (in the hexagonal coordinate system). From Fig. 1.6.5.1 (left) we can deduce ahx ¼ acb bcb ;

bhx ¼ bcb ccb ;

chx ¼ acb þ bcb þ ccb ;

which in matrix notation is

Fig. 1.6.5.1. Relative orientations of a cubic to a rhombohedral cell (hexagonal setting) (left) and of a rhombohedral cell (hexagonal setting) to a monoclinic cell (the monoclinic cell has an acute angle ).

ðahx ; bhx ; chx Þ ¼ ðacb ; bcb ; ccb ÞP 1 0

1

0

B ¼ ðacb ; bcb ; ccb Þ@ 1 0

1 1

1

1

C 1 A: 1

In the fourth column of the 4 4 matrix P1 we include the components of the origin shift, p1 ¼ ð 14; 14; 14 Þ: 0 P1 ¼

P1 o

p1 1

1

B B 1 B ¼B B 0 @

0

1

1

1

1

1

0

0

0

1 4

¼

13 13 1 3 1 3

0 @ P 1 2 p2 ¼

P 1 1 p1 ¼

Therefore,

P1 1

3 B @ 13 1 3

P1 1 ¼

P 1 1 o

2 3

0

1

C 23 A:

13

1 3

23

1 3

1 3

0

0

0

1 2 12

0 1 0

1

1 2 12

0 1

C 0A

0

0

3

1

10 1 1 0 1 1 2 1 4 0 [email protected] 12 A ¼ @ 14 A: 3 0 0

10

B B 1 P1 P 2 ¼ B B 0 @

0

1

1 1

1 1

1 4 1 4 1 4

0

0

0

1

0 1

1 0

1

0

14 1 C C 4 C ; 3 C 4 A

0

0

0

1

2 B B 1 ¼B B 1 @

1 3

13

1

For the basis vectors of the two consecutive transformations we calculate:

0

2

CB CB 1 CB CB 0 [email protected] 0 1

12

1 0

2 3 1 3 1 3

0

0

1

0

1

C 0C C C: 1C 4A 1

1

C 12 C C 0C A

1

ðamn ; bmn ; cmn ; pÞ ¼ ðacb ; bcb ; ccb ; oÞP1 P2 0 2 B B 1 B ¼ ðacb ; bcb ; ccb ; oÞB B 1 @ 0

is B B1 P 1 p B3 1 1 ¼B1 1 B3 @ 0

0

0

10 1 1 0 1 0 13 13 4 CB C B C 1 23 [email protected] 14 A ¼ @ 0 A: 3 1 1 1 14 3 3 4

0

21 3 1C 3A 1 3

ðamn ; bmn ; cmn ÞP 1 2

B ¼ ðamn ; bmn ; cmn Þ@

The column part (fourth column) of the inverse matrix P1 1 is P 1 1 p1 (cf. equation 1.2.2.6): 02

0

C C C C: 1C 4A 1

ccb ¼ 13 ahx 23 bhx þ 13 chx ;

3 B ðahx ; bhx ; chx Þ@ 13 1 3

0

1 4

bcb ¼ 13 ahx þ 13 bhx þ 13 chx ;

ðacb ; bcb ; ccb Þ ¼ ðahx ; bhx ; chx Þ P 1 1 02

0

ðahx ; bhx ; chx Þ ¼

1

The inverse matrix P 1 1 can be calculated by matrix inversion, but it is more straightforward to deduce it from Fig. 1.6.5.1; it is the matrix that converts the hexagonal basis vectors back to the cubic basis vectors: acb ¼ 23 ahx þ 13 bhx þ 13 chx ;

ðamn ; bmn ; cmn Þ ¼ ðahx ; bhx ; chx Þ P 2 0 2 B ¼ ðahx ; bhx ; chx Þ@ 1

14

0

1

1

0

1 4

1

0

3 4

0

0

1

1 C C C C; C A

which corresponds to amn ¼ 2acb bcb ccb ;

bmn ¼ bcb ccb ;

cmn ¼ acb

and an origin shift of p ¼ ð14; 14; 34 Þ in the cubic coordinate system. The corresponding (cubic to monoclinic) coordinate transformations result from

Similarly, for the second (hexagonal to monoclinic) transformation, we deduce the matrices P 2, P 1 2 and the column part of 1 P1 , P p , from the right image of Fig. 1.6.5.1: 2 2 2

53

1. SPACE GROUPS AND THEIR SUBGROUPS 0

1 xmn ¼ P1 2 P1 x cb

xmn

1

0

1 2

0

1

B C B B ymn C B 12 1 0 B C B Bz C ¼ B B mn C B 0 0 3 A @ @ 0 0 0 1 0 1 0 2 12 B 1 B0 12 2 B ¼B B1 1 1 @ 0

0

0

1 4

10

2 3

CB 1 CB CB 3 CB 0 CB 13 [email protected] 1 0 1 10

13

13

1 3

23

1 3

1 3

0 1

0

1 4

0

10

xcb

group–subgroup relation) we can expect twins with three kinds of domains, having three different orientations. An isomorphic subgroup of index 5 (i5 relation), since it is a klassengleiche symmetry reduction, will entail ﬁve kinds of antiphase domains. If the symmetry reduction includes several steps (in a chain of several maximal subgroups), the domain structure will become more complicated. With two t2 group–subgroup relations, we can expect twins of twins with two kinds of domains each. In the example of Fig. 1.2.7.1, we have a sequence of one t2 and one k2 step; therefore, as explained in a more rigorous manner in Example 1.2.7.3.4, we can expect two kinds of twin domains having two different orientations and, in addition, two kinds of antiphase domains. The actual number of observable domain kinds may be less than expected if a domain kind is not formed during nucleation. This can be controlled by the nucleation conditions; for example, an external electric ﬁeld can suppress the formation of more than one kind of differently oriented ferroelectric domains. In the physical literature, phase transitions between translationengleiche space groups are sometimes called ferroic transitions, those between klassengleiche space groups are non-ferroic. As mentioned at the end of Section 1.6.4.5, a common subgroup of two space groups usually is not suitable for relating two crystal structures. Just so, a common crystallographic subgroup is unﬁt to explain the mechanism for a reconstructive, ﬁrst-order phase transition. Such mechanisms have been proposed repeatedly (see, e.g., Capillas et al., 2007, and references therein). Like in a second-order transition, continuous atomic movements have been assumed, leading ﬁrst to a structure with the reduced symmetry of a common subgroup, followed by a symmetry increase to the space group of the second phase. However, a ﬁrst-order transition always shows hysteresis, which means that the initial and the ﬁnal phase coexist during the transition and that there is a phase boundary between them. The transition begins at a nucleation site, followed by the growth of the nucleus (see, e.g., Binder, 1987; Chaitkin & Lubensky, 1995; Chandra Shekar & Gavinda Rajan, 2001; Christian, 2002; Doherty, 1996; Gunton, 1984; Herbstein, 2006; Jena & Chaturvedi, 1992; Wayman & Bhadeshia, 1996). The reconstruction of the structure and the necessary atomic motions occur at the phase boundary between the receding old phase and the growing new phase. Any intermediate state is restricted to this interface between the two phases. It is impossible to assign a threedimensional space group to an interface. In addition, while the interface advances through the crystal, the atoms at the interface do not linger in a static state. A space group can only be assigned to a static state (time-dependent phenomena are not covered by space-group theory).

1

CB C By C 0C CB cb C C Bz C B C 14 C [email protected] cb A 1 1

xcb B C C 1 CB y C 4 CB cb C C; CB C B 34 C [email protected] zcb A 2

1

1

which is the same as xmn ¼ 12 ycb 12 zcb þ 12;

ymn ¼ 12 ycb 12 zcb þ 14;

zmn ¼ xcb þ ycb þ zcb 34:

1.6.6. Comments concerning phase transitions and twin domains When a compound forms several polymorphic forms, a Ba¨rnighausen tree can serve to show whether second-order phase transitions are feasible between them and what kinds of twin domains may be formed during such a transition. The possible kinds of domains can also be deduced for ﬁrst-order phase transitions involving a group–subgroup relation and for topotactic reactions. Second-order (continuous) phase transitions are only possible if there is a direct (not necessarily maximal) group–subgroup relation between the space groups of the two phases. If there is no such relation, for example, if two polymorphic forms can be related only by a common supergroup, the phase transformation can only be of ﬁrst order. The condition of a direct group– subgroup relation is a necessary but not a sufﬁcient condition; the phase transition can still be of ﬁrst order. The domain structure of crystalline phases that often results at solid-state phase transitions and during topotactic reactions can be transparently interpreted with the aid of symmetry considerations (Section 1.2.7; Ba¨rnighausen, 1980; Janovec & Prˇı´vratska´, 2003; van Tendeloo & Amelinckx, 1974; Wondratschek & Jeitschko, 1976). A domain structure is the result of nucleation and growth processes. The orientational relations between the phases before and after the transformation, as a rule, are not the result of a homogeneous process involving a simultaneous motion of the atoms in a single crystal. The crystalline matrix of the substrate rather governs the preferred orientation adopted by the nuclei that are formed during the course of the nucleation process. The crystallites that result from the subsequent growth of the nuclei maintain their orientations. Under these circumstances, aspect 3 of the symmetry principle, as stated in Section 1.6.2, is fully effective. A phase transition that is connected with a symmetry reduction will result in new phases that consist of (i) twin domains if the formed phase belongs to a crystal class with reduced symmetry, or (ii) antiphase domains (translational domains) if translational symmetry is lost. The total number of domains, of course, depends on the number of nucleation sites. The number of different domain kinds, however, is ruled by the index of the symmetry reduction. At a translationengleiche symmetry reduction of index 3 (t3

1.6.7. Exercising care in the use of group–subgroup relations Using the tables of this volume or from other sources or using computer programs as offered by the Bilbao Crystallographic Server (see Chapter 1.7), it may be easy to search for group– subgroup relations between space groups of crystal structures. This should only be done bearing in mind and explicitly stating a crystallographic, physical or chemical context. It is senseless to construct relations in a purely formal manner, ‘just for fun’, without a sound crystal-chemical or physical foundation. There exist no clear criteria to determine what is a ‘sound foundation’, but crystallographic, chemical and physical common sense and knowledge should always be taken into consideration.

54

1.6. RELATING CRYSTAL STRUCTURES BY GROUP–SUBGROUP RELATIONS In addition, by experience we know that setting up trees of group–subgroup relations is susceptible to pitfalls. Some sources of errors are: not taking into account necessary origin shifts; wrong origin shifts; wrong basis and/or coordinate transformations; unnecessary basis transformations just for the sake of clinging on to standard space-group settings; lack of distinction between space groups and space-group types; lack of keeping track of the development of the atomic positions from group to subgroup; using different space-group or coordinate settings for like structures. If the group–subgroup relations are correct, but origin shifts or basis transformations have not been stated, this can cause subsequent errors and misunderstandings.

Capillas, C., Perez-Mato, J. M. & Aroyo, M. I. (2007). Maximal symmetry transition paths for reconstructive phase transitions. J. Phys. Condens. Matter, 19, 275203. Chaitkin, P. M. & Lubensky, T. C. (1995). Principles of Condensed Matter Physics (reprints 2000, 2003). Cambridge University Press. Chandra Shekar, N. V. & Gavinda Rajan, K. (2001). Kinetics of pressure induced structural phase transitions – a review. Bull. Mater. Sci. 24, 1– 21. Chapuis, G. C. (1992). Symmetry relationships between crystal structures and their practical applications. Modern Perspectives in Inorganic Chemistry, edited by E. Parthe´, pp. 1–16. Dordrecht: Kluwer Academic Publishers. Christian, J. W. (2002). The Theory of Transformations in Metals and Alloys, 3rd ed. Oxford: Pergamon. Cracknell, A. P. (1975). Group Theory in Solid State Physics. New York: Taylor and Francis Ltd/Pergamon. Di Costanzo, L., Forneris, F., Geremia, S. & Randaccio, L. (2003). Phasing protein structures using the group–subgroup relation. Acta Cryst. D59, 1435–1439. Doherty, R. D. (1996). Diffusive phase transformations. Physical Metallurgy, ch. 15. Amsterdam: Elsevier. Entner, P. & Parthe´, E. (1973). PtGeSe with cobaltite structure, a ternary variant of the pyrite structure. Acta Cryst. B29, 1557–1560. Ercit, T. S., Foord, E. E. & Fitzpatrick, J. J. (2001). Ordon˜ezite from the Theodoso Soto mine, Mexico: new data and structure reﬁnement. Can. Mineral. 40, 1207–1210. Flack, H. D. (2003). Chiral and achiral structures. Helv. Chim. Acta, 86, 905–921. Foecker, A. J. & Jeitschko, W. (2001). The atomic order of the pnictogen and chalcogen atoms in equiatomic ternary compounds TPnCh (T = Ni, Pd; Pn = P, As, Sb; Ch = S, Se, Te). J. Solid State Chem. 162, 69–78. ¨ ber die Oxidation von Giovanoli, D. & Leuenberger, U. (1969). U Manganoxidhydroxid. Helv. Chim. Acta, 52, 2333–2347. Grønvold, F. & Røst, E. (1957). The crystal structure of PdSe2 and PdS2 . Acta. Cryst. 10, 329–331. Gunton, J. D. (1984). The dynamics of random interfaces in phase transitions. J. Stat. Phys. 34, 1019–1037. Herbstein, F. H. (2006). On the mechanism of some ﬁrst-order enantiotropic solid-state phase transitions: from Simon through Ubbelohde to Mnyukh. Acta Cryst. B62, 341–383. Hoffmann, W. & Ja¨niche, W. (1935). Der Strukturtyp von AlB2. Naturwissenschaften, 23, 851. Howard, C. J. & Stokes, H. T. (2005). Structures and phase transitions in perovskites – a group-theoretical approach. Acta Cryst. A61, 93–111. Iandelli, A. (1964). MX2-Verbindungen der Erdalkali- und seltenen Erdmetalle mit Gallium, Indium und Thallium. Z. Anorg. Allg. Chem. 330, 221–232. Igartua, J. M., Aroyo, M. I. & Perez-Mato, J. M. (1996). Systematic search of materials with high-temperature structural phase transitions: Application to space group P212121. Phys. Rev. B, 54, 12744–12752. International Tables for Crystallography (2005). Vol. A, Space-Group Symmetry, edited by Th. Hahn, corrected reprint of 5th ed. Dordrecht: Kluwer Academic Publishers. Izyumov, Y. A. & Syromyatnikov, V. N. (1990). Phase Transitions and Crystal Symmetry. Dordrecht: Kluwer Academic Publishers. Janovec, V. & Prˇı´vratska´, J. (2003). Domain structures. International Tables for Crystallography, Vol. D, Physical Properties of Crystals, edited by A. Authier, ch. 3.4. Dordrecht: Kluwer Academic Publishers. Jansen, M. & Feldmann, C. (2000). Strukturverwandtschaften zwischen cis-Natriumhyponitrit und den Alkalimetallcarbonaten M2CO3 dargestellt durch Gruppe-Untergruppe Beziehungen. Z. Kristallogr. 215, 343–345. Jena, A. K. & Chaturvedi, M. C. (1992). Phase Transformations in Materials. Englewood Cliffs: Prentice Hall. Jørgenson, J.-E. & Smith, R. I. (2006). On the compression mechanism of FeF3. Acta Cryst. B62, 987–992. Koch, E. & Fischer, W. (2006). Normalizers of space groups: a useful tool in crystal description, comparison and determination. Z. Kristallogr. 221, 1–14. Landau, L. D. & Lifshitz, E. M. (1980). Statistical Physics, 3rd ed., Part 1, pp. 459–471. London: Pergamon. (Russian: Statisticheskaya Fizika, chast 1. Moskva: Nauka, 1976; German: Lehrbuch der theoretischen Physik, 6. Auﬂ., Bd. 5, Teil 1, S. 436–447. Berlin: Akademie-Verlag, 1984.)

References Aizu, K. (1970). Possible species of ferromagnetic, ferroelectric, and ferroelastic crystals. Phys Rev. B, 2, 754–772. Aroyo, M. I. & Perez-Mato, J. M. (1998). Symmetry-mode analysis of displacive phase transitions using International Tables for Crystallography. Acta Cryst. A54, 19–30. Ba¨rnighausen, H. (1975). Group–subgroup relations between space groups as an ordering principle in crystal chemistry: the ‘family tree’ of perovskite-like structures. Acta Cryst. A31, part S3, 01.1–9. Ba¨rnighausen, H. (1980). Group–subgroup relations between space groups: a useful tool in crystal chemistry. MATCH Commun. Math. Chem. 9, 139–175. Baur, W. H. (1994). Rutile type derivatives. Z. Kristallogr. 209, 143–150. Baur, W. H. (2007). The rutile type and its derivatives. Crystallogr. Rev. 13, 65–113. Baur, W. H. & Fischer, R. X. (2000, 2002, 2006). Landolt–Bo¨rnstein, Numerical data and functional relationships in science and technology, New Series. Group IV, Vol. 14, Zeolite type crystal structures and their chemistry. Berlin: Springer. Baur, W. H. & McLarnan, T. J. (1982). Observed wurtzite derivatives and related tetrahedral structures. J. Solid State Chem. 42, 300–321. Becht, H. Y. & Struikmans, R. (1976). A monoclinic high-temperature modiﬁcation of potassium carbonate. Acta Cryst. B32, 3344–3346. Bernal, J. D. (1938). Conduction in solids and diffusion and chemical change in solids. Geometrical factors in reactions involving solids. Trans. Faraday Soc. 34, 834–839. Bernal, J. D. & Mackay, A. L. (1965). Topotaxy. Tschermaks mineralog. petrogr. Mitt., Reihe 3 (now Mineral. Petrol.), 10, 331–340. Billiet, Y. (1973). Les sous-groupes isosymboliques des groupes spatiaux. Bull. Soc. Fr. Mine´ral. Cristallogr. 96, 327–334. Binder, K. (1987). Theory of ﬁrst-order phase transitions. Rep. Prog. Phys. 50, 783–859. Birman, J. L. (1966a). Full group and subgroup methods in crystal physics. Phys. Rev. 150, 771–782. Birman, J. L. (1966b). Simpliﬁed theory of symmetry change in secondorder phase transitions: application to V3Si. Phys. Rev. Lett. 17, 1216– 1219. Birman, J. L. (1978). Group-theoretical methods in physics, edited by P. Kramer & A. Rieckers, pp. 203–222. New York: Springer. Bock, O. & Mu¨ller, U. (2002a). Symmetrieverwandtschaften bei Varianten des Perowskit-Typs. Acta Cryst. B58, 594–606. Bock, O. & Mu¨ller, U. (2002b). Symmetrieverwandtschaften bei Varianten des ReO3 -Typs. Z. Anorg. Allg. Chem. 628, 987–992. Brostigen, G. & Kjeskhus, G. A. (1969). Redetermined crystal structure of FeS2 (pyrite). Acta Chem. Scand. 23, 2186–2188. Brunner, G. O. (1971). An unconventional view of the ‘closest sphere packings’. Acta Cryst. A27, 388–390. Buerger, M. J. (1947). Derivative crystal structures. J. Chem.Phys. 15, 1– 16. Buerger, M. J. (1951). Phase transformations in solids, ch. 6. New York: Wiley. Bystro¨m, A., Ho¨k, B. & Mason, B. (1941). The crystal structure of zinc metaantimonate and similar compounds. Ark. Kemi Mineral. Geol. 15B, 1–8.

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1. SPACE GROUPS AND THEIR SUBGROUPS Lima-de-Faria, J., Hellner, E., Liebau, F., Makovicky, E. & Parthe´, E. (1990). Nomenclature of inorganic structure types. Report of the International Union of Crystallography Commission on Crystallographic Nomenclature Subcommittee on the Nomenclature of Inorganic Structure Types. Acta Cryst. A46, 1–11. Lotgering, F. K. (1959). Topotactical reactions with ferrimagnetic oxides having hexagonal crystal structures–I. J. Inorg. Nucl. Chem. 9, 113–123. Lyubarskii, G. Ya. (1960). Group Theory and its Applications in Physics. London: Pergamon. (Russian: Teoriya grupp i ee primenenie v ﬁzike. Moskva: Gostekhnizdat, 1957; German: Anwendungen der Gruppentheorie in der Physik. Berlin: Deutscher Verlag der Wissenschaften, 1962.) McLarnan, T. J. (1981a). Mathematical tools for counting polytypes. Z. Kristallogr. 155, 227–245. McLarnan, T. J. (1981b). The numbers of polytypes in close packings and related structures. Z. Kristallogr. 155, 269–291. McLarnan, T. J. (1981c). The combinatorics of cation-deﬁcient closepacked structures. J. Solid State Chem. 26, 235–244. McMahon, M. I., Nelmes, R. J., Wright, N. G. & Allan, D. R. (1994). Pressure dependence of the Imma phase of silicon. Phys. Rev. B, 50, 739–743. Megaw, H. D. (1973). Crystal Structures: A Working Approach. Philadelphia: Saunders. Meyer, A. (1981). Symmmetriebeziehungen zwischen Kristallstrukturen des Formeltyps AX2 , ABX4 und AB2X6 sowie deren Ordnungs- und Leerstellenvarianten. Dissertation, Universita¨t Karlsruhe. Mu¨ller, U. (1978). Strukturmo¨glichkeiten fu¨r Pentahalogenide mit Doppeloktaeder-Moleku¨len (MX5)2 bei dichtester Packung der Halogenatome. Acta Cryst. A34, 256–267. Mu¨ller, U. (1980). Strukturverwandtschaften unter den EPhþ 4 -Salzen. Acta Cryst. B36, 1075–1081. Mu¨ller, U. (1981). MX4 -Ketten aus kantenverknu¨pften Oktaedern: mo¨gliche Kettenkonﬁgurationen und mo¨gliche Kristallstrukturen. Acta Cryst. B37, 532–545. Mu¨ller, U. (1986). MX5 -Ketten aus eckenverknu¨pften Oktaedern. Mo¨gliche Kettenkonﬁgurationen und mo¨gliche Kristallstrukturen bei dichtester Packung der X-Atome. Acta Cryst. B42, 557–564. Mu¨ller, U. (1992). Berechnung der Anzahl mo¨glicher Strukturtypen fu¨r Verbindungen mit dichtest gepackter Anionenteilstruktur. I. Das Rechenverfahren. Acta Cryst. B48, 172–178. Mu¨ller, U. (1998). Strukturverwandtschaften zwischen trigonalen Verbindungen mit hexagonal-dichtester Anionenteilstruktur und besetzten Oktaederlu¨cken. Z. Anorg. Allg. Chem. 624, 529–532.

Mu¨ller, U. (2004). Kristallographische Gruppe-Untergruppe-Beziehungen und ihre Anwendung in der Kristallchemie. Z. Anorg. Allg. Chem. 630, 1519–1537. Mu¨ller, U. (2006). Inorganic Structural Chemistry, 2nd ed., pp. 212–225. Chichester: Wiley. (German: Anorganische Strukturchemie, 6. Auﬂ., 2008, S. 308–327. Wiesbaden: Teubner.) Nielsen, J. W. & Baenziger, N. C. (1953). The crystal structures of ZrBeSi and ZrBe2 . US Atom. Energy Comm. Rep. pp. 1–5. Nielsen, J. W. & Baenziger, N. C. (1954). The crystal structures of ZrBeSi and ZrBe2 . Acta Cryst. 7, 132–133. Po¨ttgen, R. & Hoffmann, R.-D. (2001). AlB2 -related intermetallic compounds – a comprehensive view based on a group–subgroup scheme. Z. Kristallogr. 216, 127–145. Ramsdell, L. D. (1925). The crystal structure of some metallic sulﬁdes. Am. Mineral. 10, 281–304. Salje, E. K. H. (1990). Phase transitions in ferroelastic and co-elastic crystals. Cambridge University Press. Sens, I. & Mu¨ller, U. (2003). Die Zahl der Substitutions- und Leerstellenvarianten des NaCl-Typs bei verdoppleter Elementarzelle (a, b, 2c). Z. Anorg. Allg. Chem. 629, 487–492. Sowa, H. & Ahsbahs, H. (1998). Pressure-induced octahedron strain in VF3 type compounds. Acta Cryst. B54, 578–584. Steger, J. J., Nahigian, H., Arnott, R. J. & Wold, A. (1974). Preparation and characterization of the solid solution series Co1xNixS (0 < x < 1). J. Solid State Chem. 11, 53–59. Stokes, H. T. & Hatch, D. M. (1988). Isotropy Subgroups of the 230 Crystallographic Space Groups. Singapore: World Scientiﬁc. Swainson, I. P., Cove, M. T. & Harris, M. J. (1995). Neutron powder diffraction study of the ferroelastic phase transition and lattice melting of sodium carbonate. J. Phys. Condens. Matter, 7, 4395–4417. Tendeloo, G. van & Amelinckx, S. (1974). Group-theoretical considerations concerning domain formation in ordered alloys. Acta Cryst. A30, 431–440. Tole´dano, J.-C. & Tole´dano, P. (1987). The Landau Theory of Phase Transitions. Singapore: World Scientiﬁc. Wayman, C. M. & Bhadeshia, H. K. (1996). Nondiffusive phase transformations. Physical Metallurgy, 4th ed., edited by R. W. Cahn & P. Haasen, ch. 16. Amsterdam: Elsevier. Wendorff, M. & Roehr, C. (2005). Reaktionen von Zink- und Cadmiumhalogeniden mit Tris(trimethylsilyl)phosphan und Tris(trimethylsilyl)arsan. Z. Anorg. Allg. Chem. 631, 338–349. Wondratschek, H. & Jeitschko, W. (1976). Twin domains and antiphase domains. Acta Cryst. A32, 664–666.

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references

International Tables for Crystallography (2011). Vol. A1, Chapter 1.7, pp. 57–69.

1.7. The Bilbao Crystallographic Server By Mois I. Aroyo, J. Manuel Perez-Mato, Cesar Capillas and Hans Wondratschek

1.7.1. Introduction

1.7.2. Databases and retrieval tools

The Bilbao Crystallographic Server, http://www.cryst.ehu.es, is a web site of crystallographic databases and programs. It can be used free of charge from any computer with a web browser via the Internet. The server is built on a core of databases and contains different shells. The set of databases includes data from the 5th edition of International Tables for Crystallography Volume A, Space-Group Symmetry (2005) (hereafter referred to as IT A) and the data for maximal subgroups of space groups as listed in Part 2 of this volume (hereafter referred to as IT A1). Access is also provided to the crystallographic data for the subperiodic layer and rod groups [International Tables for Crystallography, Volume E, Subperiodic Groups (2002)] and their maximal subgroups. A database on incommensurate structures incorporating modulated structures and composites, and a k-vector database with Brillouin-zone ﬁgures and classiﬁcation tables of the wavevectors for space groups are also available. Communication with the databases is achieved by simple retrieval tools. They allow access to the information on space groups or subperiodic groups in different types of formats: HTML, text ASCII or XML. The next shell includes programs related to group–subgroup relations of space groups. These programs use the retrieval tools for accessing the necessary space-group information and apply group-theoretical algorithms in order to obtain speciﬁc results which are not available in the databases. There follows a shell with programs on representation theory of space groups and point groups and further useful symmetry information. Parallel to the crystallographic software, a shell with programs facilitating the study of speciﬁc problems related to solid-state physics, structural chemistry and crystallography has also been developed. The server has been operating since 1998, and new programs and applications are being added (Kroumova, Perez-Mato, Aroyo et al., 1998; Aroyo, Perez-Mato et al., 2006; Aroyo, Kirov et al., 2006). The aim of the present chapter is to report on the different databases and programs of the server related to the subject of this volume. Parts of these databases and programs have already been described in Aroyo, Perez-Mato et al. (2006), and here we follow closely that presentation. The chapter is completed by the description of the new developments up to 2007. The relevant databases and retrieval tools that access the stored symmetry information are presented in Section 1.7.2. The discussion of the accompanying applications is focused on the crystallographic computing programs related to group– subgroup and group–supergroup relations between space groups (Section 1.7.3). The program for the analysis of the relations of the Wyckoff positions for a group–subgroup pair of space groups is presented in Section 1.7.4. The underlying grouptheoretical background of the programs is brieﬂy explained and details of the necessary input data and the output are given. The use of the programs is demonstrated by illustrative examples.

The databases form the core of the Bilbao Crystallographic Server and the information stored in them is used by all computer programs available on the server. The following description is restricted to the databases related to the symmetry relations between space groups; these are the databases that include spacegroup data from IT A and subgroup data from IT A1.

Copyright © 2011 International Union of Crystallography

1.7.2.1. Space-group data The programs and databases of the Bilbao Crystallographic Server use speciﬁc settings of space groups (hereafter referred to as standard or default settings) that coincide with the conventional space-group descriptions found in IT A. For space groups with more than one description in IT A, the following settings are chosen as standard: unique axis b setting, cell choice 1 for monoclinic groups; hexagonal axes setting for rhombohedral groups; and origin choice 2 (origin at 1) for the centrosymmetric groups listed with respect to two origins in IT A. The space-group database includes the following symmetry information: (i) The generators and the representatives of the general position of each space group speciﬁed by its IT A number and Hermann–Mauguin symbol; (ii) The special Wyckoff positions including the Wyckoff letter, Wyckoff multiplicity, the site-symmetry group and the set of coset representatives, as given in IT A; (iii) The reﬂection conditions including the general and special conditions; (iv) The afﬁne and Euclidean normalizers of the space groups (cf. IT A, Part 15). They are described by sets of additional symmetry operations that generate the normalizers successively from the space groups. The database includes the additional generators of the Euclidean normalizers for the general-cell metrics as listed in Tables 15.2.1.3 and 15.2.1.4 of IT A. These Euclidean normalizers are also afﬁne normalizers for all cubic, hexagonal, trigonal, tetragonal and part of the orthorhombic space-group types. For the rest of the orthorhombic space groups, the type of the afﬁne normalizer coincides with the highest-symmetry Euclidean normalizer of that space group and the corresponding additional generators form part of the database (cf. Table 15.2.1.3 of IT A). The afﬁne normalizers of triclinic and monoclinic groups are not isomorphic to groups of motions and they are not included in the normalizer database of the Bilbao Crystallographic Server. (v) The assignment of Wyckoff positions to Wyckoff sets as found in Table 14.2.3.2 of IT A. The data from the databases can be accessed using the simple retrieval tools, which use as input the number of the space group (IT A numbers). It is also possible to select the group from a table of IT A numbers and Hermann–Mauguin symbols. The output of the program GENPOS contains a list of the generators or the general positions and provides the possibility to obtain the same data in different settings either by specifying the transformation

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1. SPACE GROUPS AND THEIR SUBGROUPS matrix to the new basis or selecting one of the 530 settings listed in Table 4.3.2.1 of IT A. A list of the Wyckoff positions for a given space group in different settings can be obtained using the program WYCKPOS. The Wyckoff-position representatives for the nonstandard settings of the space groups are speciﬁed by the transformed coordinates of the representatives of the corresponding default settings. The assignments of the Wyckoff positions to Wyckoff sets are retrieved by the program WYCKSETS. This program also lists a set of coset representatives of the decompositions of the normalizers with respect to the space groups and the transformation of the Wyckoff positions under the action of these coset representatives. The programs NORMALIZER and HKLCOND give access to the data for normalizers and reﬂection conditions.

1.7.3. Group–subgroup and group–supergroup relations between space groups 1.7.3.1. Subgroups of space groups If two space groups G and H form a group–subgroup pair G > H, it is always possible to represent their relation by a chain of intermediate maximal subgroups Z k : G > Z 1 > . . . > Z n ¼ H. For a speciﬁed index of H in G there are, in general, a number of possible chains relating both groups, and a number of different subgroups Hj < G isomorphic to H. We have developed two basic tools for the analysis of the group–subgroup relations between space groups: SUBGROUPGRAPH (Ivantchev et al., 2000) and HERMANN (Capillas, 2006). Given the space-group types G and H and an index [i], both programs determine all different subgroups Hj of G with the given index and their distribution into classes of conjugate subgroups with respect to G. Owing to its importance in a number of group–subgroup problems, the program COSETS is included as an independent application. It performs the decomposition of a space group in cosets with respect to one of its subgroups. Apart from these basic tools, there are two complementary programs which are useful in speciﬁc crystallographic problems that involve group–subgroup relations between space groups. The program CELLSUB calculates the subgroups of a space group for a given multiple of the unit cell. The common subgroups of two or three space groups are calculated by the program COMMONSUBS.

1.7.2.2. Database on maximal subgroups 1.7.2.2.1. Maximal subgroups of indices 2, 3 and 4 of the space groups All maximal non-isomorphic subgroups and maximal isomorphic subgroups of indices 2, 3 and 4 of each space group can be retrieved from the database using the program MAXSUB. Each subgroup H is speciﬁed by its IT A number, the index in the group G and the transformation matrix–column pair (P, p) that relates the bases of H and G: ða0 ; b0 ; c0 ÞH ¼ ða; b; cÞG P:

ð1:7:2:1Þ 1.7.3.1.1. The program SUBGROUPGRAPH This program is based on the data for the maximal subgroups of index 2, 3 and 4 of the space groups of IT A1. These data are transformed into a graph with 230 nodes corresponding to the 230 space-group types. If two nodes in the graph are connected by an edge, the corresponding space groups form a group–maximal subgroup pair. Each one of these pairs is characterized by a group–subgroup index. The different maximal subgroups of the same space-group type are distinguished by corresponding matrix–column pairs (P, p) which give the relations between the standard coordinate systems of the group and the subgroup. The index and the set of transformation matrices are considered as attributes of the edge connecting the group with the subgroup. The speciﬁcation of the group–subgroup pair G > H leads to a reduction of the total graph to a subgraph with G as the top node and H as the bottom node, see Example 1.7.3.1.1 at the end of this section. In addition, the G > H subgraph, referred to as the general G > H graph, contains all possible groups Z k which appear as intermediate maximal subgroups between G and H. It is important to note that in the general G > H graphs the spacegroup symbols indicate space-group types, i.e. all space groups belonging to the same space-group type are represented by one node on the graph. Such graphs are called contracted. The contracted graphs have to be distinguished from the complete graphs where all space groups occurring in a group–subgroup graph are indicated by different space-group nodes. The number of the nodes in the general G > H graph may be further reduced if the index of H in G is speciﬁed. The subgraph obtained is again of the contracted type. The comparison of complete graphs and contracted graphs shows that the use of contracted graphs for the analysis of speciﬁc group–subgroup relations G > Hj can be very misleading (see Example 1.7.3.1.1, Fig. 1.7.3.2 and Fig. 1.7.3.3). The complete graphs produced by SUBGROUPGRAPH are equal for subgroups of a conjugacy class; the different orientations and/

The column p = (p1, p2, p3) of coordinates of the origin OH of H is referred to the coordinate system of G. It is important to note that, in contrast to the data listed in IT A1, the matrix–column pairs (P, p) used by the programs of the server transform the standard basis ða; b; cÞG of G to the standard basis of H (see Section 2.1.2.5 for the special rules for the settings of the subgroups used in IT A1). The different maximal subgroups are distributed in classes of conjugate subgroups. For certain applications it is necessary to represent the subgroups H as subsets of the elements of G. This is achieved by an option in MAXSUB which transforms the general-position representatives of H by the corresponding matrix–column pair (P, p)1 to the coordinate system of G. In addition, one can obtain the splittings of all Wyckoff positions of G to those of H. 1.7.2.2.2. Maximal isomorphic subgroups Maximal subgroups of index higher than 4 have indices p, p2 or 3 p , where p is a prime. They are isomorphic subgroups and are inﬁnite in number. In IT A1, the isomorphic subgroups are listed as members of series under the heading ‘Series of maximal isomorphic subgroups’. In addition, the isomorphic subgroups of indices 2, 3 and 4 are listed individually. The program SERIES provides access to the database of maximal isomorphic subgroups on the Bilbao Crystallographic Server. Apart from the parametric IT A1 descriptions of the series, its output provides the individual listings of all maximal isomorphic subgroups of indices as high as 27 for all space groups, except for the cubic ones where the maximum index is 125. The format and content of the subgroup data are similar to those of the MAXSUB access tool. In addition, there is a special tool (under ‘deﬁne a maximal index’ on the SERIES web form) that permits the online generation of maximal isomorphic subgroups of any index up to 131 for all space groups. [Note that these data are only generated online and do not form part of the (static) database of isomorphic subgroups.]

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1.7. THE BILBAO CRYSTALLOGRAPHIC SERVER or origin shifts of the conjugate subgroups Hs are manifested by the different transformation matrices ðP; pÞs listed by the program. Different chains of maximal subgroups for the group– subgroup pair G > H are obtained following the possible paths connecting the top of the graph (the group G) with the bottom (the group H). Each group–maximal subgroup pair determines one step of this chain. The index of H in G equals the product of the indices for each one of the intermediate edges. The transformation matrices relating the standard bases of G and H are obtained by multiplying the matrices of each step of the chain. Thus, for each pair of group–subgroup types with a given index there is a set of transformation matrices (P, p)j, where each matrix corresponds to a subgroup Hj isomorphic to H. Some of these subgroups could coincide. To ﬁnd the different Hj of G, the program transforms the elements of the subgroup H to the basis of the group G using the different matrices (P, p)j and compares the elements of the subgroups Hj in the basis of G. Two subgroups that are characterized by different transformation matrices are considered identical if their elements, transformed to the basis of the group G, coincide. The different subgroups Hj are distributed into classes of conjugate subgroups with respect to G by checking directly their conjugation relations with elements of G.

different transformation matrices and a link to a list of these matrices are given for each of the possible chains. The graphical representation contains the intermediate groups that connect G and H with the speciﬁed index. This graph is a subgraph of the general graph of maximal subgroups with unspeciﬁed index and is also of the contracted type. (b) Classiﬁcation of the different subgroups Hj of G. Once the index of H in G is given, the different subgroups Hj of that index are calculated and distributed into classes of conjugate subgroups of G. The subgroups of a conjugacy class form a block where each subgroup is speciﬁed by the corresponding transformation matrix– column pair ðP; pÞj that relates the standard bases of G and Hj . There is also a link to a list of the elements of the subgroups transformed to the basis of the group G, which allows the identiﬁcation of those elements of G that are retained in the subgroup. The list of transformation matrices that give the same (identical) subgroup is accessible under a separate link (see Table 1.7.3.1). The graph contains the intermediate space groups Z k i for the pair G > H but contrary to the graph of the previous step, the different isomorphic subgroups are represented by different nodes, i.e. the graph is a complete one. All isomorphic subgroups Hj are given at the bottom of the graph. Their labels are formed by the short Hermann–Mauguin symbol of the subgroup followed by a number given in parentheses which speciﬁes the class of conjugate subgroups to which the subgroup Hj belongs. Note that for group–subgroup pairs with high indices, where a lot of intermediate maximal subgroups occur, the resulting complete graph with all subgroups Hj can be very complicated and difﬁcult to overview. Alternatively, a simpler graph associated with a single speciﬁc subgroup Hj can also be obtained (the graphs are the same for all subgroups within a conjugacy class).

Input to SUBGROUPGRAPH: (i) As input, the program needs the speciﬁcation of the space groups G and H. The groups G and H can be deﬁned either by their sequential IT A numbers or by their Hermann– Mauguin symbols. The default settings of all space groups are used. (ii) If the index of H in G is speciﬁed, then the program determines the chains of maximal subgroups relating these groups and classiﬁes the isomorphic subgroups Hj into classes of conjugate subgroups. Output of SUBGROUPGRAPH: The output is illustrated by Example 1.7.3.1.1. (i) Group–subgroup pairs with non-speciﬁed indices. When the index of the subgroup H in G is not speciﬁed, the program returns a list of the possible intermediate space groups Z k relating G and H. The list is given as a table whose rows correspond to the intermediate space groups Z k, speciﬁed by their Hermann–Mauguin symbols. In addition, the table contains the maximal subgroups of Z k, speciﬁed by their IT A numbers and the corresponding indices given in brackets. This list is also represented as a contracted graph. Each space-group type in the list corresponds to one node in the graph, and the maximal subgroups are the neighbours (successors) of this node. Group–subgroup relations occurring in both directions are represented by nodes connected by two edges with opposite arrows. Maximal isomorphic subgroups are shown by loop edges (nodes connected to themselves), see Fig. 1.7.3.1. (ii) Group–subgroup pairs with speciﬁed indices. As an example, see Table 1.7.3.1 and Figs. 1.7.3.2 and 1.7.3.3.

Example 1.7.3.1.1 Consider the group–subgroup relations between the groups G ¼ P41 21 2, No. 92, and H ¼ P21 , No. 4. If no index is speciﬁed then the graph of maximal subgroups that relates P41 21 2 and P21 is represented as a table indicating the space-group types of the possible intermediate space groups Z k and the corresponding indices. The contracted general P41 21 2 > P21 graph is shown in Fig. 1.7.3.1. Two edges with opposite arrows between a group–subgroup pair correspond to group– subgroup relations in both directions, e.g. the pair P41 and P43 . Table 1.7.3.1. Group–subgroup relations for P41212 (No. 92) > P21 (No. 4), index 4 There are three different subgroups Hj ¼ ðP21 Þj of P41 21 2 distributed into two classes of conjugate subgroups. The possible chains of maximal subgroups and the corresponding matrices (P, p)j (written in concise form) are also shown. (Note that the standard setting for P21 is the unique axis b setting.) The different transformation matrices for the subgroup ðP21 Þ3 correspond to the different group–subgroup chains that relate the same subgroup to P41 21 2, cf. Fig. 1.7.3.3.

(a) Chains of maximal subgroups. If the index of the subgroup H in the group G is speciﬁed, the program returns a list of all possible chains of maximal subgroups relating G and H with this index. (The program has no access to data for maximal isomorphic subgroups with indices higher than four.) The

59

Class

Hj

Chains

(P, p)j

Class 1

ðP21 Þ1 ¼ P21 ½010 ðP21 Þ2 ¼ P21 ½100

P41 21 2 > P21 21 21 > P21 P41 21 2 > P21 21 21 > P21

a; b; c; 14 0 58 c; a; b; 14 14 38

Class 2

ðP21 Þ3 ¼ P21 ½001

P41 21 2 > P21 21 21 > P21 P41 21 2 > P41 > P21 P41 21 2 > C2221 > P21

b; c; a; 12 0 38 b; c; a; 0 12 0 a; c; b; 0 0 14

1. SPACE GROUPS AND THEIR SUBGROUPS

Fig. 1.7.3.1. General contracted graph for P41 21 2 (No. 92) > P21 (No. 4) as given by the program SUBGROUPGRAPH. The nodes of the graph correspond to the space-group types that can appear as intermediate groups in the chain of the group–subgroup pair P41 21 2 > P21 . Each edge of the graph corresponds to a maximal subgroup pair of the indicated index [i]. Isomorphic subgroups (of indices 2 and 3) are shown as loops.

Fig. 1.7.3.3. Complete graph for P41 21 2 (No. 92) > P21 (No. 4), index 4, as given by the program SUBGROUPGRAPH. The nodes represent space groups and not space-group types. The three subgroups of the type P21 are distributed into two classes of conjugate subgroups, which are indicated in parentheses after the space-group symbol. The two subgroups P21 ð1Þ with twofold screw axes along ½100 and ½010 of P41 21 2 belong to the same conjugacy class. Their complete single graphs look alike and differ considerably from the graph of P21 ð2Þ. The latter corresponds to the subgroup whose twofold screw axes point along the tetragonal axes.

Fig. 1.7.3.2. Contracted graph for the pair of space groups P41 21 2 (No. 92) > P21 (No. 4), index 4, as given by the program SUBGROUPGRAPH. The nodes of the graph correspond to space-group types. The directed edges represent the possible group–maximal subgroup pairs.

60

1.7. THE BILBAO CRYSTALLOGRAPHIC SERVER to M, or some of the classes relative to M merge together to form conjugacy classes with respect to G.

When the index [i] of the subgroup in the group is speciﬁed, the resultant graph is reduced to the chains of maximal subgroups that correspond to the value of [i]. For example, in Fig. 1.7.3.2 the contracted graph P41 21 2 > P21 of index 4 is shown. The data in Table 1.7.3.1 and the complete graph shown in Fig. 1.7.3.3 indicate that there are three different P21 subgroups of P41 21 2 > P21 of index 4, distributed over two classes of conjugate subgroups. The three different subgroups of space-group type P21 of index 4 correspond to three sets of twofold screw axes in P41 21 2: those pointing along [100] and [010] of the tetragonal cell give rise to the two conjugate subgroups, and the third one (forming a class of conjugate subgroups by itself) is along the tetragonal axis. Their full Hermann–Mauguin symbols are P21 11, P121 1 and P1121 . The corresponding transformations are listed in Table 1.7.3.1. The complete graph P41 21 2 > P21 , index 4 (Fig. 1.7.3.3) also shows that three different maximal subgroup chains end at the same P21 ðP1121 Þ subgroup, each of them speciﬁed by a different transformation matrix ðP; pÞ (Table 1.7.3.1). The three different transformation matrices are related by elements of the normalizer of the subgroup.

Input to HERMANN: The data needed are the space-group types of G and H and their index. Output of HERMANN: Essentially, the output of the program is a list of all subgroups Hj of G of index ½i ¼ ½iP ½iL , distributed into conjugacy classes with respect to G. The classes of conjugate subgroups are listed in different blocks depending on the space-group type of the iP Hermann groups M < G. The subgroups Hj in each class are listed explicitly and are distinguished by the transformation matrices ðP; pÞj. Options for transforming the elements of Hj to the basis of G and for decomposing G in right cosets with respect to Hj are available. In addition, it is possible to calculate the splittings of the Wyckoff positions of G relative to Hj (cf. Section 1.7.4, program WYCKSPLIT). There is an optional link to the program SYMMODES, which carries out a symmetry analysis of the possible distortions compatible with Hj for the symmetry break G ! Hj . Example 1.7.3.1.2 Consider the pair of group–subgroup types P422 (No. 89) > P21 (No. 4), index [i] = 8 (Fig. 1.7.3.4). The factorization of the index into ½iP ¼ 4 and ½iL ¼ 2 follows from the ratio of the orders of the point groups of G and H. The two spacegroup types P2, No. 3, and C2, No. 5, satisfy the conditions for i¼8 Hermann subgroups for the pair P422 > P21 . There are three P2 Hermann subgroups of G distributed in two conjugacy classes: the subgroups with twofold axes along [100] and [010] form a conjugacy class, and P2½001 ðP2Þ is a normal subgroup in P422. The two C2 Hermann subgroups with twofold axes along [110] and ½110 form a single conjugacy class. Each Hermann subgroup has just one normal subgroup of P21 type of index ½iL ¼ 2. Altogether there are ﬁve different P21 subgroups of P422 of index [i] = 8, distributed in three conjugacy classes, following the conjugacy relationships of the corresponding Hermann groups.

1.7.3.1.2. The program HERMANN The method of calculation of the subgroups and their distribution into classes of conjugate subgroups used in SUBGROUPGRAPH is not adequate for group–subgroup pairs of indices greater than 50. The program HERMANN has been developed to treat the cases of such considerable reduction of symmetry. It is a modiﬁcation of SUBGROUPGRAPH and is based on Hermann’s theorem (cf. Lemma 1.2.8.1.1). Consider a group–subgroup pair G > H, with H a general subgroup of index [i]. The existence and the uniqueness of Hermann’s group M, G > M > H, implies the possibility of factorizing the group– subgroup chain G > H and its index [i] into two subchains of smaller indices with ½i ¼ ½iP ½iL . The ﬁrst one, the soiP called translationengleiche or t-chain G > M, is related to the reduction of the point-group symmetry in the subgroup. The iL second one is known as the klassengleiche or k-chain M > H, and it takes account of the loss of translations. (The ½iL index is equal to the cell-multiplication factor of the volumes of the primitive cells of the lattices.) The program HERMANN calculates all subgroups Hj of G of index [i] and distributes them into conjugacy classes with respect to G. In addition, the program indicates the corresponding Hermann groups. It is important to note that for a given pair of i space-group types G > H and index [i], Hermann groups M of different space-group types can exist that belong to the same crystal class, cf. Example 1.7.3.1.2. However, there is a unique Hermann group for any group–subgroup pair of speciﬁc space groups. The space-group types of the possible Hermann groups M for i the pair G > H are determined by the following conditions: (i) the groups M are subgroups of G of index ½iP where iP is equal to the ratio of the point-group orders of G and H; (ii) the point groups of M coincide with the point group of H, P H ; (iii) the groups M have subgroups H of index ½iL ¼ ½i=½iP . The complete set of the application of subgroups Hj of G is calculated by the consecutive iP iL SUBGROUPGRAPH to the two subchains G > M and M > H. At this stage, the subgroups Hj are distributed into conjugacy classes with respect to the corresponding Hermann groups only. Finally, the program determines the conjugacy classes of Hj with respect to G: these either coincide with conjugacy classes relative

Fig. 1.7.3.4. Group–subgroup graph for P422 (No. 89) > P21 (No. 4), index 8. The nodes represent space groups and not space-group types. The directional symbols [100] etc. state the orientations of the monoclinic axes referred to the basis of P422. Horizontal lines connect subgroups conjugate in P422. There are two space-group types of Hermann subgroups in the intermediate level of the ﬁgure: P2 and C2. The distribution of the ﬁve P21 subgroups into conjugacy classes with respect to P422 follows the conjugacy-class distribution of the corresponding Hermann subgroups.

61

1. SPACE GROUPS AND THEIR SUBGROUPS are 23 36 ¼ 24 cosets in the decomposition of G : T H, i.e. some of the coset representatives of T G0 belong to the same cosets with respect to T H. The distribution of the coset representatives of G : T G0 into cosets of G with respect to T H is different for the right- and left-coset decomposition. Consider the three coset representatives of G : T G0 corresponding to the threefold rotation, namely ð3þ ; 000Þ, ð3þ ; 23 13 13Þ and ð3þ ; 13 23 23Þ. In the basis of the subgroup, the rotational part

1.7.3.1.3. The program COSETS The coset decomposition of a group G with respect to a subgroup H < G is a basic step in many problems that involve group–subgroup relations between space groups, e.g. the distribution of subgroups into conjugacy classes (cf. Section 1.2.6.3) or the determination of supergroups of space groups (cf. Section 2.1.7.4). The procedure for the coset decomposition G : H is well deﬁned: one transforms G to the basis of H by the transformation matrix ðP; pÞ that relates the two bases, and then one distributes the transformed elements of G into left g r H or right Hg s cosets with respect to H (cf. Section 1.2.4.2). The elements g r and g s are called the coset representatives. Consider the space-group pair G > H with the corresponding transformation matrix ðP; pÞ. The coset decomposition G : H can be easily achieved if both G and H are decomposed into cosets with respect to the translation subgroup T H0 of H, consisting of integer translations only (i.e. the coset decomposition is performed with respect to a primitive lattice of H). The determination of the coset representatives of G : T H0 for the right-coset decomposition is simpliﬁed by the fact that two elements of G that differ by an integer translation t H of H belong to the same coset T H0 g q . This is, however, not in general true for the left-coset decomposition: in that case, two elements of G belong to the same coset g q T H0 if they differ by the translation W q t H where g q ¼ ðW q ; wq Þ. In other words, two elements of G (with the same rotational part) belong to the same coset 0 g q T H if the difference q between their translational parts satisﬁes ðW q Þ1 q 2 T H0 . Given the space-group pair G > H with the corresponding transformation matrix ðP; pÞ, the program COSETS decomposes G into left or right cosets with respect to H.

0

0 ð3þ Þ ¼ @ 1 0

0

1 2

3

P ¼ @ 13 1 3

0 1 0

2 1 A; 0

0 0 P 1 ¼ @ 12 1 12 0

0

0

1

12 32 A 1 12 2

and the translational parts by P 1 t j to ð000Þ, ð100Þ and ð2 12 12Þ, correspondingly. The elements with translational parts ð000Þ, ð100Þ belong to the same coset T H ð3þ ; 000Þ in the case of rightcoset decomposition, and ð3þ ; 2 12 12Þ can be taken as a representative of a different coset [the program chooses the element (3þ ; 0 12 12Þ as a coset representative]. In the case of left-coset decomposition, the coset representatives are ð3þ ; 000Þ and ð3þ ; 100Þ: the elements ð3þ ; 100Þ and ð3þ ; 2 12 12Þ belong to the same coset ð3þ ; 100ÞT H as the difference in their translational parts q ¼ ð1; 12 ; 12Þ satisﬁes the condition q ¼ W q t H , with W q ¼ 3þ and t H ¼ ð1; 0; 0Þ. The output of COSETS gives the distribution of the 24 coset representatives of G : T H into the six cosets of the decomposition of R3m with respect to P21 =c. In the case of right-coset decomposition, the elements fð1; 000Þ, ð1; 0 12 12Þ, ð3þ ; 000Þ, ð3þ ; 0 12 12Þ, ð3 ; 000Þ, ð3 ; 0 12 12Þg can be selected as coset representatives. The elements fð1; 000Þ, ð1; 0 12 12Þ, ð3þ ; 000Þ, ð3 ; 000Þg are valid coset representatives also for the left-coset decomposition of R3m with respect to P21 =c, while ð3þ ; 0 12 12Þ; ð3 ; 0 12 12Þ have to be substituted by ð3þ ; 100Þ, ð3 ; 100Þ. 1.7.3.1.4. The program CELLSUB For several applications, it is of interest to determine the subgroups of a space group for a speciﬁc multiple of the cell, i.e. for a given ½iL index. This happens, for example, in the search for possible low-symmetry phases after a phase transition from a known high-symmetry phase with experimental data indicating the reduction of translational symmetry. The program CELLSUB calculates the different subgroups of a space group G for a given maximal index ½ðiL Þmax in two steps:

Example 1.7.3.1.3 Consider the group–subgroup pair R3m (No. 166) > P21/c (No. 14) of index 6, with the transformation matrix 0

1

ð3þ Þ ¼ @ 12

Output of COSETS: The output data consist of the list of the cosets of the decomposition of G with respect to H. The ﬁrst coset corresponds to the subgroup H represented by the set of its general-position triplets, fh v ; v ¼ 1; . . . ; f jP H jg, where jP H j is the order of the point group P H and f is the so-called centring factor that equals the number of lattice points per cell. The triplets of the rest of the cosets are of the form ðg s hv Þ for the left-coset decomposition or ðhv g r Þ for the right-coset decomposition, with s, r = 1, . . . , jG : Hj.

1

1 0 0A 1

is transformed by P 1 ð3þ ÞP to

Input to COSETS: The data that are needed are the group G, the subgroup H and the transformation matrix ðP; pÞ that relates the default settings of G and H. The user can choose between right- or left-coset decomposition of G with respect to H.

02

1 1 0

(i) It determines all possible space-group types H which form group–subgroup pairs with G, G > H of index ½i ¼ ½iP ½iL , such that the index ½iL is equal to or smaller than a given value ½ðiL Þmax . (ii) For each pair of space-group types G > H and their index [i] satisfying the above condition, CELLSUB ﬁnds all possible subgroups Hj speciﬁed by the matrix–column pairs (P, p)j and classiﬁes them into classes of conjugate subgroups.

1

3 0A 1

relating the bases deﬁned for the default settings of the group and the subgroup (hexagonal axes setting for R3m, and the unique axis b setting for P21 =c). The decomposition of P21 =c with respect to T H consists of four cosets. (For P space groups, T H0 coincides with T H .) There are 36 coset representatives of G : T G0 , with T G0 consisting of integer translations only. From the determinant of the transformation matrix it follows that there

The method for obtaining the different subgroup types and indices of a given space group G is similar to that used in the SUBGROUPGRAPH module. It is also based on the data for maximal subgroups of the space groups in IT A1. Given the

62

1.7. THE BILBAO CRYSTALLOGRAPHIC SERVER group G, the program constructs a graph of maximal subgroups, imposing the additional condition ½iL ½ðiL Þmax .

(i) A dedicated module of the program CELLSUB (cf. Section 1.7.3.1.4) calculates the sets of subgroups fH1;r ; i1;r g of G1 and fH2;s ; i2;s g of G2 with ½ðiL Þm ½ðiL Þmax . Here, the indices r and s distinguish the different space-group types of the subgroups H1 and H2 . (ii) The intersection of the sets fH1;r ; i1;r g and fH2;s ; i2;s g for H1;r ﬃ H2;s ﬃ H then gives the set of the space-group types of the common subgroups fHg of G1 and G2 with ½ðiL Þm ½ðiL Þmax , m = 1, 2. (iii) The program COMMONSUBS selects and lists those subgroups of the set fHg whose indices ½i1 and ½i2 in G1 and G2 satisfy the condition given by equation (1.7.3.1).

Input to CELLSUB: (i) The space group G. It can be speciﬁed either by its sequential IT A number or by its Hermann–Mauguin symbol. (ii) The (maximal) ½ðiL Þmax index. By default (full-list option), all subgroups with an index ½iL smaller than the given ½ðiL Þmax are shown. As an option, it is possible to choose only the subgroups with a speciﬁed ½ðiL Þmax index. Output of CELLSUB: (i) A list of the space–group types of the subgroups H with the corresponding index [i], the index ½iP and the index ½iL . The subgroups are further classiﬁed into k-subgroups, t-subgroups (for the special case of ½iL ¼ 1) or general subgroups. (ii) For every space-group type H of the list, a link to the SUBGROUPGRAPH module provides the following data: the chains of maximal subgroups relating G and H with the given index, the classiﬁcation of the different subgroups Hj of G in classes of conjugate subgroups, the graphical representations and all the beneﬁts of the program SUBGROUPGRAPH.

The generalization of the procedure for the case of common subgroups of three groups G is straightforward and has been also implemented in the program COMMONSUBS. Input to COMMONSUBS: The necessary data include the speciﬁcation of the space-group types G, the index ½ðiL Þmax and the number of formula units per conventional unit cell1 (or the ratio of the two ½iL indices for the special case of common subgroups of two space groups). The search for common subgroups can be further restricted by specifying the point group, the crystal class or the type of centring desired for the common subgroup.

1.7.3.1.5. The program COMMONSUBS Two space groups G1 and G2 which are not in a group–subgroup relation may be related by common subgroups Hj with G1 > Hj < G2 . Given G1 and G2 , the program COMMONSUBS determines these groups Hj. Such group–subgroup relations have a subjective component: in general the lattices of both space groups do not ﬁt ideally. There will be some misﬁt between the lattice parameters of H1, a subgroup of G1 and those of H2, a subgroup of G2. The decision as to how much misﬁt could be tolerated depends on the speciﬁc structural criteria applicable to the problem studied. The higher the proportion of common symmetry between G1 and G2 , i.e. the smaller the indices ½i1 ¼ jG1 : Hj and ½i2 ¼ jG2 : Hj are, the more promising is the search for a relation between the two crystal structures. Owing to the theorem of Hermann both the indices ½i1 and ½i2 may be split into the pointgroup index ½ðiP Þm and the lattice index ½ðiL Þm , such that ½im ¼ ½ðiP Þm ½ðiL Þm , m = 1, 2. The point-group indices are ﬁnite, ½ðiP Þm 48; the lattice index may have any value in principle. Large indices, however, mean little common symmetry and thus low probability of structural relevance. Therefore, it is reasonable to limit the value of ½ðiL Þm and thus of ½i1 and ½i2 by introducing a maximal value ½ðiL Þmax . If two structures with space groups G1 and G2 can be compared within a common subgroup H, then the number of formula units ZH of a primitive unit cell of H should be the same for both structures. This means that ZH ¼ Z1 ½ðiL Þ1 ¼ Z2 ½ðiL Þ2 , where Z1 and Z2 are the numbers of the formula units of the primitive unit cells of the crystal structures 1 (with space group G1 ) and 2 (with space group G2 ). It follows that ½i2 ¼ ½i1 ðZ1 =Z2 Þ ð½ðiP Þ2 =½ðiP Þ1 Þ or ½i2 ¼ ½i1

Z1 jP G2 j ; Z2 jP G1 j

Output of COMMONSUBS: (i) A list of common subgroups H speciﬁed by their Hermann– Mauguin symbols, the point groups P H, the indices [i] of H in the space groups G, and the corresponding factors ½ðiP Þ and ½ðiL Þ. Optional links to the databases give access to additional information related to Hm, its maximal subgroups or the point group P H. (ii) The output for the case of common subgroups of two space groups gives more details on the group–subgroup relations for the two branches G1 > H and G2 > H. Apart from the speciﬁcation of the common subgroups H and their indices ½ði1 Þ and ½ði2 Þ in G1 and G2 , one obtains a list of representatives of the conjugacy classes of the subgroups H with respect to G1 and G2 . There are also links to the programs WYCKSPLIT (for the splittings of the Wyckoff positions for G > H, cf. Section 1.7.4.1) and SYMMODES (Capillas et al., 2003) (for the group-theoretical study of the possible phasetransition mechanism related to the symmetry break G > H). A list of all subgroups of a conjugacy class and the corresponding transformation matrices are obtained via links to the programs HERMANN (cf. Section 1.7.3.1.2) and SUBGROUPGRAPH (cf. Section 1.7.3.1.1). The existence of a common subgroup does not automatically mean the existence of a structural relation between the corresponding crystal structures. In general, the existence of a common supergroup of two space groups is mostly taken as more indicative of a structural relation than that of a common subgroup, provided the indices of the common supergroup are comparable with those of the common subgroup. However, the following example shows the utility of the common-subgroup approach in the search for the possible symmetry of an intermediate phase between two phases with no group–subgroup relation between their space groups.

ð1:7:3:1Þ

where jPj is the order of the point group P. Given the space groups G1 and G2 , the formula units per primitive cells Z1 and Z2 , and ½ðiL Þmax , the program COMMONSUBS calculates the common subgroups in several steps:

1

The number of formula units per conventional unit cell Zc is simply related to the number Z of formula units per primitive unit cell via the centring factor f, Z ¼ Zc =f , with f = 1 for primitive lattices, 2 for I-, C-, B- or A-centred lattices, 3 for R-centred lattices and 4 for F-centred lattices.

63

1. SPACE GROUPS AND THEIR SUBGROUPS Example 1.7.3.1.4 The perovskite-like ferroelectric compounds PbZr1xTixO3 exhibit a morphotropic2 phase boundary around x = 0.45–0.50. At compositions with x < 0.47 they are rhombohedral, at x > 0.47 tetragonal. For x = 0.48 a tetragonal-to-monoclinic phase transition has been observed at ~300 K, the space group changing from P4mm, No. 99, to Cm, No. 8 (Noheda et al., 1999, 2000). The monoclinic structure results from the tetragonal one by shifts of the Pb and Zr/Ti atoms along the tetragonal [110] direction. The monoclinic structure can also be envisaged as a distorted variant of the rhombohedral phase, space group R3m, No. 160. There is no group–subgroup relation between P4mm and R3m, but Cm is a common subgroup of both. In this way, the monoclinic structure can be considered as providing a ‘bridge’ between the rhombohedral and tetragonal regions of the morphotropic phase boundary. The application of COMMONSUBS for G1 ¼ P4mm with Zc;1 ¼ 1, G2 ¼ R3m with Zc;2 ¼ 3, and iL ¼ 1 (i.e. no cell multiplication) yields exactly Cm as the common monoclinic subgroup.

1.7.3.2.1. The programs MINSUP and SUPERGROUPS In analogy to the case of minimal supergroups (cf. Section 2.1.7), the determination of all supergroups Gr of a given spacegroup type G and index [i] of a space group H can be done by inverting the data for the subgroups Hs of G of index [i]. In the following we outline the basic arguments of this procedure. Let G be a space group and H < G be one of its subgroups of index [i]. Then all subgroups Hs of G of the same index [i] and isomorphic to H can be calculated by a dedicated module of SUBGROUPGRAPH (cf. Section 1.7.3.1.1). The number of subgroups with index [i] is ﬁnite for any space group. Therefore, such a list is always ﬁnite. Let H < G be a member of this list. We are looking for all supergroups Gr > H of index [i] that are isomorphic to G, Gr ﬃ G. Then the supergroups Gr are also afﬁne equivalent to G, i.e. there must be a mapping ar 2 A such that a1 r G ar ¼ Gr , where A is the group of all reversible afﬁne mappings. Different supergroups Gr are obtained if ar 62 N A ðGÞ. As in the case of minimal supergroups (cf. Section 2.1.7), there are two cases to be distinguished:

Structural relations established through a common subgroup are also being used to model ﬁrst-order transformations between phases with no group–subgroup relation between their symmetry groups. The local symmetry of a common subgroup of the two end symmetries is supposed to describe approximately the symmetry constraints of the local transient states taking place during the transformation [see e.g. Capillas et al. (2007) and the references therein]. In some cases these intermediate conﬁgurations can even be stabilized and appear as stable intermediate phases in the phase diagram. The program COMMONSUBS can be useful in both types of searches.

(1) The ﬁrst candidates for the mapping ar are the elements of the afﬁne normalizer of H, i.e. ar 2 N A ðHÞ. Following Koch (1984) (see also Lemma 2.1.7.4.1), other supergroups Gr will be obtained by the transformation of G with the representatives of the cosets in the decomposition of the group N A ðHÞ relative to the group D ¼ N A ðHÞ \ N A ðGÞ. (2) There could exist further supergroups Gr > H; Gr ﬃ G of the same index [i] with ar 62 N A ðHÞ, i.e. ar is not an element of the afﬁne normalizer of the group H. Summarizing: Any supergroup of H, Gr > H; Gr ﬃ G may be found by the following procedure: (i) determine all subgroups Hs < G of the same index and distribute them into classes of conjugate subgroups with respect to G. From each class of conjugate subgroups, choose a representative Hr , speciﬁed by ðP; pÞr ;3 (ii) apply ðP; pÞ1 r to the group G in order to obtain the group Gr ; and (iii) test whether Gr is already among the determined supergroups of H. If it is not, then Gr is a new supergroup of H and further supergroups may be generated by the coset representatives of the decomposition of N A ðHÞ relative to ðN A ðHÞ \ N A ðGÞÞ as explained above. The procedure described above for the determination of supergroups is also applied to the determination of minimal supergroups G of H (cf. Section 2.1.7). In this case, the distinct maximal subgroups Hr , representatives of the classes of conjugate subgroups Hs with respect to G, are retrieved directly from the maximal-subgroup database of the server.

1.7.3.2. Supergroups of space groups The problem of the determination of the supergroups of a given space group is of rather general interest. For several applications it is not sufﬁcient to know only the space-group types of the supergroups of a given group; it is instead necessary to have available all different supergroups Gr > H that are isomorphic to G and are of the same index [i]. In the literature there are few papers treating the supergroups of space groups in detail (Koch, 1984; Wondratschek & Aroyo, 2001). In IT A one ﬁnds only listings of minimal supergroups of space groups which, in addition, are not explicit: they only provide for each space group H the list of those space-group types in which H occurs as a maximal subgroup (cf. Section 2.1.6). It is not trivial to determine all supergroups Gr > H if only the types of the minimal supergroups are known. The Bilbao Crystallographic Server offers two basic programs that solve this problem for a given ﬁnite index [i] (Ivantchev et al., 2002): (i) MINSUP, which gives all minimal supergroups of index 2, 3 and 4 of a given space group, and (ii) SUPERGROUPS, which calculates all different supergroups of a given space-group type and a given index. As in the case of subgroups, we have developed two complementary programs that involve the calculation of supergroups of space groups: (i) the program CELLSUPER, for calculating the supergroups of a space group for a given ½iL index, and (ii) COMMONSUPER for the computation of common supergroups of two or more space groups.

Input to MINSUP and SUPERGROUPS: (i) The program MINSUP needs as input the IT A number (or the Hermann–Mauguin symbol) of the group for which the minimal supergroups have to be determined. The type of supergroup is chosen from a table (returned by the program) which contains the IT A number of the minimal supergroup, its Hermann–Mauguin symbol and the index of the group in the supergroup. There is also a link to a list of the transformation matrices that relate the basis of the supergroup with that of the subgroup. (ii) For the determination of all supergroups of a given type, it is necessary to select the type of the normalizers of the group and the supergroup. By default the Euclidean normalizers of general cell metrics are used as listed in Tables 15.2.1.3 and

2 A morphotropic transition is an abrupt change in the structure of a solid solution with variation in composition.

3

64

To simplify the algorithm, the condition ar 62 N A ðGÞ is substituted by ar 62 G.

1.7. THE BILBAO CRYSTALLOGRAPHIC SERVER Table 1.7.3.2. P422, No. 89, supergroups of P222, No. 16 (a = b = c), index 2, as determined by MINSUP

15.2.1.4 of IT A. The afﬁne normalizers of the space groups (except triclinic and monoclinic) are also accessible. For a translation lattice with metrics of apparent higher symmetry, the users may themselves provide the set of additional generators for the speciﬁc Euclidean normalizer (cf. Table 15.2.1.3 of IT A and Koch & Mu¨ller, 1990). (iii) The program SUPERGROUPS takes as input the IT A numbers of the space groups G and H and the index of H in G. The transformation matrices relating the bases of G and H necessary for the determination of the supergroups Gr are retrieved from the IT A1 database. In case of a non-minimal supergroup, the program SUBGROUPGRAPH determines the transformation matrix (or matrices) for the corresponding chains of maximal subgroups that relate G and H. As in the case of MINSUP, the space-group normalizers used by default are the Euclidean normalizers. It is also possible for the user to use the afﬁne normalizers given in IT A or to provide a speciﬁc one.

The different supergroups are distinguished by the transformation matrices (P, p)r and the coset representatives of the decomposition of ðP422Þr with respect to H. (The unit element is taken as a coset representative in all cases and is not listed). The locations of the axes 4 are referred to the orthorhombic cell. Supergroup

(P, p)j

Coset representative

ðP422Þ1 ðP422Þ2 ðP422Þ3 ðP422Þ4 ðP422Þ5 ðP422Þ6

a; b; c; 0 0 0 a; b; c; 12 0 0 b; c; a; 0 0 0 b; c; a; 0 12 0 c; a; b; 0 0 0 c; a; b; 12 0 0

ð4z j 0 0 0Þ ð4z j 12 12 0Þ ð4y j 0 0 0Þ ð4y j 12 0 12Þ ð4x j 0 0 0Þ ð4x j 0 12 12Þ

Location of 4 00z 1 0 z 2

0y0 1 y 0 2

x00 x 12 0

supergroups, as calculated by MINSUP, are listed in Table 1.7.3.2. They are distinguished by their transformation matrix– column pairs ðP; pÞr and the coset representatives of the decomposition of Gr with respect to H. The existence of the six different supergroups becomes obvious if we consider the type and location of the symmetry elements corresponding to the listed coset representatives of the different supergroups (Table 1.7.3.2). Owing to the specialized metrics of P222, the fourfold axis of P422 can be chosen along any of the three orthorhombic axes. Accordingly, the six supergroups are distributed into three pairs. Comparison of the space-group diagrams of P422 and P222 (Fig. 1.7.3.5) shows that the two supergroups for each orientation of the fourfold axis correspond exactly to the two possible locations of the fourfold axis in the orthorhombic cell.

Output of MINSUP and SUPERGROUPS: For the two supergroup programs the results contain: (i) The transformation matrix (P, p)r that relates the basis of the supergroup with that of the subgroup. (ii) One representative from each coset in the decomposition of the supergroup Gr with respect to the group H. The full cosets of the decomposition Gr : H are also accessible. The elements of Gr are listed with respect to the basis of the subgroup H. From the considerations given above it should have become clear that the aim of the presented procedure and the supergroup programs is to solve the following ‘purely’ group-theoretical problem: Given a group–subgroup pair of space groups G > H, determine all supergroups Gr of H isomorphic to G. The procedure does not include any preliminary checks on the compatibility of the metric of the studied space group with that of a supergroup. Depending on the particular case some of the supergroups obtained are not space groups but just afﬁne groups isomorphic to space groups (see Koch, 1984). As an example consider the cubic supergroups of P21 21 21, No. 19: only if the three basis vectors of P21 21 21 have equal length can one speak of supergroups of the cubic space-group type P21 3, No. 198. However, for each group P21 21 21 there exist afﬁne analogues of P21 3 as supergroups.

Example 1.7.3.2.2 Here we consider the supergroups Gr ¼ a1 r G ar of H with ar 62 N A ðHÞ. Consider the group Pnma, No. 62, and its minimal supergroups of type Cmcm, No. 63, of index 2. The group Cmcm has two maximal Pnma subgroups: ðPnmaÞ1 speciﬁed by ðP; pÞ1 ¼ ðb; c; aÞ, and ðPnmaÞ2 with ðP; pÞ2 ¼ ðc; a; b; 14 ; 14 ; 0Þ. The Euclidean and afﬁne normalizers of Cmcm and Pnma are identical and correspond to the group Pmmm with the additional translations ðx þ 12 ; y; zÞ, ðx; y þ 12 ; zÞ and ðx; y; z þ 12Þ. Accordingly, the application of the normalizer procedure to any of the two group–subgroup pairs will not generate further equivalent supergroups. The ﬁrst pair Cmcm > ðPnmaÞ1 gives rise to the minimal supergroup Bbmm ðCmcm; b; c; aÞ. The second supergroup Amma ðCmcm; c; a; bÞ can only be obtained considering the second group–subgroup pair. Both supergroups are related by a cyclic

Example 1.7.3.2.1 Here we consider supergroups Gr ¼ a1 r G a r of H with ar 2 N A ðHÞ. As an example we consider the group–supergroup pair H < G with H ¼ P222, No. 16, and the supergroup G ¼ P422, No. 89, of index [i] = 2. Further, we suppose that the group P222 has specialized cell metrics speciﬁed as a = b = c. In the subgroup data of P422 there is only one entry for the subgroup P222 of index 2. We are interested in all P422 supergroups of index 2 of the group P222. The afﬁne normalizer of P422 coincides with its Euclidean normalizer and it has the translations ðx þ 12 ; y þ 12 ; zÞ, ðx; y; z þ 12Þ and the inversion as additional generators (cf. IT A, Table 15.2.1.4). The Euclidean normalizer of P222 with a = b = c coincides with its afﬁne normalizer. It corresponds to the cubic group Pm3m with the additional generating translations ðx þ 12 ; y; zÞ, ðx; y þ 12 ; zÞ and ðx; y; z þ 12Þ (cf. IT A, Table 15.2.1.3). The decomposition of N ðHÞ with respect to the intersection of the two normalizers contains six cosets, i.e. the group P222 has six supergroups P422 isomorphic to each other. The different

Fig. 1.7.3.5. Space-group diagrams for (a) P222, No. 16, with specialized cell metrics (see the text) and (b) P422, No. 89. For explanations of the space-group diagrams, see IT A Chapter 1.4.

65

1. SPACE GROUPS AND THEIR SUBGROUPS rotation of the three axes which is not in the normalizer of H (or G).

The input data for COMMONSUPER include the speciﬁcation of H1 and H2 , the numbers of formula units per conventional unit cell, and the maximum lattice index ½ðiL Þmax . The output data of COMMONSUPER are:

The number of supergroups of a space group H of a ﬁnite index is not always ﬁnite. This is the case of a space group H whose normalizer N ðHÞ contains continuous translations in one, two or three independent directions (see IT A, Part 15). As typical examples one can consider the inﬁnitely many centrosymmetric supergroups of the polar groups: there are no restrictions on the location of the additional inversion centre on the polar axis. For such group–supergroup pairs there could be up to three parameters r, s and t in the origin-shift column of the transformation matrix and in the translational part of the coset representatives. The parameters can have any value and each value corresponds to a different supergroup of the same spacegroup type.

(i) The space-group types of the common supergroups G of H1 and H2 with the indices ½i1 and ½i2 , ½ðiL Þ1 and ½ðiL Þ2 , and ½ðiP Þ1 and ½ðiP Þ2 . Optional links to the programs POINT and GENPOS give access to data for the point group P G and the general positions of the supergroup G. (ii) A link to the SUPERGROUPS module (see Section 1.7.3.2.1) enables the calculation of all different supergroups Gr of H1 and H2 of a space-group type G and indices ½i1 and ½i2 . Each supergroup Gr is speciﬁed by the corresponding transformation matrix relating the conventional bases of the supergroup and the group, and the representatives of the coset decomposition of Gr relative to H1 or H2 . Example 1.7.3.2.3 The program COMMONSUPER is useful in the search for structural relationships between structures whose symmetry groups H1 and H2 are not group–subgroup related. The derivation of the two structures as different distortions from a basic structure is a clear manifestation of such relationships. The symmetry group of the basic structure is a common supergroup of H1 and H2 . Consider the ternary intermetallic compound CeAuGe. At 8.7 GPa a ﬁrst-order phase transition is observed from a hexagonal arrangement (space group P63 mc, No. 186, two formula units per unit cell, Z1 ¼ 2) into an orthorhombic high-pressure modiﬁcation of symmetry Pnma, No. 62, Z2 ¼ 4 (Brouskov et al., 2005). There is no group– subgroup relation between the symmetry groups of the highand low-pressure structures. For ½ðiL Þmax ¼ 4 the program ﬁnds two common supergroups of H1 ¼ P63 mc, Z1 ¼ 2 and H2 ¼ Pnma, Z2 ¼ 4: (i) the group P63 =mmc with ½i1 ¼ 2 and ½i2 ¼ 6, and (ii) P6=mmm, with ½i1 ¼ 4 and ½i2 ¼ 12. The common basic structure of the AlB2 type, proposed by Brouskov et al. (2005), corresponds to the common supergroup P6=mmm found by COMMONSUPER.

1.7.3.2.2. The program CELLSUPER The program CELLSUPER is an application similar to CELLSUB (cf. Section 1.7.3.1.4): in this case, the search is for the space-group types of supergroups Gs of H of a given maximum lattice index ½ðiL Þmax . The algorithm is similar to that of CELLSUB: using the data for the index and space-group types of minimal supergroups, the program constructs a tree of minimal supergroups starting from H and imposing the condition ½iL ½ðiL Þmax . The input data of CELLSUPER coincide with those of CELLSUB with the only difference being that they are referred to the low-symmetry group H. The output data include: (i) The space-group types of the supergroups Gs of H with the corresponding indices [i], [iP] and [iL]. The supergroups are classiﬁed into t-supergroups, k-supergroups and general supergroups. (ii) A link to the SUPERGROUPS module (cf. Section 1.7.3.2.1) enables the calculation of all different supergroups ðGs Þr of H of the space-group type Gs and index [i]. Each supergroup ðGs Þr is speciﬁed by the corresponding transformation matrix relating the conventional bases of the supergroup and the group, and the representatives of the coset decomposition ðGs Þr : H.

1.7.4. Relations of Wyckoff positions for a group–subgroup pair of space groups

1.7.3.2.3. The program COMMONSUPER The program COMMONSUPER calculates the space-group types of common supergroups G of two space groups H1 and H2 for a given maximal lattice index ½ðiL Þmax . The procedure used is analogous to the one implemented in the program COMMONSUBS (cf. Section 1.7.3.1.5). The two sets of supergroups of H1 and of H2 are determined by the program CELLSUPER. The intersection of the sets of supergroups gives the set of the spacegroup types of the common supergroups fGg of H1 and H2 with ½iL ½ðiL Þmax . A relation between the indices ½i1 ¼ jGj=jH1 j and ½i2 ¼ jGj=jH2 j is obtained by imposing the structural requirement of equal numbers of formula units in the (primitive) unit cell of the common supergroup G obtained from the numbers of the formula units Z1 and Z2 of H1 and H2 : ½i2 ¼ ½i1

Z2 jP H1 j : Z1 jP H2 j

Consider two group–subgroup-related space groups G > H. Atoms that are symmetrically equivalent under G, i.e. belong to the same orbit of G, may become non-equivalent under H, (i.e. the orbit splits) and/or their site symmetries may be reduced. The orbit relations induced by the symmetry reduction are the same for all orbits belonging to a Wyckoff position, so one can speak of Wyckoff-position relations or splitting of Wyckoff positions. Theoretical aspects of the relations of the Wyckoff positions for a group–subgroup pair of space groups G > H have been treated in detail by Wondratschek (1993) (see also Section 1.5.3). A compilation of the Wyckoff-position splittings for all space groups and all their maximal subgroups is published as Part 3 of this volume. However, for certain applications it is easier to have the appropriate computer tools for the calculations of the Wyckoff-position splittings for G > H: for example, when H is not a maximal subgroup of G, or when the space groups G > H are related by transformation matrices different from those listed in the tables of Part 3. The program WYCKSPLIT (Kroumova, Perez-Mato & Aroyo, 1998) calculates the Wyckoff-position splittings for any group–subgroup pair. In addition, the program

ð1:7:3:2Þ

The program COMMONSUPER selects and lists those supergroups of the set fGg whose indices ½i1 ¼ jGj=jH1 j and ½i2 ¼ jGj=jH2 j satisfy the above condition.

66

1.7. THE BILBAO CRYSTALLOGRAPHIC SERVER

ð1 ; 0; 3Þ of P42 =mnm, No. Fig. 1.7.4.1. Sequence of calculations of WYCKSPLIT for the splitting of the Wyckoff positions 2a m:mm ð0; 0; 0Þ and 4d 4:: 2 4 136, with respect to its subgroup Cmmm, No. 65, of index 2. ðOG ÞH are the orbits of P42 =mnm in the basis of Cmmm.

provides further information on Wyckoff-position splittings that is not listed in Part 3, namely the relations between the representatives of the orbit of G and the corresponding representatives of the suborbits of H.

(ii) the assignment of the orbits OH ðXj Þ to the Wyckoff positions W Hl of H; (iii) the determination of the correspondence between the points Xjm of the suborbits OH ðXj Þ and the representatives of W Hl .

1.7.4.1. The program WYCKSPLIT

The direct determination of the suborbits OH ðXj Þ is not an easy task. The restrictions on the site-symmetry groups S H ðXj Þ which follow from the reduction-factor lemma (cf. Section 1.5.3) are helpful but in many cases not sufﬁcient for the determination of the suborbits. The solution used in our approach is based on the general-position decomposition, equation (1.7.4.1). It is important to note that each of the suborbits of the general position gives exactly one suborbit OH ðXj Þ when the variable parameters of OH ðX0;j Þ are substituted by the corresponding parameters (ﬁxed or variable) of the special position. The assignment of the suborbits to the Wyckoff positions of H is done by comparing the multiplicities of the orbits, the number of the variable parameters [the number of the variable parameters of W Hl is equal to or greater than that of OH ðXj Þ] and the values of the ﬁxed parameters. If there is more than one Wyckoff position

To simplify the notation, we assume in the following that the group G, its Wyckoff-position representatives and the points of the orbits are referred to the basis of the subgroup H. (1) Splitting of the general position. Consider the group– subgroup chain of space groups G > H of an index [i]. The general-position orbits OG ðX0 Þ have unique splitting schemes: they are split into [i] suborbits OH ðX0;j Þ of the general position of the subgroup, i.e. they all are of the same multiplicity: OG ðX0 Þ ¼ OH ðX0;1 Þ [ . . . [ OH ðX0;i Þ:

ð1:7:4:1Þ

This property is a direct corollary of the relation between the index [i] and the so-called reduction factors of the sitesymmetry groups S G ðXÞ and S H ðXÞ of a point X in G and H (Wondratschek, 1993; see also Section 1.5.3). The determination of the splitting of the general-position orbit OG ðX0 Þ is then reduced to the selection of the [i] points ðX0;j Þ belonging to the [i] independent suborbits OH ðX0;j Þ of H, equation (1.7.4.1). Owing to the one-to-one mapping between the general-position points of OG ðX0 Þ and the elements g of G, the right cosets Hg j of the decomposition of G with respect to H (cf. Deﬁnition 1.2.4.2.1) correspond to the suborbits OH ðX0;j Þ. In this way, the representatives of these cosets can be chosen as the [i] points X0;j in the decomposition of OG ðX0 Þ. (2) Splitting of a special position. The calculation of the splitting of a special Wyckoff position W G involves the following steps:

Table 1.7.4.1. Wyckoff positions of Cmmm (No. 65) with multiplicities 2 and 8 Each Wyckoff position is speciﬁed by its multiplicity and Wyckoff letter, site symmetry and a coordinate triplet of a representative element.

(i) the determination of the suborbits OH ðXj Þ into which the special-Wyckoff-position orbit OG ðXÞ has split;

67

Wyckoff multiplicity and letter

Site symmetry

Representative element

2d 2c 2b 2a 8q 8p 8o 8n 8m

mmm mmm mmm mmm ..m ..m .m. m.. ..2

ð0; 0; 12Þ ð12 ; 0; 12Þ ð12 ; 0; 0Þ ð0; 0; 0Þ ðx; y; 12Þ ðx; y; 0Þ ðx; 0; zÞ ð0; y; zÞ ð14 ; 14 ; zÞ

1. SPACE GROUPS AND THEIR SUBGROUPS of H satisfying these conditions, then the assignment is done by a direct comparison of the points of the suborbit OH ðXj Þ with those of a special W Hl orbit obtained by substitution of the variable parameters by arbitrary numbers. The determination of the explicit correspondences between the points of OH ðXj Þ and the representatives of W Hl is done by comparing the values of the ﬁxed parameters and the variable-parameter relations in both sets.

2a m:mm ð0; 0; 0Þ are shown in Fig. 1.7.4.1. First it is necessary to transform the representatives of W G to the basis of H, which gives the orbits ðOG ÞH ð0; 0; 0Þ and ðOG ÞH ð14 ; 14 ; 34Þ. The substitution of the values x = 0, y = 0, z = 0 in the coordinate triplets of the decomposed general position of G (cf. the corresponding output of WYCKSPLIT) gives two suborbits of multiplicity 2 2a;1 1 1 for the 2a position: OH ð0; 0; 0Þ and O2a;2 H ð0; 2 ; 2Þ. The assign2a;j ment of the suborbits OH to the Wyckoff positions of H (cf. Table 1.7.4.1) is straightforward. Summarizing: the Wyckoff position 2a m:mm ð0; 0; 0Þ splits into two independent positions of Cmmm with no site-symmetry reduction:

The program WYCKSPLIT calculates the splitting of the Wyckoff positions for a group–subgroup pair G > H given the corresponding transformation relating the coordinate systems of G and H.

2a m:mm ð0; 0; 0Þ ! 2a mmm ð0; 0; 0Þ [ 2c mmm ð0; 12 ; 12Þ:

Input to WYCKSPLIT: The program needs as input the following information:

No splitting occurs for the case of the special 4d position orbit: 1 1 3 the result is one orbit of multiplicity 8, O4d;1 H ð4 ; 4 ; 4Þ. The 4d;1 1 1 3 assignment of OH ð4 ; 4 ; 4Þ is also obvious: there are ﬁve Wyckoff positions of Cmmm of multiplicity 8 but four of them are discarded as they have ﬁxed parameters 0 or 12 (Table 4d;1 1.7.4.1). The orbit OH belongs to the Wyckoff position 8m ::2 ð14 ; 14 ; zÞ. As expected, the sum of the site-symmetry reduction factors equals the index of Cmmm in P42 =mnm for both cases (cf. Section 1.5.3). The loss of the fourfold inversion axis results in the appearance of an additional degree of freedom corresponding to the variable parameter of 8m ::2 ð14 ; 14 ; zÞ.

(i) The speciﬁcation of the space-group types G and H by their IT A numbers. (ii) The transformation matrix–column pair ðP; pÞ that relates the basis of G to that of H. The user can input a speciﬁc transformation or follow a link to the IT A1 database for the maximal subgroups of G. In the case of a non-maximal subgroup, the program SUBGROUPGRAPH provides the transformation matrix (or matrices) for a speciﬁed index of H in G. The transformations are checked for consistency with the default settings of G and H used by the program. The Wyckoff positions W G to be split can be selected from a list. In addition, it is possible to calculate the splitting of any orbit OG ðXÞ speciﬁed by the coordinate triplet of one of its points.

References

Output of WYCKSPLIT: (i) Splittings of the selected Wyckoff positions W G into Wyckoff positions W Hl of the subgroup, speciﬁed by their multiplicities and Wyckoff letters. (ii) The correspondence between the representatives of the Wyckoff position and the representatives of its suborbits is presented in a table where the coordinate triplets of the representatives of W G are referred to the bases of the group and of the subgroup.

Aroyo, M. I., Kirov, A., Capillas, C., Perez-Mato, J. M. & Wondratschek, H. (2006). Bilbao Crystallographic Server. II. Representations of crystallographic point groups and space groups. Acta Cryst. A62, 115– 128. Aroyo, M. I., Perez-Mato, J. M., Capillas, C., Kroumova, E., Ivantchev, S., Madariaga, G., Kirov, A. & Wondratschek, H. (2006). Bilbao Crystallographic Server: I. Databases and crystallographic computing programs. Z. Kristallogr. 221, 15–27. Brouskov, V., Hanﬂand, M., Po¨ttgen, R. & Schwarz, U. (2005). Structural phase transitions of CeAuGe at high pressure. Z. Kristallogr. 220, 122– 127. Capillas, C. (2006). Me´todos de la cristalografı´a computacional en el ana´lisis de transiciones de fase estructurales. PhD thesis, Universidad del Paı´s Vasco, Spain. Capillas, C., Kroumova, E., Aroyo, M. I., Perez-Mato, J. M., Stokes, H. T. & Hatch, D. M. (2003). SYMMODES: a software package for grouptheoretical analysis of structural phase transitions. J. Appl. Cryst. 36, 953–954. Capillas, C., Perez-Mato, J. M. & Aroyo, M. I. (2007). Maximal symmetry transition paths for reconstructive phase transitions. J. Phys. Condens. Matter, 19, 275203. International Tables for Crystallography (2005). Vol. A, Space-Group Symmetry, edited by Th. Hahn, 5th ed. Dordrecht: Kluwer Academic Publishers. International Tables for Crystallography (2002). Vol. E, Subperiodic Groups, edited by V. Kopsky´ & D. B. Litvin. Dordrecht: Kluwer Academic Publishers. Ivantchev, S., Kroumova, E., Aroyo, M. I., Perez-Mato, J. M., Igartua, J. M., Madariaga, G. & Wondratschek, H. (2002). SUPERGROUPS – a computer program for the determination of the supergroups of space groups. J. Appl. Cryst. 35, 511–512. Ivantchev, S., Kroumova, E., Madariaga, G., Pe´rez-Mato, J. M & Aroyo, M. I. (2000). SUBGROUPGRAPH: a computer program for analysis of group–subgroup relations between space groups. J. Appl. Cryst. 33, 1190–1191. Koch, E. (1984). The implications of normalizers on group–subgroup relations between space groups. Acta Cryst. A40, 593–600. Koch, E. & Mu¨ller, U. (1990). Euklidische Normalisatoren fu¨r trikline und monokline Raumgruppen bei spezieller Metrik des Translationengitters. Acta Cryst. A46, 826–831.

WYCKSPLIT can treat group or subgroup data in unconventional settings if the transformation matrices to the corresponding conventional settings are given. Example 1.7.4.1.1 To illustrate the calculation of the Wyckoff-position splitting we consider the group–subgroup pair P42 =mnm (No. 136) > Cmmm (No. 65) of index 2, see Fig. 1.7.4.1. The relation between the conventional bases ða; b; cÞ of the group and of the subgroup ða0 ; b0 ; c0 Þ is retrieved by the program MAXSUB and is given by a0 ¼ a b, b0 ¼ a þ b; c0 ¼ c. The general position of P42 =mnm splits into two suborbits of the general position of Cmmm: 16k 1 ðx; y; zÞ ! 16r 1 ðx1 ; y1 ; z1 Þ [ 16r 1 ðx2 ; y2 ; z2 Þ: This splitting is directly related to the coset decomposition of P42 =mnm with respect to Cmmm. As coset representatives, i.e. as points which determine the splitting of the general position, one can choose X0;1 ¼ ðx1 ; y1 ; z1 Þ and X0;2 ¼ ðx2 ; y2 ; z2 Þ ¼ ðy; x þ 12 ; z þ 12ÞÞ (referred to the basis of the subgroup). The splitting of any special Wyckoff position is obtained from the splitting of the general position. The consecutive steps of ð1 ; 0; 3Þ and the splittings of the special positions 4d 4:: 2 4

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1.7. THE BILBAO CRYSTALLOGRAPHIC SERVER Noheda, B., Gonzalo, J. A., Cross, L. E, Guo, R., Park, S.-E., Cox, D. E. & Shirane, G. (2000). Tetragonal-to-monoclinic phase transition in a ferroelectric perovskite: The structure of Pb(Zr0.52Ti0.48)O3. Phys. Rev. B, 61, 8687–8695. Wondratschek, H. (1993). Splitting of Wyckoff positions (orbits). Mineral. Petrol. 48, 87–96. Wondratschek, H. & Aroyo, M. I. (2001). The application of Hermann’s group M in group–subgroup relations between space groups. Acta Cryst. A57, 311–320.

Kroumova, E., Perez-Mato, J. M. & Aroyo, M. I. (1998). WYCKSPLIT: a computer program for determination of the relations of Wyckoff positions for a group–subgroup pair. J. Appl. Cryst. 31, 646. Kroumova, E., Perez-Mato, J. M., Aroyo, M. I., Ivantchev, S., Madariaga, G. & Wondratschek, H. (1998). The Bilbao Crystallographic Server: a web site with crystallographic tools using the International Tables for Crystallography. Abstracts of the 18th European Crystallographic Meeting, Prague, Czech Republic. Noheda, B., Cox, D. E., Shirane, G., Gonzalo, J. A., Cross, L. E. & Park, S.-E. (1999). A monoclinic ferroelectric phase in the Pb(Zr1xTix)O3 solid solutions. Appl. Phys. Lett. 74, 2059.

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references

International Tables for Crystallography (2011). Vol. A1, Chapter 2.1, pp. 72–96.

2.1. Guide to the subgroup tables and graphs By Hans Wondratschek and Mois I. Aroyo; Yves Billiet (Section 2.1.5)

2.1.1. Contents and arrangement of the subgroup tables

(1) The short (international) Hermann–Mauguin symbol for the plane group or space group. These symbols will be henceforth referred to as ‘HM symbols’. HM symbols are discussed in detail in Chapter 12.2 of IT A with a brief summary in Section 2.2.4 of IT A. (2) The plane-group or space-group number as introduced in International Tables for X-ray Crystallography, Vol. I (1952). These numbers run from 1 to 17 for the plane groups and from 1 to 230 for the space groups. (3) The full (international) Hermann–Mauguin symbol for the plane or space group, abbreviated ‘full HM symbol’. This describes the symmetry in up to three symmetry directions (Blickrichtungen) more completely, see Table 12.3.4.1 of IT A, which also allows comparison with earlier editions of International Tables. (4) The Schoenﬂies symbol for the space group (there are no Schoenﬂies symbols for the plane groups). The Schoenﬂies symbols are primarily point-group symbols; they are extended by superscripts for a unique designation of the space-group types, cf. IT A, Sections 12.1.2 and 12.2.2.

In this chapter, the subgroup tables, the subgroup graphs and their general organization are discussed. In the following sections, the different types of data are explained in detail. For every plane group and every space group there is a separate table of maximal subgroups and minimal supergroups. The subgroup data are listed either individually, or as members of (inﬁnite) series, or both. The supergroup data are not as complete as the subgroup data. However, most of them can be obtained by proper evaluation of the subgroup data, as shown in Section 2.1.7. In addition, there are graphs of translationengleiche and klassengleiche subgroups which contain for each space group all kinds of subgroups, not just the maximal ones. The presentation of the plane-group and space-group data in the tables of Chapters 2.2 and 2.3 follows the style of the tables of Parts 6 (plane groups) and 7 (space groups) in Vol. A of International Tables for Crystallography (2005), henceforth abbreviated as IT A. The data comprise: Headline Generators selected General position I Maximal translationengleiche subgroups II Maximal klassengleiche subgroups I Minimal translationengleiche supergroups II Minimal non-isomorphic klassengleiche supergroups.

2.1.2.2. Data from IT A 2.1.2.2.1. Generators selected As in IT A, for each plane group and space group G a set of symmetry operations is listed under the heading ‘Generators selected’. From these group elements, G can be generated conveniently. The generators in this volume are the same as those in IT A. They are explained in Section 2.2.10 of IT A and the choice of the generators is explained in Section 8.3.5 of IT A. The generators are listed again in this present volume because many of the subgroups are characterized by their generators. These (often nonconventional) generators of the subgroups can thus be compared with the conventional ones without reference to IT A.

For the majority of groups, the data can be listed completely on one page. Sometimes two pages are needed. If the data extend less than half a page over one full page and data for a neighbouring space-group table ‘overﬂow’ to a similar extent, then the two overﬂows are displayed on the same page. Such deviations from the standard sequence are indicated on the relevant pages by a remark Continued on . . . . The two overﬂows are separated by a solid line and are designated by their headlines. The sequence of the plane groups and space groups G in this volume follows exactly that of the tables of Part 6 (plane groups) and Part 7 (space groups) in IT A. The format of the subgroup tables has also been chosen to resemble that of the tables of IT A as far as possible. Graphs for translationengleiche and klassengleiche subgroups are found in Chapters 2.4 and 2.5. Examples of graphs of subgroups can also be found in Section 10.1.4.3 of IT A, but only for subgroups of point groups. The graphs for the space groups are described in Section 2.1.8.

2.1.2.2.2. General position Like the generators, the general position has also been copied from IT A, where an explanation can be found in Section 2.2.11. The general position in IT A is the ﬁrst block under the heading ‘Positions’, characterized by its site symmetry of 1. The elements of the general position have the following meanings: (1) they are coset representatives of the space group G with respect to its translation subgroup. The other elements of a coset are obtained from its representative by combination with translations of G; (2) they form a kind of shorthand notation for the matrix description of the coset representatives of G; (3) they are the coordinates of those symmetry-equivalent points that are obtained by the application of the coset representatives on a point with the coordinates x; y; z; (4) their numbers refer to the geometric description of the symmetry operations in the block ‘Symmetry operations’ of the space-group tables of IT A.

2.1.2. Structure of the subgroup tables Some basic data in these tables have been repeated from the tables of IT A in order to allow the use of the subgroup tables independently of IT A. These data and the main features of the tables are described in this section. 2.1.2.1. Headline The headline contains the speciﬁcation of the space group for which the maximal subgroups are considered. The headline lists from the outside margin inwards: Copyright © 2011 International Union of Crystallography

Many of the subgroups H < G in these tables are characterized by the elements of their general position. These elements are

72

2.1. GUIDE TO THE SUBGROUP TABLES AND GRAPHS speciﬁed by numbers which refer to the corresponding numbers in the general position of G. Other subgroups are listed by the numbers of their generators, which again refer to the corresponding numbers in the general position of G. Therefore, the listing of the general position of G as well as the listing of the generators of G is essential for the structure of these tables. For examples, see Sections 2.1.3 and 2.1.4.

maximal isomorphic subgroups are listed, whereas in IT A only a few of them are described. In addition, the subgroups are described here in more detail. The sequence of the subgroups within each block follows the value of the index; subgroups of lowest index are listed ﬁrst. Subgroups having the same index are listed according to their lattice relations to the lattice of the original group G, cf. Section 2.1.4.3. Subgroups with the same lattice relations are listed in decreasing order of space-group number. Conjugate subgroups have the same index and the same spacegroup number. They are grouped together and connected by a brace on the left-hand side. Conjugate classes of maximal subgroups and their lengths are therefore easily recognized. In the series of maximal isomorphic subgroups, braces are inapplicable so there the conjugacy classes are stated explicitly. The block designations are:

2.1.2.3. Speciﬁcation of the setting All 17 plane-group types1 and 230 space-group types are listed and described in IT A. However, whereas each plane-group type is represented exactly once, 44 space-group types, i.e. nearly 20%, are represented twice. This means that the conventional setting of these 44 space-group types is not uniquely determined and must be speciﬁed. The same settings underlie the data of this volume, which follows IT A as much as possible. There are three reasons for listing a space-group type twice:

(1) In the block I Maximal translationengleiche subgroups, all maximal translationengleiche subgroups are listed, see Section 2.1.3. None of them are isomorphic. (2) Under the heading II Maximal klassengleiche subgroups, all maximal klassengleiche subgroups are listed in up to three separate blocks, each of them marked by a bullet, . Maximal non-isomorphic subgroups can only occur in the ﬁrst two blocks, whereas maximal isomorphic subgroups are only found in the last two blocks.

(1) Each of the 13 monoclinic space-group types is listed twice, with ‘unique axis b’ and ‘unique axis c’, where b or c is the direction distinguished by symmetry (monoclinic axis). The tables of this Part 2 always refer to the conventional cell choice, i.e. ‘cell choice 1’, whereas in IT A for each setting three cell choices are shown. In the graphs of Chapters 2.4 and 2.5, the monoclinic space groups are designated by their short HM symbols. Note on standard monoclinic space-group symbols: In this volume, as in IT A, the monoclinic space groups are listed for two settings. Nevertheless, the short symbol for the setting ‘unique axis b’ has always been used as the standard (short) HM symbol. It does not carry any information about the setting of the particular description. As in IT A, no other short symbols are used for monoclinic space groups and their subgroups in the present volume. (2) Twenty-four orthorhombic, tetragonal or cubic space-group types are listed with two different origins. In general, the origin is chosen at a point of highest site symmetry (‘origin choice 1’); for exceptions see IT A, Section 8.3.1. If there are centres of inversion and if by this rule the origin is not at an inversion centre, then the space group is described once more with the origin at a centre of inversion (‘origin choice 2’). (3) There are seven trigonal space groups with a rhombohedral lattice. These space groups are described in a hexagonal basis (‘hexagonal axes’) with a rhombohedrally centred hexagonal lattice as well as in a rhombohedral basis with a primitive lattice (‘rhombohedral axes’).

Loss of centring translations. This block is described in Section 2.1.4.2 in more detail. Subgroups in this block are always non-isomorphic. The block is empty (and is then omitted) for space groups that are designated by an HM symbol starting with the letter P. Enlarged unit cell. In this block, those maximal klassengleiche subgroups H < G of index 2, 3 and 4 are listed for which the conventional unit cell of H is larger than that of G, see Section 2.1.4.3. These subgroups may be nonisomorphic or isomorphic, see Section 2.1.5. Therefore, it may happen that a maximal isomorphic klassengleiche subgroup of index 2, 3 or 4 is listed twice: once here explicitly and once implicitly as a member of a series. Series of maximal isomorphic subgroups. Maximal klassengleiche subgroups H < G of indices 2, 3 and 4 may be isomorphic while those of index i > 4 are always isomorphic to G. The total number of maximal isomorphic klassengleiche subgroups is inﬁnite. These inﬁnitely many subgroups cannot be described individually but only by a (small) number of inﬁnite series. In each series, the individual subgroups are characterized by a few integer parameters, see Section 2.1.5.

If there is a choice of setting for the space group G, the chosen setting is indicated under the HM symbol in the headline. If a subgroup H < G belongs to one of these 44 space-group types, its ‘conventional setting’ must be deﬁned. The rules that are followed in this volume are explained in Section 2.1.2.5.

(3) After the data for the subgroups, the data for the supergroups are listed. The data for minimal non-isomorphic supergroups are split into two main blocks with the headings I Minimal translationengleiche supergroups and II Minimal non-isomorphic klassengleiche supergroups.

2.1.2.4. Sequence of the subgroup and supergroup data

(4) The latter block is split into the listings

As in the subgroup data of IT A, the sequence of the maximal subgroups is as follows: subgroups of the same kind are collected in a block. Each block has a heading. Compared with IT A, the blocks have been partly reorganized because in this volume all

Additional centring translations and Decreased unit cell. (5) Minimal isomorphic supergroups are not listed because they can be read from the data for the maximal isomorphic subgroups.

1

The clumsy terms ‘plane-group type’ and ‘space-group type’ are frequently abbreviated by the shorter terms ‘plane group’ and ‘space group’ in what follows, as is often done in crystallography. Occasionally, however, it is essential to distinguish the individual group from its ‘type of groups’.

For details, see Section 2.1.6.

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2. MAXIMAL SUBGROUPS OF THE PLANE GROUPS AND SPACE GROUPS 2.1.2.5. Special rules for the setting of the subgroups

The monoclinic axis c and the glide reﬂection a are retained because P112=a is the conventional full HM symbol for unique axis c, cell choice 1. ½2 b0 ¼ 2b; c0 ¼ 2c, all four subgroups A112=m. The monoclinic axis c and the A centring are retained because A112=m is the conventional full HM symbol for this setting. ½2 a0 ¼ 2a; b0 ¼ 2b, both subgroups C112=e. The monoclinic axis c is retained. The glide reﬂection is designated by ‘e’ (simultaneous a- and b-glide reﬂection in the same plane perpendicular to c). The nonconventional C centring is converted into the conventional primitive setting P, by which the e-glide reﬂection also becomes an a-glide reﬂection.

The multiple listing of 44 space-group types has implications for the subgroup tables. If a subgroup H < G belongs to one of these types, its ‘conventional setting’ must be deﬁned. In many cases there is a natural choice; sometimes, however, such a choice does not exist, and the appropriate conventions have to be stated. The three reasons for listing a space group twice will be discussed in this section, cf. Section 2.1.2.3.

2.1.2.5.1. Monoclinic subgroups Rules (a) If the monoclinic axis of H is the b or c axis of the basis of G, then the setting of H is also ‘unique axis b’ or ‘unique axis c’. In particular, if G is monoclinic, then the settings of G and H agree. (b) If the monoclinic axis of H is neither b nor c in the basis of G, then for H the setting ‘unique axis b’ is chosen. (c) The cell choice is always ‘cell choice 1’ with the symbols C and c for unique axis b, and A and a for unique axis c.

Example 2.1.2.5.3 G ¼ Pban, No. 50; origin choice 1. I Maximal (monoclinic) translationengleiche subgroups ½2 P112=n: conventional unique axis c; nonconventional glide reﬂection n. The monoclinic axis c is retained but the glide reﬂection n is adjusted to a glide reﬂection a in order to conform to the conventional symbol P112=a of cell choice 1. ½2 P12=a1: conventional unique axis b; nonconventional glide reﬂection a. The monoclinic axis b is retained but the glide reﬂection a is adjusted to a glide reﬂection c of the conventional symbol P12=c1, cell choice 1. ½2 P2=b11: nonconventional monoclinic unique axis a; nonconventional glide reﬂection b. The monoclinic axis a is transformed to the conventional unique axis b; the glide reﬂection b is adjusted to the conventional symbol P12=c1 of the setting unique axis b, cell choice 1.

Remarks (see also the following examples): Rule (a) is valid for the many cases where the setting of H is ‘inherited’ from G. In particular, this always holds for isomorphic subgroups. Rule (b) is applied if G is orthorhombic and the monoclinic axis of H is the a axis of G and if H is a monoclinic subgroup of a trigonal group. Rule (b) is not natural, but speciﬁes a preference for the setting ‘unique axis b’. This seems to be justiﬁed because the setting ‘unique axis b’ is used more frequently in crystallographic papers and the standard short HM symbol is also referred to it. Rule (c) implies a choice of that cell which is most explicitly described in the tables of IT A. By this choice, the centring type and the glide vector are ﬁxed to the conventional values of ‘cell choice 1’.

2.1.2.5.2. Subgroups with two origin choices Altogether, 24 orthorhombic, tetragonal and cubic space groups with inversions are listed twice in IT A. There are three kinds of possible ambiguities for such groups with two origin choices:

The necessary adjustment is performed through a coordinate transformation, i.e. by a change of the basis and by an origin shift, see Section 2.1.3.3.

(a) Only the original group G is listed with two origin choices in IT A, G(1) and G(2), but the subgroup H < G is listed with one origin. Then the matrix parts P for the transformations ðP; p1 Þ and ðP; p2 Þ of the coordinate systems of G(1) and G(2) to that of H are the same but the two columns of origin shift differ, namely p1 from G(1) to H and p2 from G(2) to H. They are related to the shift u between the origins of G(1) and G(2). However, the transformations from both settings of the space group G to the setting of the space group H are not unique and there is some choice in the transformation matrix and the origin shift. The transformation has been chosen such that

Example 2.1.2.5.1 G ¼ P12=m1, No. 10; unique axis b. II Maximal klassengleiche subgroups, Enlarged unit cell ½2 a0 ¼ 2a, both subgroups P12=a1. The monoclinic axis b is retained but the glide reﬂection a is converted into a glide reﬂection c (P12=c1 is the conventional HM symbol for cell choice 1). ½2 b0 ¼ 2b; c0 ¼ 2c, all four subgroups A12=m1. The monoclinic axis b is retained but the A centring is converted into the conventional C centring (C12=m1 is the conventional HM symbol for cell choice 1). ½2 a0 ¼ 2a; c0 ¼ 2c, both subgroups B12=e1. The monoclinic axis b is retained. The glide reﬂection is designated by ‘e’ (simultaneous c- and a-glide reﬂection in the same plane perpendicular to b). The nonconventional B centring is converted into the conventional primitive setting P, by which the e-glide reﬂection also becomes a c-glide reﬂection.

(i) it transforms the nonconventional description of the space group H to a conventional one; (ii) the description of the crystal structure in the subgroup H is as similar as possible to that in the supergroup G. If it is not possible to achieve the latter aim, a transformation with simple matrix and column parts has been chosen which fulﬁls the ﬁrst condition.

Example 2.1.2.5.2 G ¼ P112=m, No. 10; unique axis c. II Maximal klassengleiche subgroups, Enlarged unit cell ½2 a0 ¼ 2a, both subgroups P112=a.

Example 2.1.2.5.4 G ¼ Pban, No. 50, origin choice 1 and origin choice 2. I Maximal translationengleiche subgroups

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2.1. GUIDE TO THE SUBGROUP TABLES AND GRAPHS There are seven maximal t-subgroups of Pban, No. 50, four of which are orthorhombic, H ¼ Pba2, Pb2n, P2an and P222, and three of which are monoclinic, H ¼ P112=n, P12=a1 and P2=b11. In the orthorhombic subgroups, the centres of inversion of G are lost but at least one kind of twofold axis is retained. Therefore, no origin shift for H is necessary from the setting ‘origin choice 1’ of G(1), where the origin is placed at the intersection of the three twofold axes. For the column part of the transformation ðP; p1 Þ, p1 ¼ o holds. For the monoclinic maximal t-subgroups of Pban the origin is shifted from the intersection of the three twofold axes in G(1) to an inversion centre of H. On the other hand, the origin is situated on an inversion centre for origin choice 2 of G(2), as is the origin in the conventional description of the three monoclinic maximal t-subgroups. For them the origin shift is p2 ¼ o, while there is a nonzero column p2 for the orthorhombic subgroups.

2.1.2.5.3. Space groups with a rhombohedral lattice The seven trigonal space groups with a rhombohedral lattice are often called rhombohedral space groups. Their HM symbols begin with the lattice letter R and they are listed with both hexagonal axes and rhombohedral axes. Rules (a) A rhombohedral subgroup H of a rhombohedral space group G is listed in the same setting as G: if G is referred to hexagonal axes, so is H; if G is referred to rhombohedral axes, so is H. (b) If G is a non-rhombohedral trigonal or a cubic space group, then a rhombohedral subgroup H < G is always referred to hexagonal axes. (c) A non-rhombohedral subgroup H of a rhombohedral space group G is referred to its conventional setting. Remarks Rule (a) provides a clear deﬁnition, in particular for the axes of isomorphic subgroups. Rule (b) has been followed in the subgroup tables because the rhombohedral setting is rarely used in crystallography. Rule (c) implies that monoclinic subgroups of rhombohedral space groups are referred to the setting ‘unique axis b’.

(b) Both G and its subgroup H < G are listed with two origins. Then the origin choice of H is the same as that of G. This rule always applies to isomorphic subgroups as well as in some other cases. Example 2.1.2.5.5 Maximal k-subgroups H = Pnnn, No. 48, of the space group G = Pban, No. 50. There are two such subgroups with the lattice relation c0 ¼ 2c. Both G and H are listed with two origins such that the origin choices of G and H are either the same or are strongly related.

There is a peculiarity caused by the two settings of the rhombohedral space groups. The rhombohedral lattice appears to be centred in the hexagonal axes setting, whereas it is primitive in the rhombohedral axes setting. Therefore, there are trigonal subgroups of a rhombohedral space group G which are listed in the block ‘Loss of centring translations’ for the hexagonal axes setting of G but are listed in the block ‘Enlarged unit cell’ when G is referred to rhombohedral axes. Although unnecessary and not done for other space groups with primitive lattices, the line

(c) The group G is listed with one origin but the subgroup H < G is listed with two origins. This situation is restricted to maximal k-subgroups with the only exception being Ia3d > I41 =acd, where there are three conjugate t-subgroups of index 3. In all cases the subgroup H is referred to origin choice 2. This rule is followed in the subgroup tables because it gives a better chance of retaining the origin of G in H. If there are two origin choices for H, then H has inversions and these are also elements of the supergroup G. The (unique) origin of G is placed on one of the inversion centres. For origin choice 2 in H, the origin of H may agree with that of G, although this is not guaranteed. In addition, origin choice 2 is often preferred in structure determinations.

Loss of centring translations

none

is listed for the rhombohedral axes setting. Example 2.1.2.5.7 G ¼ R3, No. 146. Maximal klassengleiche subgroups of index 2 and 3. Comparison of the data for the settings ‘hexagonal axes’ and ‘rhombohedral axes’. The data for the general position and the generators are omitted. hexagonal axes Loss of centring translations ½3 P32 ð145Þ ½3 P31 ð144Þ ½3 P3 ð143Þ Enlarged unit cell ½2 a0 ¼ b; b0 ¼ a þ b; c0 ¼ 2c R3 ð146Þ b; a þ b; 2c ...

Example 2.1.2.5.6 Maximal k-subgroups of Pccm, No. 49. In the block Enlarged unit cell, ½2 a0 ¼ 2a one ﬁnds two subgroups Pcna (50, Pban). One of them has the origin of G, the origin of the other subgroup is shifted by 12 ; 0; 0 and is placed on one of the inversion centres of G that is removed from the ﬁrst subgroup. The analogous situation is found in the block [2] b0 ¼ 2b, where the two subgroups of space-group type Pncb (50, Pban) show the analogous relation. In the next block, [2] a0 ¼ 2a; b0 ¼ 2b, the four subgroups Ccce (68) behave similarly. For G = Pmma, No. 51, the same holds for the two subgroups of the type Pmmn (59) in the block [2] b0 ¼ 2b. On the other hand, for G = Immm, No. 71, in the block ‘Loss of centring translations’ three subgroups of type Pmmn (59) and one of type Pnnn (48) are listed. All of them need an origin shift of 14 ; 14 ; 14 because they have lost the inversion centres of the origin of G.

0; 1=3; 0 1=3; 1=3; 0

rhombohedral axes Loss of centring translations Enlarged unit cell ½2 a0 ¼ a þ c; b0 ¼ a þ b; c0 ¼ b þ c R3 ð146Þ a þ c; a þ b; b þ c ½3 a0 ¼ a b; b0 ¼ b c; c0 ¼ a þ b þ c P32 ð145Þ a b; b c; a þ b þ c P31 ð144Þ a b; b c; a þ b þ c P3 ð143Þ a b; b c; a þ b þ c

75

none

0; 1=3; 1=3 1=3; 0; 1=3

2. MAXIMAL SUBGROUPS OF THE PLANE GROUPS AND SPACE GROUPS The sequence of the blocks has priority over the classiﬁcation by increasing index. Therefore, in the setting ‘hexagonal axes’, the subgroups of index 3 precede the subgroup of index 2. In the tables, the lattice relations are simpler for the setting ‘hexagonal axes’. The complete general position is listed for the maximal k-subgroups of index 3 in the setting ‘hexagonal axes’; only the generator is listed for rhombohedral axes.

matrix, shift: These entries contain information on the transformation of H from the setting of G to the standard setting of H. They are explained in Section 2.1.3.3. In general, the numbers in the list ‘Sequence’ of H follow the order of the numbers in the group G, i.e. they rise monotonically. Sometimes this sequence is modiﬁed because those entries which have the same additional translations are joined together, see, e.g. the maximal k-subgroups of Fm3m with ‘Loss of centring translations’. In addition, in a class of conjugate subgroups, the monotonically rising order may be obeyed only for the ﬁrst member of the conjugacy class. The order of the other members is then determined by the conjugation of the ﬁrst member. (In IT A the monotonically rising order of the numbers is kept in all conjugate subgroups.)

2.1.3. I Maximal translationengleiche subgroups (t-subgroups) 2.1.3.1. Introduction In this block, all maximal t-subgroups H of the plane groups and the space groups G are listed individually. Maximal t-sugroups are always non-isomorphic. For the sequence of the subgroups, see Section 2.1.2.4. There are no lattice relations for t-subgroups because the lattice is retained. Therefore, the sequence is determined only by the rising value of the index and by the decreasing space-group number.

Example 2.1.3.2.1 G ¼ Pm3m, No. 221, tetragonal t-subgroups I Maximal translationengleiche subgroups ( ½3 P4=m12=m ð123; P4=mmmÞ 1; 2; 3; 4; 13; 14; 15; 16; . . . ½3 P4=m12=m ð123; P4=mmmÞ 1; 4; 2; 3; 18; 19; 17; 20; . . . ½3 P4=m12=m ð123; P4=mmmÞ 1; 3; 4; 2; 22; 24; 23; 21; . . . Comments: If W 1 ; W 2 ; W 3 ; . . . is the order of the ﬁrst sequence, then the second sequence follows the order C 1 W 1 C, C1 W 2 C, C 1 W 3 C; . . .. Here the C means a threefold rotation and is the conjugating element; for the second subgroup C ¼ ð9Þ y; z; x of the general position of Pm3m; for the third subgroup C ¼ ð5Þ z; x; y. In this example the columns w of the symmetry operations (and thus of the conjugating elements) are the zero columns o and could be omitted.

2.1.3.2. A description in close analogy with IT A The listing is similar to that of IT A and presents on one line the following information for each subgroup H: ½i HMS1 (No., HMS2)

sequence

matrix

shift

Conjugate subgroups are listed together and are connected by a left brace. The symbols have the following meaning: [i] HMS1

No. HMS2 sequence

matrix

shift

index of H in G; HM symbol of H referred to the coordinate system and setting of G. This symbol may be nonconventional; space-group No. of H; conventional HM symbol of H if HMS1 is not a conventional HM symbol; sequence of numbers; the numbers refer to those coordinate triplets of the general position of G that are retained in H, cf. Remarks; for general position cf. Section 2.1.2.2.2; matrix part of the transformation to the conventional setting corresponding to HMS2, cf. Section 2.1.3.3; column part of the transformation to the conventional setting corresponding to HMS2, cf. Section 2.1.3.3.

The description of the subgroups can be explained by the following four examples. Example 2.1.3.2.2 G ¼ C1m1, No. 8, unique axis b I Maximal translationengleiche subgroups ½2 C1 ð1; P1Þ 1þ Comments: HMS1: C1 is not a conventional HM symbol. Therefore, the conventional symbol P1 is added as HMS2 after the spacegroup number 1 of H. sequence: ‘1þ’ means x; y; z; x þ 12 ; y þ 12 ; z. Example 2.1.3.2.3 G ¼ Fdd2, No. 43 I Maximal translationengleiche subgroups ... ½2 F112 ð5; A112Þ ð1; 2Þþ Comments: HMS1: F112 is not a conventional HM symbol; therefore, the conventional symbol A112 is added to the space-group No. 5 as HMS2. The setting unique axis c is inherited from G. sequence: (1, 2)þ means:

Remarks In the sequence column for space groups with centred lattices, the abbreviation ‘(numbers)þ’ means that the coordinate triplets speciﬁed by ‘numbers’ are to be taken plus those obtained by adding each of the centring translations, see the comments following Examples 2.1.3.2.2 and 2.1.3.2.3. The symbol HMS2 is omitted if HMS1 is a conventional HM symbol. The following deviations from the listing of IT A are introduced in these tables:

x; y; z; x; y þ 12 ; z þ 12 ; x ; y ; z; x ; y þ 12 ; z þ 12 ;

x þ 12 ; y; z þ 12 ; x þ 12 ; y ; z þ 12 ;

x þ 12 ; y þ 12 ; z; x þ 12 ; y þ 12 ; z;

Example 2.1.3.2.4 G ¼ P 42 =nmc ¼ P42 =n 21 =m 2=c, No. 137, origin choice 2 I Maximal translationengleiche subgroups ... ½2 P2=n 21 =m 1 (59, Pmmn) 1; 2; 5; 6; 9; 10; 13; 14

No.: the space-group No. of H is added. HMS2: In order to specify the setting clearly, the full HM symbol is listed for monoclinic subgroups, not the standard (short) HM symbol as in IT A.

76

2.1. GUIDE TO THE SUBGROUP TABLES AND GRAPHS are designated by boldface italic upper-case letters (W 0 is analogous to W ): 0 1 0 1 P11 P12 P13 W11 W12 W13 W ¼ @ W21 W22 W23 A; P ¼ @ P21 P22 P23 A; W31 W32 W33 P31 P32 P33

Comments: HMS1: The sequence of the directions in the HM symbol for a tetragonal space group is c, a, a b. From the parts 42 =n, 21 =m and 2=c of the full HM symbol of G, only 2=n, 21 =m and 1 remain in H. Therefore, HMS1 is P2=n 21 =m 1, and the conventional symbol Pmmn is added as HMS2. No.: The space-group number of H is 59. The setting origin choice 2 of H is inherited from G. sequence: The coordinate triplets of G retained in H are: (1) x; y; z; (2) x þ 12 ; y þ 12 ; z; (5) x; y þ 12 ; z; (6) x þ 12 ; y; z; (9) x; y; z; etc.

0

Q11 Q ¼ @ Q21 Q31

Q12 Q22 Q32

1 Q13 Q23 A: Q33

Let a; b; c ¼ ðaÞT be the row of basis vectors of G and a0 ; b0 ; c0 ¼ ða0 ÞT the basis of H, then the basis ða0 ÞT is expressed in the basis ðaÞT by the system of equations

Example 2.1.3.2.5 G ¼ P31 12, No. 151 I Maximal translationengleiche subgroups ½2 P31 11 ð144; P31 Þ 1; 2; 3 ( ½3 P112 ð5; C121Þ 1; 6 b; 2a b; c ½3 P112 ð5; C121Þ 1; 4 a b; a b; c 0; 0; 1=3 ½3 P112 ð5; C121Þ 1; 5 a; a þ 2b; c 0; 0; 2=3 Comments: brace: The brace on the left-hand side connects the three conjugate monoclinic subgroups. HMS1: P112 is not the conventional HM symbol for unique axis c but the constituent ‘2’ of the nonconventional HM symbol refers to the directions 2a b, a b and a þ 2b, in the hexagonal basis. According to the rules of Section 2.1.2.5, the standard setting is unique axis b, as expressed by the HM symbol C121. HMS2: Note that the conventional monoclinic cell is centred. matrix, shift: The entries in the columns ‘matrix’ and ‘shift’ are explained in the following Section 2.1.3.3 and evaluated in Example 2.1.3.3.2.

a0 ¼ P11 a þ P21 b þ P31 c b0 ¼ P12 a þ P22 b þ P32 c

ð2:1:3:1Þ

0

c ¼ P13 a þ P23 b þ P33 c or

0

P11 ða0 ; b0 ; c0 ÞT ¼ ða; b; cÞT @ P21 P31

P12 P22 P32

1 P13 P23 A: P33

ð2:1:3:2Þ

In matrix notation, this is ða0 ÞT ¼ ðaÞT P:

ð2:1:3:3Þ

The column p of coordinates of the origin O0 of H is referred to the coordinate system of G and is called the origin shift. The matrix–column pair (P, p) describes the transformation from the coordinate system of G to that of H, for details, cf. IT A, Part 5. Therefore, P and p are listed in the subgroup tables in the columns ‘matrix’ and ‘shift’, cf. Section 2.1.3.2. The column ‘matrix’ is empty if there is no change of basis, i.e. if P is the unit matrix I. The column ‘shift’ is empty if there is no origin shift, i.e. if p is the column o consisting of zeroes only. A change of the coordinate system, described by the matrix– column pair ðP; pÞ, changes the point coordinates from the column x to the column x 0 . The formulae for this change do not contain the pair ðP; pÞ itself, but the related pair ðQ; qÞ ¼ ðP 1 ; P 1 pÞ:

2.1.3.3. Basis transformation and origin shift Each t-subgroup H < G is deﬁned by its representatives, listed under ‘sequence’ by numbers each of which designates an element of G. These elements form the general position of H. They are taken from the general position of G and, therefore, are referred to the coordinate system of G. In the general position of H, however, its elements are referred to the coordinate system of H. In order to allow the transfer of the data from the coordinate system of G to that of H, the tools for this transformation are provided in the columns ‘matrix’ and ‘shift’ of the subgroup tables. The designation of the quantities is that of IT A Part 5 and is repeated here for convenience. The transformation described in this section is not restricted to translationengleiche subgroups but is applied to klassengleiche subgroups as well. In the following, columns and rows are designated by boldface italic lower-case letters. Point coordinates x; x0, translation parts w; w0 of the symmetry operations and shifts p; q ¼ P 1 p are represented by columns. The sets of basis vectors ða; b; cÞ ¼ ðaÞT and ða0 ; b0 ; c0 Þ ¼ ða0 ÞT are represented by rows [indicated by ð. . .ÞT , which means ‘transposed’]. The quantities with unprimed symbols are referred to the coordinate system of G, those with primes are referred to the coordinate system of H. The following columns will be used (w0 is analogous to w): 0 1 0 1 0 01 0 1 0 1 x p1 q1 w1 x w ¼ @ w 2 A; x ¼ @ y A ; x 0 ¼ @ y 0 A; p ¼ @ p 2 A; q ¼ @ q 2 A: z w3 z0 p3 q3

x 0 ¼ Qx þ q ¼ P 1 x P 1 p ¼ P 1 ðx pÞ:

ð2:1:3:4Þ

Not only the point coordinates but also the matrix–column pairs for the symmetry operations are changed by a change of the coordinate system. A symmetry operation W is described in the coordinate system of G by the system of equations2 x~ ¼ W11 x þ W12 y þ W13 z þ w1 y~ ¼ W21 x þ W22 y þ W23 z þ w2

ð2:1:3:5Þ

z~ ¼ W31 x þ W32 y þ W33 z þ w3 ; or x~ ¼ W x þ w ¼ ðW ; wÞ x;

ð2:1:3:6Þ

i.e. by the matrix–column pair (W, w). The symmetry operation W will be described in the coordinate system of the subgroup H by the equation 2

Please note that in equation (2.1.3.6) the matrix W is multiplied by the column x from the right-hand side whereas in equation (2.1.3.3) the matrix P is multiplied by the row ðaÞT from the left-hand side. Therefore, the running index in W is the second one, whereas in P it is the ﬁrst one.

0

The ð3 3Þ matrices W and W of the symmetry operations, as well as the matrix P for a change of basis and its inverse Q ¼ P 1,

77

2. MAXIMAL SUBGROUPS OF THE PLANE GROUPS AND SPACE GROUPS 0

0

x~ ¼ W x 0 þ w 0 ¼ ðW 0 ; w 0 Þ x 0 ;

Analogously, the (3 1) columns x and x0 must be augmented to (4 1) columns by a ‘1’ in the fourth row in order to enable matrix multiplication with the augmented matrices: 0 1 0 01 x x ByC B y0 C C; x0 ¼ B 0 C: x¼B @zA @z A

ð2:1:3:7Þ

and thus by the pair ðW 0 ; w 0 Þ. This pair can be calculated from the pair ðW ; wÞ by the equations W 0 ¼ Q W P ¼ P 1 W P

ð2:1:3:8Þ

and w 0 ¼ q þ Qw þ QW p ¼ P 1 ðw þ W p pÞ ¼ P 1 ðw þ ðW IÞpÞ: ð2:1:3:9Þ

1

The three equations (2.1.3.4), (2.1.3.8) and (2.1.3.9) are replaced by the two equations

These equations are rather complicated and unpleasant. They become simple when using augmented matrices and columns. In this case the formulae are reduced formally to normal matrix multiplication [the formalism is simpler but the necessary calculations are not, because the inversion of a (4 4) matrix is tedious if done by hand]. The matrices P, Q, W and W 0 may be combined with the corresponding columns p, q, w and w0 to form (4 4) matrices (callled augmented matrices):3 0 1 P11 P12 P13 p1 B C B P21 P22 P23 p2 C C B P¼B C; B P31 P32 P33 p3 C A @ 0 0 0 1 1 0 Q11 Q12 Q13 q1 C B B Q21 Q22 Q23 q2 C C B Q ¼ P1 ¼ B C; B Q31 Q32 Q33 q3 C A @ 0 0 0 1 1 0 W11 W12 W13 w1 C B B W21 W22 W23 w2 C C B W¼B C; B W31 W32 W33 w3 C A @ 0 0 0 1 1 0 0 0 0 W12 W13 w01 W11 C B 0 0 0 B W21 W22 W23 w02 C C B 0 W ¼B C: 0 0 0 0 C B W31 W W w 32 33 3A @ 0 0 0 1

and

W0 ¼ QWP ¼ P1 WP:

ð2:1:3:12Þ

Index HM & No: ½2 P1n1 ð7; P1c1Þ ½2 Pm11 ð6; P1m1Þ ½2 P1121 ð4Þ This means that are 0 0 0 P1 ¼ @ 0 1 1 0

sequence 1; 3 1; 4 1; 2

matrix c; b; a c c; a; b

shift

1=4; 0; 0

the transformation matrices and origin shifts 1 0 0 1 0 A; P 2 ¼ @ 0 1 1

1 0 0

1 0 0 1 1 A; P 3 ¼ @ 0 0 0

0 1 0

1 0 0A 1

011 0 1 0 1 0 0 4 p1 ¼ @ 0 A; p2 ¼ @ 0 A; p3 ¼ @ 0 A: 0 0 0 The ﬁrst subgroup is monoclinic, the symmetry direction is the b axis, which is standard. However, the glide direction 12 ða þ cÞ is nonconventional. Therefore, the basis of G is transformed to a basis of the subgroup H such that the b axis is retained, the glide direction becomes the c0 axis and the a0 axis is chosen such that the basis is a right-handed one, the angle 0 90 and the transformation matrix P is simple. This is done by the chosen matrix P 1 . The origin shift is the o column. With equations (2.1.3.8) and (2.1.3.9), one obtains for the glide reﬂection x; y; z 12, which is x; y; z þ 12 after standardization by 0 wj < 1. For the second monoclinic subgroup, the symmetry direction is the (nonconventional) a axis. The rules of Section 2.1.2.5 require a change to the setting ‘unique axis b’. A cyclic permutation of the basis vectors is the simplest way to achieve this. The reﬂection x; y; z is now described by x; y; z. Again there is no origin shift. The third monoclinic subgroup is in the conventional setting ‘unique axis c’, but the origin must be shifted onto the 21 screw axis. This is achieved by applying equation (2.1.3.9) with p3 , which changes x þ 12 ; y; z þ 12 of Pmn21 to x; y; z þ 12 of P1121.

ða0 ÞT ¼ ða0 ; b0 ; c0 j s0H ÞT

and ða0 ÞT ¼ ða0 ; b0 ; c0 j pÞT :

The relation between ða0 ÞT and ðaÞT is given by equation (2.1.3.10), which replaces equation (2.1.3.3), ða0 ÞT ¼ ðaÞT P:

ð2:1:3:11Þ

Example 2.1.3.3.1 Consider the data listed for the t-subgroups of Pmn21, No. 31:

with s0H ¼ p þ sG . As the vector sG one can take the zero vector sG ¼ o, which results in s0H ¼ p ¼ p1 a þ p2 b þ p3 c, i.e. ðaÞT ¼ ða; b; c j oÞT

x0 ¼ Qx ¼ P1 x

and

The coefﬁcients of these augmented matrices are integer, rational or real numbers. The (3 1) rows (a)T and (a0 )T must be augmented to (4 1) rows by appending some vectors sG and s0H , respectively, as fourth entries in order to enable matrix multiplication with the augmented matrices: ðaÞT ¼ ða; b; c j sG ÞT

1

Example 2.1.3.3.2 Evaluation of the t-subgroup data of the space group P31 12, No. 151, started in Example 2.1.3.2.5. The evaluation is now continued with the columns ‘sequence’, ‘matrix’ and ‘shift’. They are used for the transformation of the elements of H to their conventional form. Only the monoclinic t-subgroups are

ð2:1:3:10Þ

3 The horizontal and vertical lines in the augmented matrices are useful to facilitate recognition of their coefﬁcients; they have no mathematical meaning.

78

2.1. GUIDE TO THE SUBGROUP TABLES AND GRAPHS (and a for P 6 ) correspond to the cell-centring translation vectors of the subgroup cells.

of interest here because the trigonal subgroup is already in the standard setting. One takes from the tables of subgroups in Chapter 2.3 Index HM & No: sequence matrix ( ½3 P112 ð5; C121Þ 1; 6 b; 2a b; c ½3 P112 ð5; C121Þ 1; 4 a b; a b; c ½3 P112 ð5; C121Þ 1; 5 a; a þ 2b; c

Comments: This example reveals that the conjugation of conjugate subgroups does not necessarily imply the conjugation of the representatives of these subgroups in the general positions of IT A. The three monoclinic subgroups C121 in this example are conjugate in the group G by the 31 screw rotation. Conjugation of the representatives (4) and (6) by the 31 screw rotation of G results in the column w5 ¼ 0; 0; 43, which is standardized according to the rules of IT A to w5 ¼ 0; 0; 13. Thus, the conjugacy relation is disturbed by the standardization of the representative (5).

shift 0; 0; 1=3 0; 0; 2=3

Designating the three matrices by P 6, P 4 , P 5 , one obtains 0

1 0 0 2 0 1 1 P 6 ¼ @ 1 1 0 A; P 4 ¼ @ 1 1 0 0 1 0 0

1 0 1 0 1 1 0 0 A; P 5 ¼ @ 0 2 0 A 1 0 0 1

with the corresponding inverse matrices 0

1 0 1 12 1 0 2 12 1 Q6 ¼ @ 2 0 0 A; Q4 ¼ @ 12 12 0 0 1 0 0

1 0 0 1 0 A; Q5 ¼ @ 0 1 0

1 12 0 1 0A 2 0 1

2.1.4. II Maximal klassengleiche subgroups (k-subgroups) 2.1.4.1. General description The listing of the maximal klassengleiche subgroups (maximal k-subgroups) H of the space group G is divided into the following three blocks for practical reasons:

and the origin shifts 0 1 0 1 0 1 0 0 0 p6 ¼ @ 0 A; p4 ¼ @ 0 A; p5 ¼ @ 0 A: 1 2 0 3 3

Loss of centring translations. Maximal subgroups H of this block have the same conventional unit cell as the original space group G. They are always non-isomorphic and have index 2 for plane groups and index 2, 3 or 4 for space groups. Enlarged unit cell. Under this heading, maximal subgroups of index 2, 3 and 4 are listed for which the conventional unit cell has been enlarged. The block contains isomorphic and nonisomorphic subgroups with this property. Series of maximal isomorphic subgroups. In this block all maximal isomorphic subgroups of a space group G are listed in a small number of inﬁnite series of subgroups with no restriction on the index, cf. Sections 2.1.2.4 and 2.1.5.

For the three new bases this means a06 ¼ b; b06 ¼ 2a b; c06 ¼ c a04 ¼ a b; b04 ¼ a b; c04 ¼ c and a05 ¼ a; b05 ¼ a þ 2b; c05 ¼ c: All these bases span ortho-hexagonal cells with twice the volume of the original hexagonal cell because for the matrices detðP i Þ ¼ 2 holds. In the general position of G ¼ P31 12, No.151, one ﬁnds

The description of the subgroups is the same within the same block but differs between the blocks. The partition into these blocks differs from the partition in IT A, where the three blocks are called ‘maximal non-isomorphic subgroups IIa’, ‘maximal non-isomorphic subgroups IIb’ and ‘maximal isomorphic subgroups of lowest index IIc’. The kind of listing in the three blocks of this volume is discussed in Sections 2.1.4.2, 2.1.4.3 and 2.1.5 below.

ð1Þ x; y; z; ð4Þ y; x; z þ 23; ð5Þ x þ y; y; z þ 13; ð6Þ x; x y; z:

These entries represent the matrix–column pairs ðW ; wÞ: 0 1 0 1 0 1 0 1 0 1 0 0 0 0 1 0 ð1Þ @ 0 1 0 A; @ 0 A; ð4Þ @ 1 0 0 A; @ 0 A; 2 0 0 1 0 0 0 1 3 0

1 ð5Þ @ 0 0

1 1 0

1 0 1 0 0 0 A; @ 0 A; 1 1 3

0

1 ð6Þ @ 1 0

0 1 0

2.1.4.2. Loss of centring translations

1 0 1 0 0 0 A; @ 0 A: 1 0

Consider a space group G with a centred lattice, i:e: a space group whose HM symbol does not start with the lattice letter P but with one of the letters A, B, C, F, I or R. The block contains those maximal subgroups of G which have fully or partly lost their centring translations and thus are not t-subgroups. The conventional unit cell is not changed. Only in space groups with an F-centred lattice can the centring be partially lost, as is seen in the list of the space group Fmmm, No. 69. On the other hand, for F23, No. 196, the maximal subgroups P23, No. 195, or P21 3, No. 198, have lost all their centring translations. For the block ‘Loss of centring translations’, the listing in this volume is the same as that for t-subgroups, cf. Section 2.1.3. The centring translations are listed explicitly where applicable, e.g. for space group C2, No. 5, unique axis b

Application of equations (2.1.3.8) on the matrices W k and (2.1.3.9) on the columns wk of the matrix–column pairs results in 0 1 0 1 0 1 0 0 W 04 ¼ W 05 ¼ W 06 ¼ @ 0 1 0 A; w04 ¼ w06 ¼ o; w05 ¼ @ 0 A: 1 0 0 1 All translation vectors of G are retained in the subgroups but the volume of the cells is doubled. Therefore, there must be centring-translation vectors in the new cells. For example, the application of equation (2.1.3.9) with ðP 6 ; p6 Þ to the translation of G with the vector a, i.e. w ¼ ð1; 0; 0Þ, results in the column w0 ¼ ð12 ; 12 ; 0Þ, i.e. the centring translation 12 ða0 þ b0 Þ of the subgroup. Either by calculation or, more easily, from a small sketch one sees that the vectors b for P 4 , a þ b for P 5

½2 P121 1 ð4Þ

1; 2 þ ð12 ; 12 ; 0Þ

1=4; 0; 0:

In this line, the representatives 1; 2 þ ð12 ; 12 ; 0Þ of the general position are x; y; z x þ 12 ; y þ 12 ; z.

79

2. MAXIMAL SUBGROUPS OF THE PLANE GROUPS AND SPACE GROUPS The listing differs from that in IT A in only two points:

Columns 1 and 2: HM symbol and space-group number of the subgroup; cf. Section 2.1.3.2. Column 3: generators, here the pairs

(1) the full HM symbol is taken as the conventional symbol for monoclinic space groups, whereas in IT A the short HM symbol is the conventional one; (2) the information needed for the transformation of the data from the setting of the space group G to that of H is added. In this example, the matrix is the unit matrix and is not listed; the column of origin shift is 14 ; 0; 0. This transformation is analogous to that of t-subgroups and is described in detail in Section 2.1.3.3. The sequence of the subgroups in this block is one of decreasing space-group number of the subgroups.

Under the heading ‘Enlarged unit cell’, those maximal ksubgroups H are listed for which the conventional unit cell is enlarged relative to the unit cell of the original space group G. All maximal k-subgroups with enlarged unit cell of index 2, 3 or 4 of the plane groups and of the space groups are listed individually. The listing is restricted to these indices because 4 is the highest index of a maximal non-isomorphic subgroup, and the number of these subgroups is ﬁnite. Maximal subgroups of higher indices are always isomorphic to G and their number is inﬁnite. The description of the subgroups with enlarged unit cell is more detailed than in IT A. In the block IIb of IT A, different maximal subgroups of the same space-group type with the same lattice relations are represented by the same entry. For example, the eight maximal subgroups of the type Fmmm, No. 69, of space group Pmmm, No. 47, are represented by one entry in IT A. In the present volume, the description of the maximal subgroups in the block ‘Enlarged unit cell’ refers to each subgroup individually and contains for each of them a set of space-group generators and the transformation from the setting of the space group G to the conventional setting of the subgroup H. Some of the isomorphic subgroups listed in this block may also be found in IT A in the block ‘Maximal isomorphic subgroups of lowest index IIc’. Subgroups with the same lattice are collected in small blocks. The heading of each such block consists of the index of the subgroup and the lattice relations of the sublattice relative to the original lattice. Basis vectors that are not mentioned are not changed.

a; 3b; c a; 3b; c a; 3b; c

x; y; z þ 12; x; y þ 2; z þ 12; x; y þ 4; z þ 12;

x; y; z þ 12; x; y; z þ 12; x; y; z þ 12;

Often the lattice relations above the data describing the subgroups are the same as the basis vectors in column 4, as in this example. They differ in particular if the sublattice of H is nonconventionally centred. Examples are the H-centred subgroups of trigonal and hexagonal space groups. The sequence of the subgroups is determined (1) by the index of the subgroup such that the subgroups of lowest index are given ﬁrst; (2) within the same index by the kind of cell enlargement; (3) within the same cell enlargement by the No. of the subgroup, such that the subgroup of highest space-group number is given ﬁrst.

2.1.4.3.1. Enlarged unit cell, index 2 For sublattices with twice the volume of the unit cell, the sequence of the different cell enlargements is as follows: (1) Triclinic space groups: (i) a0 ¼ 2a, (ii) b0 ¼ 2b, (iii) c0 ¼ 2c, (iv) b0 ¼ 2b; c0 ¼ 2c; A-centring, (v) a0 ¼ 2a; c0 ¼ 2c; B-centring, (vi) a0 ¼ 2a; b0 ¼ 2b; C-centring, (vii) a0 ¼ 2a; b0 ¼ 2b; c0 ¼ 2c; F-centring.

Example 2.1.4.3.1 This example is taken from the table of space group C2221, No. 20.

[3] b0 ¼ 3b ( C2221 ð20Þ h2; 3i C2221 ð20Þ h3; 2 þ ð0; 2; 0Þi C2221 ð20Þ h3; 2 þ ð0; 4; 0Þi

x; y; z þ 12; x þ 2; y; z þ 12; x þ 4; y; z þ 12;

for the six lines listed in the same order. Column 4: basis vectors of H referred the basis vectors of G. 3a; b; c means a0 ¼ 3a, b0 ¼ b, c0 ¼ c; a; 3b; c means a0 ¼ a, b0 ¼ 3b, c0 ¼ c. Column 5: origin shifts, referred to the coordinate system of G. These origin shifts by o, a and 2a for the ﬁrst triplet of subgroups and o, b and 2b for the second triplet of subgroups are translations of G. The subgroups of each triplet are conjugate, indicated by the left braces.

2.1.4.3. Enlarged unit cell

Enlarged unit cell [3] a0 ¼ 3a ( 3a; b; c C2221 ð20Þ h2; 3i C2221 ð20Þ hð2; 3Þ þ ð2; 0; 0Þi 3a; b; c C2221 ð20Þ hð2; 3Þ þ ð4; 0; 0Þi 3a; b; c

x; y; z þ 12; x þ 2; y; z þ 12; x þ 4; y; z þ 12;

(2) Monoclinic space groups: (a) with P lattice, unique axis b: (i) b0 ¼ 2b, (ii) c0 ¼ 2c, (iii) a0 ¼ 2a, (iv) a0 ¼ 2a; c0 ¼ 2c; B-centring, (v) a0 ¼ 2a; b0 ¼ 2b; C-centring, (vi) b0 ¼ 2b; c0 ¼ 2c; A-centring, (vii) a0 ¼ 2a; b0 ¼ 2b; c0 ¼ 2c; F-centring. (b) with P lattice, unique axis c: (i) c0 ¼ 2c, (ii) a0 ¼ 2a, (iii) b0 ¼ 2b, (iv) a0 ¼ 2a; b0 ¼ 2b; C-centring, (v) b0 ¼ 2b; c0 ¼ 2c; A-centring,

1; 0; 0 2; 0; 0

0; 1; 0 0; 2; 0

The entries mean:

80

2.1. GUIDE TO THE SUBGROUP TABLES AND GRAPHS 0

0

(vi) a ¼ 2a; c ¼ 2c; B-centring, (vii) a0 ¼ 2a; b0 ¼ 2b; c0 ¼ 2c; F-centring. (c) with C lattice, unique axis b: There are three sublattices of index 2 of a monoclinic C lattice. One has lost its centrings such that a P lattice with the same unit cell remains. The subgroups with this sublattice are listed under ‘Loss of centring translations’. The block with the other two sublattices consists of c0 ¼ 2c, C-centring and I-centring. The sequence of the subgroups in this block is determined by the space-group number of the subgroup. (d) with A lattice, unique axis c: There are three sublattices of index 2 of a monoclinic A lattice. One has lost its centrings such that a P lattice with the same unit cell remains. The subgroups with this sublattice are listed under ‘Loss of centring translations’. The block with the other two sublattices consists of a0 ¼ 2a, A-centring and I-centring. The sequence of the subgroups in this block is determined by the No. of the subgroup.

(v) (vi) (vii) (viii) (ix) (x) (xi) (xii) (xiii)

a0 ¼ 3a, c0 ¼ 2a þ c, a0 ¼ 3a, b0 ¼ a þ b, c0 ¼ a þ c, a0 ¼ 3a, b0 ¼ 2a þ b, c0 ¼ a þ c, a0 ¼ 3a, b0 ¼ a þ b, c0 ¼ 2a þ c, a0 ¼ 3a, b0 ¼ 2a þ b, c0 ¼ 2a þ c, b0 ¼ 3b, b0 ¼ 3b, c0 ¼ b þ c, b0 ¼ 3b, c0 ¼ 2b þ c, c0 ¼ 3c.

(2) Monoclinic space groups: (a) Space groups P121, P121 1, P1m1, P12=m1, P121 =m1 (unique axis b): (i) b0 ¼ 3b, (ii) c0 ¼ 3c, (iii) a0 ¼ a c, c0 ¼ 3c, (iv) a0 ¼ a 2c, c0 ¼ 3c, (v) a0 ¼ 3a. (b) Space groups P112, P1121 , P11m, P112=m, P1121 =m (unique axis c): (i) c0 ¼ 3c, (ii) a0 ¼ 3a, (iii) a0 ¼ 3a; b0 ¼ a þ b, (iv) a0 ¼ 3a; b0 ¼ 2a þ b, (v) b0 ¼ 3b. (c) Space groups P1c1, P12=c1, P121 =c1 (unique axis b): (i) b0 ¼ 3b, (ii) c0 ¼ 3c, (iii) a0 ¼ 3a, (iv) a0 ¼ 3a; c0 ¼ 2a þ c, (v) a0 ¼ 3a; c0 ¼ 4a þ c. (d) Space groups P11a, P112=a, P1121 =a (unique axis c): (i) c0 ¼ 3c, (ii) a0 ¼ 3a, (iii) b0 ¼ 3b, (iv) a0 ¼ a 2b; b0 ¼ 3b, (v) a0 ¼ a 4b; b0 ¼ 3b. (e) All space groups with C lattice (unique axis b): (i) b0 ¼ 3b, (ii) c0 ¼ 3c, (iii) a0 ¼ a 2c; c0 ¼ 3c, (iv) a0 ¼ a 4c; c0 ¼ 3c, (v) a0 ¼ 3a. (f) All space groups with A lattice (unique axis c): (i) c0 ¼ 3c, (ii) a0 ¼ 3a, (iii) a0 ¼ 3a; b0 ¼ 2a þ b, (iv) a0 ¼ 3a; b0 ¼ 4a þ b, (v) b0 ¼ 3b.

(3) Orthorhombic space groups: (a) Orthorhombic space groups with P lattice: Same sequence as for triclinic space groups. (b) Orthorhombic space groups with C (or A) lattice: Same sequence as for monoclinic space groups with C (or A) lattice. (c) Orthorhombic space groups with I and F lattice: There are no subgroups of index 2 with enlarged unit cell. (4) Tetragonal space groups: (a) Tetragonal space groups with P lattice: (i) c0 ¼ 2c. (ii) a0 ¼ 2a; b0 ¼ 2b; C-centring. The conventional setting results in a P lattice. (iii) a0 ¼ 2a; b0 ¼ 2b; c0 ¼ 2c; F-centring. The conventional setting results in an I lattice. (b) Tetragonal space groups with I lattice: There are no subgroups of index 2 with enlarged unit cell. (5) For trigonal and hexagonal space groups, c0 ¼ 2c holds. For rhombohedral space groups referred to hexagonal axes, a0 ¼ b, b0 ¼ a þ b, c0 ¼ 2c or a0 ¼ a þ b, b0 ¼ a, c0 ¼ 2c holds. For rhombohedral space groups referred to rhombohedral axes, a0 ¼ a þ c, b0 ¼ a þ b, c0 ¼ b þ c or a0 ¼ 2a, b0 ¼ 2b, c0 ¼ 2c, F-centring holds. (6) Only cubic space groups with a P lattice have subgroups of index 2 with enlarged unit cell. For their lattices the following always holds: a0 ¼ 2a, b0 ¼ 2b, c0 ¼ 2c, F-centring. 2.1.4.3.2. Enlarged unit cell, index 3 or 4 With a few exceptions for trigonal, hexagonal and cubic space groups, k-subgroups with enlarged unit cells and index 3 or 4 are isomorphic. To each of the listed sublattices belong either one or several conjugacy classes with three or four conjugate subgroups or one or several normal subgroups. Only the sublattices with the numbers (5)(a)(v), (5)(b)(i), (5)(c)(ii), (6)(iii) and (7)(i) have index 4, all others have index 3. The different cell enlargements are listed in the following sequence:

(3) Orthorhombic space groups: (i) a0 ¼ 3a, (ii) b0 ¼ 3b, (iii) c0 ¼ 3c. (4) Tetragonal space groups: (i) c0 ¼ 3c. (5) Trigonal space groups: (a) Trigonal space groups with hexagonal P lattice: (i) c0 ¼ 3c, (ii) a0 ¼ 3a; b0 ¼ 3b; H-centring, (iii) a0 ¼ a b; b0 ¼ a þ 2b; c0 ¼ 3c, R lattice, (iv) a0 ¼ 2a þ b; b0 ¼ a þ b; c0 ¼ 3c, R lattice, (v) a0 ¼ 2a; b0 ¼ 2b.

(1) Triclinic space groups: (i) a0 ¼ 3a, (ii) a0 ¼ 3a, b0 ¼ a þ b, (iii) a0 ¼ 3a, b0 ¼ 2a þ b, (iv) a0 ¼ 3a, c0 ¼ a þ c,

81

2. MAXIMAL SUBGROUPS OF THE PLANE GROUPS AND SPACE GROUPS (b) Trigonal space groups with rhombohedral R lattice and hexagonal axes: (i) a0 ¼ 2b; b0 ¼ 2a þ 2b. (c) Trigonal space groups with rhombohedral R lattice and rhombohedral axes: (i) a0 ¼ a b; b0 ¼ b c; c0 ¼ a þ b þ c, (ii) a0 ¼ a b þ c; b0 ¼ a þ b c; c0 ¼ a þ b þ c.

2.1.5.2. Basis transformation The conventional basis of the unit cell of each isomorphic subgroup in the series has to be deﬁned relative to the basis of the original space group. For this deﬁnition the prime p is frequently sufﬁcient as a parameter. Example 2.1.5.2.1 The isomorphic subgroups of the space group P42 22, No. 93, can be described by two series with the bases of their members:

(6) Hexagonal space groups: (i) c0 ¼ 3c, (ii) a0 ¼ 3a; b0 ¼ 3b; H-centring, (iii) a0 ¼ 2a; b0 ¼ 2b.

½p a; b; pc ½p2 pa; pb; c. In other cases, one or two positive integers, here called q and r, deﬁne the series and often the value of the prime p.

(7) Cubic space groups with P lattice: (i) a0 ¼ 2a; b0 ¼ 2b; c0 ¼ 2c, I lattice.

Example 2.1.5.2.2 In space group P6, No. 174, the series qa rb; ra þ ðq þ rÞb; c is listed. The values of q and r have to be chosen such that while q > 0, r > 0, p ¼ q2 þ r2 þ qr is prime.

2.1.5. Series of maximal isomorphic subgroups

By Y. Billiet Example 2.1.5.2.3 In the space group P1121 =m, No. 11, unique axis c, the series pa; qa þ b; c is listed. Here p and q are independent and q may take the p values 0 q < p for each value of the prime p.

2.1.5.1. General description Maximal subgroups of index higher than 4 have index p, p2 or p , where p is prime, are necessarily isomorphic subgroups and are inﬁnite in number. Only a few of them are listed in IT A in the block ‘Maximal isomorphic subgroups of lowest index IIc’. Because of their inﬁnite number, they cannot be listed individually, but are listed in this volume as members of series under the heading ‘Series of maximal isomorphic subgroups’. In most of the series, the HM symbol for each isomorphic subgroup H < G will be the same as that of G. However, if G is an enantiomorphic space group, the HM symbol of H will be either that of G or that of its enantiomorphic partner. 3

2.1.5.3. Origin shift Each of the sublattices discussed in Section 2.1.4.3.2 is common to a conjugacy class or belongs to a normal subgroup of a given series. The subgroups in a conjugacy class differ by the positions of their conventional origins relative to the origin of the space group G. To deﬁne the origin of the conventional unit cell of each subgroup in a conjugacy class, one, two or three integers, called u, v or w in these tables, are necessary. For a series of subgroups of index p, p2 or p3 there are p, p2 or p3 conjugate subgroups, respectively. The positions of their origins are deﬁned by the p or p2 or p3 permitted values of u or u, v or u, v, w, respectively.

Example 2.1.5.1.1 Two of the four series of isomorphic subgroups of the space group P41, No. 76, are (the data for the generators are omitted): [p] c0 ¼ pc P43 ð78Þ prime p > 2; p = 4n 1 a; b; pc no conjugate subgroups prime p > 4; p = 4n + 1 a; b; pc P41 ð76Þ no conjugate subgroups

Example 2.1.5.3.1 The space group G, P42c, No. 112, has two series of maximal isomorphic subgroups H. For one of them the lattice relations are ½p2 a0 ¼ pa; b0 ¼ pb, listed as pa; pb; c. The index is p2. For each value of p there exist exactly p2 conjugate subgroups with origins in the points u; v; 0, where the parameters u and v run independently: 0 u < p and 0 v < p.

On the other hand, the corresponding data for P43 , No. 78, are [p]

c0 ¼ pc P43 ð78Þ P41 ð76Þ

prime p > 4; p = 4n + 1 no conjugate subgroups prime p > 2; p = 4n 1 no conjugate subgroups

In another type of series there is exactly one (normal) subgroup H for each index p; the location of its origin is always chosen at the origin 0; 0; 0 of G and is thus not indicated as an origin shift.

a; b; pc a; b; pc

Example 2.1.5.3.2 Consider the space group Pca21, No. 29. Only one subgroup exists for each value of p in the series a; b; pc. This is indicated in the tables by the statement ‘no conjugate subgroups’.

Note that in both tables the subgroups of the type P43, No. 78, are listed ﬁrst because of the rules on the sequence of the subgroups. If an isomorphic maximal subgroup of index i 4 is a member of a series, then it is listed twice: as a member of its series and individually under the heading ‘Enlarged unit cell’. Most isomorphic subgroups of index 3 are the ﬁrst members of series but those of index 2 or 4 are rarely so. An example is the space group P42, No. 77, with isomorphic subgroups of index 2 (not in any series) and 3 (in a series); an exception is found in space group P4, No. 75, where the isomorphic subgroup for c0 ¼ 2c is the ﬁrst member of the series ½p c0 ¼ pc.

2.1.5.4. Generators The generators of the p (or p2 or p3 ) conjugate isomorphic subgroups H are obtained from those of G by adding translational components. These components are determined by the parameters p (or q and r, if relevant) and u (and v and w, if relevant). Example 2.1.5.4.1 Space group P21 3, No. 198. In the series deﬁned by the lattice relations pa; pb; pc and the origin shift u; v; w there exist

82

2.1. GUIDE TO THE SUBGROUP TABLES AND GRAPHS 02 1 0 1 exactly p conjugate subgroups for each value of p. The 13 13 1 0 1 3 1 generators of each subgroup are deﬁned by the parameter p 23 A and P ¼ @ 1 1 1 A P 1 ¼ @ 13 3 1 1 1 and the triplet u; v; w in combination with the generators (2), 0 1 1 3 3 3 (3) and (5) of G. Consider the subgroup characterized by the basis 7a; 7b; 7c and by the origin shift u ¼ 3; v ¼ 4; w ¼ 6. are listed in IT A, Table 5.1.3.1, see also Figs. 5.1.3.6(a) and (c) One obtains from the generator (2) x þ 12 ; y; z þ 12 of G the in IT A. corresponding generator of H by adding the translation vector Applying equations (2.1.5.3), (2.1.5.1) and (2.1.5.2), one gets ðp2 12 þ 2uÞa þ 2vb þ ðp2 12Þc to the translation vector 12 a þ 12 c of the generator (2) of G and obtains 192 a þ 8b þ 72 c, so that this ða0rh ÞT ¼ ða0hex ÞT P 1 ¼ ðahex ÞT XP 1 ¼ ðarh ÞT Y ¼ ðahex ÞT P 1 Y: 19 7 generator of H is written x þ 2 ; y þ 8; z þ 2. ð2:1:5:4Þ 3

From equation (2.1.5.4) it follows that

2.1.5.5. Special series For most space groups, there is only one description of their series of the isomorphic subgroups. However, if a space group is described twice in IT A, then there are also two different descriptions of these series. This happens for monoclinic space groups with the settings unique axis b and unique axis c, for some orthorhombic, tetragonal and cubic space groups with origin choices 1 and 2 and for trigonal space groups with rhombohedral lattices with hexagonal axes and rhombohedral axes.

XP 1 ¼ P 1 Y or Y ¼ PXP 1 :

One obtains Y from equation (2.1.5.5) by matrix multiplication, 0 pþ2 Y¼

2.1.5.5.1. Monoclinic space groups In the monoclinic space groups, the series in the listings ‘unique axis b’ and ‘unique axis c’ are closely related by a simple cyclic permutation of the axes a, b and c, see IT A, Section 2.2.16.

1

p1 3 p1 C ; 3 A pþ2 3

c0rh ¼ 13 ½ðp 1Þarh þ ðp 1Þbrh þ ðp þ 2Þcrh : The column of the origin shift uhex ¼ 0; 0; u in hexagonal axes must be transformed by urh ¼ Puhex . The result is the column urh ¼ u; u; u in rhombohedral axes.

2.1.5.5.3. Space groups with two origin choices Space groups with two origin choices are always described in the same basis, but origin 1 is shifted relative to origin 2 by the shift vector s. For most space groups with two origins, the appearance of the two series related by the origin shift is similar; there are only differences in the generators. Example 2.1.5.5.2 Consider the space group Pnnn, No. 48, in both origin choices and the corresponding series deﬁned by pa; b; c and u; 0; 0. In origin choice 1, the generator (5) of G is described by the ‘coordinates’ x þ 12 ; y þ 12 ; z þ 12. The translation part ðp2 12Þa of the third generator of H stems from the term 12 in the ﬁrst ‘coordinate’ of the generator (5) of G. Because ðp2 12Þa must be a translation vector of G, p is odd. Such a translation part is not found in the generators (2) and (3) of H because the term 12 does not appear in the ‘coordinates’ of the corresponding generators of G. The situation is inverted in the description for origin choice 2. The translation term ðp2 12Þa appears in the ﬁrst and second generator of H and not in the third one because the term 12 occurs in the ﬁrst ‘coordinate’ of the generators (2) and (3) of G but not in the generator (5). The term 2u appears in both descriptions. It is introduced in order to adapt the generators to the origin shift u; 0; 0.

ð2:1:5:1Þ

in analogy to Part 5, IT A. ðahex ÞT is the row of basis vectors of the conventional hexagonal basis. The matrix X is deﬁned by 0 1 1 0 0 X ¼ @ 0 1 0 A: 0 0 p With rhombohedral axes, equation (2.1.5.1) would be written ð2:1:5:2Þ

with the matrix Y to be determined. The transformation from hexagonal to rhombohedral axes is described by ðarh ÞT ¼ ðahex ÞT P 1 ;

p1 3 pþ2 3 p1 3

a0rh ¼ 13 ½ðp þ 2Þarh þ ðp 1Þbrh þ ðp 1Þcrh ; b0rh ¼ 13 ½ðp 1Þarh þ ðp þ 2Þbrh þ ðp 1Þcrh ;

Example 2.1.5.5.1 Space group R3, No. 148. The second series is described with hexagonal axes by the basis transformation a; b; pc, i.e. a0hex ¼ ahex ; b0hex ¼ bhex ; c0hex ¼ p chex ; and the origin shift 0; 0; u. We discuss the basis transformation ﬁrst. It can be written

ða0rh ÞT ¼ ðarh ÞT Y;

3 B p1 @ 3 p1 3

and from Y for the bases of the subgroups with rhombohedral axes

2.1.5.5.2. Trigonal space groups with rhombohedral lattice In trigonal space groups with rhombohedral lattices, the series with hexagonal axes and with rhombohedral axes appear to be rather different. However, the ‘rhombohedral’ series are the exact transcript of the ‘hexagonal’ series by the same transformation formulae as are used for the different monoclinic settings. However, the transformation matrices P and P 1 in Part 5 of IT A are more complicated in this case.

ða0hex ÞT ¼ ðahex ÞT X

ð2:1:5:5Þ

In other space groups described in two origin choices, surprisingly, the number of series is different for origin choice 1 and origin choice 2.

ð2:1:5:3Þ

where the matrices

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2. MAXIMAL SUBGROUPS OF THE PLANE GROUPS AND SPACE GROUPS Example 2.1.5.5.3 In the tetragonal space group I41 =amd, No. 141, for origin choice 1 there is one series of maximal isomorphic subgroups of index p2 , p prime, with the bases pa; pb; c and origin shifts u; v; 0. For origin choice 2, there are two series with the same bases pa; pb; c but with the different origin shifts u; v; 0 and 1 2 þ u; v; 0. What are the reasons for these results? For origin choice 1, the term 12 appears in the ﬁrst and second ‘coordinates’ of all generators (2), (3), (5) and (9) of G. This term 12 is the cause of the translation vectors ðp2 12Þa and ðp2 12Þb in the generators of H. For origin choice 2, fractions 14 and 34 appear in all ‘coordinates’ of the generator (3) y þ 14 ; x þ 34 ; z þ 14 of G. As a consequence, translational parts with vectors ðp4 þ 14Þa and ð3p4 54Þb appear if p = 4n 1. On the other hand, translational parts with vectors ðp4 14Þa; ð3p4 34Þb are introduced in the generators of H if p = 4n + 1 holds. Another consequence of the fractions 14 and 34 occurring in the generator (3) of G is the difference in the origin shifts. They are 1 2 þ u; v; 0 for p = 4n 1 and u; v; 0 for p = 4n + 1. Thus, the one series in origin choice 1 for odd p is split into two series in origin choice 2 for p = 4n 1 and p = 4n + 1.4

themselves, see Example 2.1.6.2.2. In the following we restrict the considerations to supergroups G of a space group H which are themselves space groups. This holds, for example, for the symmetry relations between crystal structures when the symmetries of both structures can be described by space groups. Quasicrystals, incommensurate phases etc. are thus excluded. Even under this restriction, supergroups show much more variable behaviour than subgroups do. One of the reasons for this complication is that the search for subgroups H < G is restricted to the elements of the space group G itself, whereas the search for supergroups G > H has to take into account the whole (continuous) group E of all isometries. For example, there are only a ﬁnite number of subgroups H of any space group G for any given index i. On the other hand, there may not only be an inﬁnite number of supergroups G of a space group H for a ﬁnite index i but even an uncountably inﬁnite number of minimal supergroups of H. Example 2.1.6.1.2 Let H ¼ P1. Then there is an inﬁnite number of t-supergroups P1 of index 2 because there is no restriction for the sites of the centres of inversion and thus of the conventional origin of P1. In the tables of this volume, the minimal translationengleiche supergroups G of a space group H are not listed individually but the type of G is listed by index, conventional HM symbol and space-group number if H is listed as a translationengleiche subgroup of G in the subgroup tables. Not listed is the number of supergroups belonging to one entry. Non-isomorphic klassengleiche supergroups are listed individually. For them, nonconventional HM symbols are listed in addition; for klassengleiche supergroups with ‘Decreased unit cell’, the lattice relations are added. More details, such as the representatives of the general position or the generators as well as the transformation matrix and the origin shift, would only duplicate the subgroup data. In this Section 2.1.6, the kind of listing is described explicitly. The data for maximal subgroups H are complete for all space groups G. Therefore, it is possible to derive:

2.1.6. The data for minimal supergroups 2.1.6.1. Description of the listed data In the previous sections, the relation H < G between space groups was seen from the viewpoint of the group G. In this case, H was a subgroup of G. However, the same relation may be viewed from the group H. In this case, G > H is a supergroup of H. As for the subgroups of G, cf. Section 1.2.6, different kinds of supergroups of H may be distinguished. Deﬁnition 2.1.6.1.1. Let H < G be a maximal subgroup of G. Then G > H is called a minimal supergroup of H. If H is a translationengleiche subgroup of G then G is a translationengleiche supergroup (t-supergroup) of H. If H is a klassengleiche subgroup of G, then G is a klassengleiche supergroup (k-supergroup) of H. If H is an isomorphic subgroup of G, then G is an isomorphic supergroup of H. If H is a general subgroup of G, then G & is a general supergroup of H.

(1) all minimal translationengleiche supergroups G of H if the point-group symmetry of H is at least orthorhombic (i.e. neither triclinic nor monoclinic); (2) all minimal klassengleiche supergroups G for each space group H.

Following from Hermann’s theorem, Lemma 1.2.8.1.2, a minimal supergroup of a space group is either a translationengleiche supergroup (t-supergroup) or a klassengleiche supergroup (k-supergroup). A proper minimal t-supergroup always has an index i, 1 < i < 5, and is never isomorphic. A minimal k-supergroup with index i, 1 < i < 5, may be isomorphic or nonisomorphic; for indices i > 4 a minimal k-supergroup can only be an isomorphic k-supergroup. The propositions, theorems and their corollaries of Sections 1.4.6 and 1.4.7 for maximal subgroups are correspondingly valid for minimal supergroups. Subgroups of space groups of ﬁnite index are always space groups again. This does not hold for supergroups. For example, the direct product G of a space group H with a group of order 2 is not a space group, although H < G is a subgroup of index 2 of G. Moreover, supergroups of space groups may be afﬁne groups which are only isomorphic to space groups but not space groups

In Section 2.1.7 the procedure is described by which the supergroup data can be obtained from the subgroup data. This procedure is not trivial and care has to be taken to really obtain the full set of supergroups. In Section 2.1.7 one can also ﬁnd the reasons why this procedure is not applicable when the space group H belongs to a triclinic or monoclinic point group. Isomorphic supergroups are not indicated at all because they are implicitly contained in the subgroup data. Their derivation from the subgroup data is discussed in Section 2.1.7.2. Like the subgroup data, the supergroup data are also partitioned into blocks. 2.1.6.2. I Minimal translationengleiche supergroups For each space group H, under this heading are listed those space-group types G for which H appears as an entry under the heading I Maximal translationengleiche subgroups. The listing consists of the index in brackets [ . . . ], the conventional HM symbol and the space-group number (in parentheses). The space groups are ordered by ascending space-group number. If this line

4 F. Ga¨hler (private communication) has shown that such a splitting can be avoided if one allows the prime p to enter the formulae for the origin shifts. In these tables we have not made use of this possibility in order to keep the origin shifts in the same form for all space groups G.

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2.1. GUIDE TO THE SUBGROUP TABLES AND GRAPHS is empty, the heading is printed nevertheless and the content is announced by ‘none’, as in P6=mmm, No. 191. Note that the real setting of the supergroup and thus its HM symbol may be nonconventional.

and one of Pnnn; there are three minimal supergroups of type Pccm, namely Pccm, Pmaa and Pbmb, as well as three minimal supergroups of type Pban, viz. Pban, Pncb and Pcna. There are six minimal supergroups of the type P422 and four minimal supergroups of the type P23; they are space groups if the lattice conditions are fulﬁlled. The number of different supergroups will be calculated in Examples 2.1.7.3.1 and 2.1.7.4.2 by the procedure described in Section 2.1.7.4.

Example 2.1.6.2.1 Let H ¼ P21 21 2, No. 18. Among the entries of the block one ﬁnds the space groups of the crystal class mmm: ½2 Pbam, No. 55; ½2 Pccn, No. 56; ½2 Pbcm, No. 57; ½2 Pnnm, No. 58; ½2 Pmmn, No. 59 and ½2 Pbcn, No. 60, designated by their standard HM symbols. However, the full HM symbols P 21 =b 21 =a 2=m, P 21 =c 21 =c 2=n, P 2=b 21 =c 21 =m, P 21 =n 21 =n 2=m, P 21 =m 21 =m 2=n and P 21 =b 2=c 21 =n reveal that only the four HM symbols Pbam, Pccn, Pnnm and Pmmn of these six entries describe supergroups of P21 21 2. The symbols P 2=b 21 =c 21 =m and P 21 =b 2=c 21 =n represent four supergroups of P21 21 2, namely G ¼ P 21 =b 21 =m 2=a, P 21 =m 21 =a 2=b, P 21 =c 21 =n 2=b and P 21 =n 21 =c 2=a. This is not obvious but will be derived, as well as the origin shift if necessary, with the procedure described in Examples 2.1.7.3.2 and 2.1.7.4.3.

2.1.6.3. II Minimal non-isomorphic klassengleiche supergroups If G is a k-supergroup of H, G and H always belong to the same crystal family and there are no lattice restrictions for H. As mentioned above, in the tables of this volume only nonisomorphic minimal k-supergroups are listed among the supergroup data; no isomorphic minimal supergroups are given. The block II Minimal non-isomorphic klassengleiche supergroups is divided into two subblocks with the headings Additional centring translations and Decreased unit cell. If both subblocks are empty, only the heading of the block is listed, stating ‘none’ for the content of the block, as in P6=mmm, No. 191. If at least one of the subblocks is non-empty, then the heading of the block and the headings of both subblocks are listed. An empty subblock is then designated by ‘none’; in the other subblock the supergroups are listed. The kind of listing depends on the subblock. Examples may be found in the tables of P222, No. 16, and Fd3c, No. 228. As discussed in Section 2.1.7.1, there is exactly one supergroup for each of the non-isomorphic k-supergroup entries of H, although often not in the conventional setting. A transformation of the general-position representatives or of the generators to the conventional setting may be necessary to obtain the standard HM symbol of G in the same way as in Examples 2.1.7.3.1 and 2.1.7.3.2, which refer to translationengleiche supergroups. Under the heading ‘Additional centring translations’, the supergroups are listed by their indices and either by their nonconventional HM symbols, with the space-group numbers and the conventional HM symbols in parentheses, or by their conventional HM symbols and only their space-group numbers in parentheses. Examples are provided by space group Pbca, No. 61, with both subblocks non-empty and by space group P222, No. 16, with supergroups only under the heading ‘Additional centring translations’. Not only the HM symbols but also the centrings themselves may be nonconventional. In this volume, the nonconventional centrings tetragonal c (c4gm as a supergroup of p4gm) and h (h31m as a supergroup of p31m) are used for HM symbols of plane groups, tetragonal C (C4m2 as a supergroup of P4m2), Rrev ‘reverse’, different from the conventional Robv ‘obverse’ (Rrev 3 as supergroup of P3), and H (H312 as supergroup of P312) are used for HM symbols of space groups. Under the heading ‘Decreased unit cell’ each supergroup is listed by its index and by its lattice relations, where the basis vectors a0, b0 and c0 refer to the supergroup G and the basis vectors a, b and c to the original group H. After these data are listed either the nonconventional HM symbol, followed by the spacegroup number and the conventional HM symbol in parentheses, or the conventional HM symbol with the space-group number in parentheses. Examples are provided again by space group Pbca, No. 61, with both subblocks occupied and space group F43m, No. 216, with an empty subblock ‘Additional centring translations’ but data under the heading ‘Decreased unit cell’.

The supergroups listed in this block represent space groups only if the lattice conditions of H fulﬁl the lattice conditions for G. This is not a problem if group H and supergroup G belong to the same crystal family,5 cf. Example 2.1.6.2.1. Otherwise the lattice parameters of H have to be checked correspondingly, as in Example 2.1.6.2.2. Example 2.1.6.2.2 Space group H ¼ P222, No. 16. For the minimal supergroups G ¼ Pmmm, No. 47, Pnnn, No. 48, Pccm, No. 49, and Pban, No. 50, there is no lattice condition because P222 and all these supergroups belong to the same orthorhombic crystal family and thus are space groups. If, however, a space group of the types G ¼ P422, No. 89, P42 22, No. 93, P42m, No. 111, or P42c, No. 112, is considered as a supergroup of H ¼ P222, two of the three independent lattice parameters a, b, c of P222 must be equal (or in crystallographic practice, approximately equal). These must be a and b if c is the tetragonal axis, b and c if a is the tetragonal axis or c and a if b is the tetragonal axis. In the latter two cases, the setting of P222 has to be transformed to the c-axis setting of P422. For the cubic supergroup P23, No. 195, all three lattice parameters of P222 must be (approximately) equal. If they are not, elements of G are not isometries and G is an afﬁne group which is only isomorphic to a space group. The lattice conditions are useful in the search for supergroups G > H which are space groups, i.e. form the symmetry of crystal structures. Whereas a subgroup H < G does not become noticeable in the lattice parameters of a space group G, a space group G > H of another crystal family must be indicated by the lattice parameters of the space group H. Thus it may be an important advantage if the conditions of temperature, pressure or composition allow the start of the search for possible phase transitions of the low-symmetry phase. As mentioned already, the number of the minimal t-supergroups cannot be taken or concluded from the subgroup tables. It is different in the different cases of Example 2.1.6.2.2 above. The space group P222 has one minimal supergroup of the type Pmmm 5 For the term ‘crystal family’ see Section 1.2.5.2 or, for more details, IT A, Section 8.2.7.

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2. MAXIMAL SUBGROUPS OF THE PLANE GROUPS AND SPACE GROUPS group G is isomorphic to the space group H or not. Therefore, the last paragraph of Section 2.1.7.2 also holds for the nonisomorphic k-subgroup pairs of P21 22, P221 2 and P2221 among the subgroups of P222.

2.1.7. Derivation of the minimal supergroups from the subgroup tables The minimal supergroups of the space groups, in particular the translationengleiche or t-supergroups, are not fully listed in the tables. However, with the exception of the translationengleiche supergroups of triclinic and monoclinic space groups H, the listing is sufﬁcient to derive the supergroups with the aid of the subgroup tables. For this derivation the coset decomposition ðX : T ðX ÞÞ of a space group X relative to its translation subgroup T ðX Þ as well as the normalizers N ðHÞ and N ðGÞ of the space groups H and G play a decisive role. The coset decomposition of a group relative to a subgroup has been deﬁned in Section 1.2.4.2; the coset decomposition of a space group relative to its translation subgroup in Sections 1.2.5.1 and 1.2.5.4. The notions of the afﬁne and the Euclidean normalizer have been introduced in Section 1.2.6.3. In the next Sections 2.1.7.1 and 2.1.7.2 the minimal k-supergroups including the isomorphic supergroups will be derived by inversion of the subgroup data. In Section 2.1.7.3 one minimal t-supergroup of each type will be found from the corresponding subgroup data, also by inversion. Starting from this supergroup other minimal t-supergroups can be obtained. This procedure is described in Sections 2.1.7.4 and 2.1.7.5.

Example 2.1.7.1.1 Consider the minimal k-supergroups of the space group P21 21 2, No. 18. Four entries for ‘Additional centring translations’ and two for ‘Decreased unit cell’ are listed in the supergroup data of H ¼ P21 21 2; the missing entry for c0 ¼ 12 c results in a k-supergroup isomorphic to H and is thus not listed among the supergroup data of P21 21 2. The supergroups with ‘Additional centring translations’ shall be looked at in more detail. The supergroup C222 is obtained directly by adding the C-centring to the symmetry operations of H. Adding the A- and B-centrings to H ¼ P21 21 2 results in supergroups of the type C2221. For C2221 , in the a and b directions 2 and 2 1 axes alternate, whereas in the c direction there are only 2 1 axes. Adding the A-centring ð0; 1=2; 1=2Þþ to P21 21 2 results in alternating 2 and 2 1 axes in the directions b and c but there are only 2 1 axes in the direction of a; A21 22 is obtained. Adding the B-centring ð1=2; 0; 1=2Þþ to P21 21 2 results in alternating 2 and 2 1 axes in the directions a and c but there are only 2 1 axes in the b direction; B221 2 is obtained. These relations can also be derived in another way. Transformation of the relation P21 221 < C2221 with its matrix–column pair as listed in the subgroup table of C2221 in Chapter 2.3 results in the relation P21 21 2 < A21 22. On the other hand, the relation P221 21 < C2221 is transformed by its matrix–column pair to P21 21 2 < B221 2. It is often easy to construct the supergroup from the drawing of the original space group H in IT A by adding the centring vectors or the additional basis vectors. This happens, for example, for the supergroup I222, No. 23, where the origin shift by ð0; 0; 1=4Þ is obvious from the comparison of the drawings of P21 21 2 and I222. This agrees with the data in the subgroup table of I222.

2.1.7.1. Determination of the non-isomorphic minimal k-supergroups by inverting the subgroup data All non-isomorphic klassengleiche maximal subgroups of a space group G are listed individually, whereas the inﬁnite number of isomorphic k-subgroups of G are listed essentially by series. Therefore, it is more transparent to deal with the isomorphic supergroups separately (in Section 2.1.7.2). The non-isomorphic klassengleiche supergroups are considered here. The procedure for deriving the k-supergroups Gj of a space group H is simpler than that for the derivation of the t-supergroups, described in Sections 2.1.7.3, 2.1.7.4 and 2.1.7.5. The data for k-supergroups of H are more detailed and, unlike the t-supergroups, there is only one k-supergroup per entry for the supergroup data of H. To show this, one considers the coset decomposition of the group H with respect to the normal subgroup T ðHÞ of all its translations, cf. Section 8.1.6 of IT A. The set of cosets with respect to this decomposition forms a group, the factor group H=T ðHÞ. Each coset, i.e. each element of the factor group H=T ðHÞ, consists of all those elements (symmetry operations) of H which have the same matrix part in common and differ in their translation parts only. In a klassengleiche supergroup G > H the coset decomposition of H is retained; only the set of translations is increased in G relative to H, T ðGÞ > T ðHÞ. With the additional translations, each coset of H is extended to a coset of G. The cosets are independent of the chosen coset representatives. Thus, as the coset representatives of H always belong to the elements of G, the coset representatives of H can be taken as the coset representatives of G and the elements of G are uniquely determined. It follows that for each maximal k-subgroup H which is listed among the subgroups of G, G is the only minimal k-supergroup for the corresponding extension of the lattice translations (there may be other lattice extensions in addition which result in other supergroups). If different k-subgroups Hj of G and their lattice extensions are conjugate under the Euclidean normalizer of G, then G is the common minimal k-supergroup of these subgroups Hj . This result is independent of whether the minimal k-super-

The completeness of the data for the minimal k-supergroups depends on the completeness of the listed lattice extensions, i.e. on the completeness of the listed possible centrings as well as of the possible decreased unit cells of the lattices. These data are well known for the small indices 2, 3 and 4 occurring in these group–subgroup relations. 2.1.7.2. The isomorphic minimal supergroups It is not necessary to list the isomorphic minimal supergroups, i.e. those minimal supergroups G which belong to the space-group type of H. Therefore, a block ‘series of isomorphic minimal supergroups’ does not occur among the supergroup data. The derivation of the isomorphic minimal supergroups G from the data in the subgroup tables is straightforward. For each index, one looks for the listed isomorphic normal subgroups H and for the classes of conjugate isomorphic subgroups Hq in the subgroup table of G. If some of these items are conjugate under the Euclidean normalizer of G, then only one item of this conjugacy class has to be taken into consideration as a representative. For each of these representatives there is one corresponding supergroup of H. As for the isomorphic maximal subgroups, the indices of the minimal supergroups are p, p2 or p3, p prime. However, the large conjugacy classes of isomorphic maximal subgroups always belong to single isomorphic minimal supergroups.

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2.1. GUIDE TO THE SUBGROUP TABLES AND GRAPHS (P1, p1) also transforms the group G1 from its standard description to that referred to the coordinate system of H1. This transformed group G 01 is one supergroup G 01 > H1 from which one can start to derive other supergroups of this type, if there are any, cf. Sections 2.1.7.4 and 2.1.7.5. The same procedure has to be applied to any other maximal t-subgroup Hk < G1 and to the other listed t-supergroups G > H. The calculations can be veriﬁed by viewing the space-group diagrams of the corresponding space groups in IT, Volume A.

Example 2.1.7.2.1 Consider the p2 conjugate isomorphic subgroups H of a space group P42c, No. 112, in the series ½ p2 a0 ¼ pa; b0 ¼ pb, prime p ﬁxed. The same supergroup G belongs to each of these subgroups H. The indices u and v, designating the members of a conjugacy class of subgroups Hk < G, may have any of their admissible values or may be set to zero. Choosing other values of u and/or v means dealing with the same supergroup G but transformed by an element of G. Because the parameters u and/or v appear only in the translation parts of the (4 4) symmetry matrices, this may mean a shift of the origin in the description of G. If for practical reasons the origin of G will be chosen as the origin of Hk, i.e. at different points of G for the different groups Hk, then the (same) group G is described relative to different origins. These origins are then chosen in different translationally equivalent points of G.

Example 2.1.7.3.1 Space group H1 ¼ P222, No.16. In continuation of Example 2.1.6.2.2, one ﬁnds no data for the matrix P and the column p listed in the entry for the subgroup P222 in the subgroup tables of Pmmm, No. 47, Pnnn, No. 48, origin choice 1, Pban, No. 50, origin choice 1, P422, No. 89, P42 22, No. 93, P42m, No. 111, and P23, No. 195. This means P = I and p = o. Thus, P222 is a subgroup of these space groups and the standard settings agree, i.e. the generators of G1 can be taken directly from those of H1, adding the last generator of G1. This is conﬁrmed by the space-group diagrams. Regarding the tetragonal and cubic supergroups, the lattice restrictions for the space group H1 have to be obeyed, cf. Example 2.1.6.2.2. Only an origin shift but no transformation matrix P is listed in the subgroup tables of the supergroups Pnnn origin choice 2, p ¼ ð1=4; 1=4; 1=4Þ; Pccm, No. 49, p ¼ ð0; 0; 1=4Þ; Pban origin choice 2, p ¼ ð1=4; 1=4; 0Þ and P42c, No. 112, p ¼ ð0; 0; 1=4Þ. Thus, the r matrix parts of the r non-translational generators of H1 ; r ¼ 2, are retained, W 01r ¼ W 1r , and equation (2.1.3.9) is reduced to

Space groups Hj which are conjugate under the Euclidean normalizer of the supergroup G have the supergroup G in common if they are complemented by the corresponding conjugate sets of translations. For example, both members of each of the subgroup pairs P222 of index [2] in the subgroup table of the space group P222, No. 16, for a0 ¼ 2a or b0 ¼ 2b or c0 ¼ 2c have their minimal supergroup G in common because they are conjugate under the Euclidean normalizer Pð1=2; 1=2; 1=2Þmmm.6 If for some temperature, pressure and composition of a substance a = b = c holds for the lattice parameters of P222, N E ðHÞ ¼ N A ðHÞ ¼ Pð1=2; 1=2; 1=2Þm3m, i.e. the Euclidean normalizer is equal to the afﬁne normalizer, making the three supergroups G conjugate in the normalizer. Such relations have little importance in practice if the space group describes the symmetry of a substance. This substance has the crystal symmetry P222 independent of the accidentally higher lattice symmetry.

w 0 ¼ w þ ðW IÞp;

ð2:1:7:1Þ

where ðW 01r ; w 01r Þ are those (two) generators of Gj which stem from H. The third generator of G1, the inversion or rotoinversion, has to be transformed correspondingly. The column parts w01r of the r = 2 generators 2 z and 2 y of the four space groups Pnnn origin choice 2, Pccm, Pban origin choice 2 and P42c are then (normalized to values between 0 wr < 1) the same as those of the group H, i.e. they are generators of supergroups of H. The columns of the inversions are

2.1.7.3. Determination of one minimal t-supergroup by inverting the subgroup data A proper translationengleiche supergroup G of a space group H cannot be isomorphic to H because it belongs to another crystal class. Therefore, it is not necessary to include the word ‘nonisomorphic’ in the header. The minimal t-supergroups G of H have indices i, 1 < i < 5; only the conventional HM symbol of the supergroup G together with the index and the space-group number are listed in the supergroup data of the group H. However, in the subgroup data of G the subgroups Hk are explicitly listed with their indices, their (nonconventional and) conventional HM symbols, their spacegroup numbers, their general positions and their transformation matrices P and columns p. Suppose the supergroups Gj are listed on the line ‘I Minimal translationengleiche supergroups’ of the space group H. In order to determine all supergroups Gm > H, one takes one of the listed supergroups G, say G1. In the subgroup table of G1 one ﬁnds a subgroup H1 < G1, isomorphic to H (at least one must exist, otherwise G1 would not be listed among the minimal supergroups of H). The transformation (P1, p1), listed with the subgroup H1, transforms the symmetry operations of H1 from the coordinate system of G1 to the standard coordinate system of H1. Such transformations are described by equations (2.1.3.8) and (2.1.3.9) in Section 2.1.3.3. The matrix–column pair

0 1 0 1 0 1 0 1 1 1 0 0 2 2 B1C @0A @ 1 A @ ; 2 and 0 A; @ 2 A; 1 1 1 0 2 2 2 respectively. This describes the position of 1 or 4 relative to the origin of the P222 framework in the corresponding spacegroup diagrams of IT A. The coordinates of the inversion centre are half the coefﬁcients of the columns. The supergroup Pnnn, origin choice 2, is the same as Pnnn, origin choice 1; only the setting is different. The space-group diagrams in IT A display the agreement of the frameworks of the rotation axes in H1 and Gj . The supergroup P42c is a space group only if the corresponding lattice relations for H1 are fulﬁlled. Example 2.1.7.3.2 Continued from Example 2.1.6.2.1. For H1 ¼ P21 21 2, No. 18, and its orthorhombic types of supergroups, the places for the matrix P and the column p under P21 21 2 in the subgroup data of Pbam, No. 55, and Pmmn, No. 59, origin choice 1 are empty. In analogy to the preceding example, P21 21 2 is a subgroup of these space groups in the standard setting.

6

Here we make use of the notation L(U, V, W)P N for the HM symbol of the normalizer N . L is the lattice letter, P N is the point-group part of the HM symbol of N and U, V, W determine the basis vectors a0i of N referred to the basis vectors ak of the lattice of G: a0 ¼ Ua, b0 ¼ Vb, c0 ¼ Wc. Such nomenclature can be used conveniently if the lattice relations are simple, as they are in these examples.

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2. MAXIMAL SUBGROUPS OF THE PLANE GROUPS AND SPACE GROUPS The equation P ¼ I also holds for the subgroup data of Pccn, No. 56, Pnnm, No. 58, and Pmmn, No. 59, origin choice 2, but p ¼ ð1=4; 1=4; 1=4Þ, p ¼ ð0; 0; 1=4Þ and p ¼ ð1=4; 1=4; 0Þ, respectively. Thus, the reduced equation (2.1.7.1) holds for the generators of H1 and the inversion. Again the column parts of the generators 2 z and 2 y of the groups Pccn, Pnnm and Pmmn, origin choice 2, are the same as those of the group H1, if normalized to values between 0 wr < 1, i.e. they are generators of supergroups of H. The columns of the inversions are 0 1 0 1 011 1 0 2 2 B1C @ A @ 2 A; 0 and @ 12 A; 1 1 0 2 2

one ﬁnds bx at x ¼ 1=4, my at y ¼ 1=4 and az at z ¼ 0. This results in the HM symbol P 21 =b 21 =m 2=a ¼ Pbma for this supergroup; see also the diagram in IT A. For Pbcn, 0

0 @ P¼ 0 1

1 0 0

1 0 1 A; 0

0 1 0 @ p ¼ 1 A: 4 1 4

A procedure analogous to that applied for Pbcm yields for Pbcn, No. 60, the standard setting of P21 21 2 and the inversion centre at (1/4, 0, 1/4). Again by combination of the (screw) rotations with the inversion one gets nx at x ¼ 0, cy at y ¼ 1=4 and az at z ¼ 1=4. The nonconventional HM symbol of this supergroup of P21 21 2 is thus Pnca or P 21 =n 21 =c 2=a.

respectively. As in the previous example, this describes the position of 1 relative to the origin of the P21 21 2 framework in the corresponding space-group diagrams of IT A. The coordinates of the inversion centre are half the coefﬁcients of the columns. The supergroup Pmmn, origin choice 2, is the same as Pmmn, origin choice 1; only the setting is different. Again the space-group diagrams in IT A display the agreement of the frameworks of rotation and screw rotation axes in H1 and Gj . Concerning Pbcm, No. 57, in its subgroup table one ﬁnds the line of P21 21 2 with the representatives 1, 2, 3, 4 of the general position and 0 1 0 1 0 0 1 0 I 6¼ P ¼ @ 1 0 0 A; p ¼ @ 14 A: 0 1 0 0

2.1.7.4. Derivation of further minimal t-supergroups by using normalizers Up to now one minimal t-supergroup G > H per entry using the tables of maximal subgroups has been found, see Examples 2.1.7.3.1 of P222 and 2.1.7.3.2 of P21 21 2. The question arises as to whether this list is complete or whether further t-supergroups Gj > H exist which belong to the space-group type of G and are represented by the same entry of the supergroup data. If these supergroups belong to the same space-group type then they are isomorphic and are thus conjugate under the group A of all afﬁne mappings according to the theorem of Bieberbach. Then there must be an afﬁne mapping a 2 A such that a1 Ga ¼ Gj . To ﬁnd these mappings, one makes use of the afﬁne normalizers NA ðGÞ and NA ðHÞ and considers their intersection D ¼ NA ðGÞ \ NA ðHÞ (Koch, 1984). One of the two diagrams of Fig. 2.1.7.1 will describe the situation because G is a minimal supergroup of H. Let N G ðHÞ be the normalizer of the group H in the group G, i.e. the set of all elements of G which leave H invariant. Whereas in general G N G ðHÞ H holds, for a minimal supergroup G either G ¼ N G ðHÞ (right diagram) or N G ðHÞ ¼ H (left diagram) holds.

The representative 1 is described by ðW ; wÞ ¼ ðI; oÞ; it is invariant under the transformation. Using equations (2.1.3.8) and (2.1.3.9) and the above values for P and p for the representatives 2, 3 and 4 of the subgroup P221 21, these will be transformed to the matrix–column pairs of 2z, 21y and 21x in the standard form of P21 21 2. The inversion is transformed to an inversion with the column ð1=2; 0; 0Þ, i.e. the centre of inversion has the coordinates ð1=4; 0; 0Þ. Combining the (screw) rotations with the inversion, one obtains the reﬂection and the glide reﬂections, referred to the coordinate system of P21 21 2. With the application of the formulae of IT A, Chapter 11.2,

Lemma 2.1.7.4.1. Let G1 > H1 be a minimal t-supergroup of a space group H1, let N ðG1 Þ and N ðH1 Þ be their afﬁne normalizers, and D ¼ N ðG1 Þ \ N ðH1 Þ is the intersection of these normalizers. Then in minimal supergroups Gj > H1 exist, isomorphic to G1, where in ¼ jN ðH1 Þ : Dj is the index of D in N ðH1 Þ. If in is ﬁnite, then the representatives am, m ¼ 1; . . . ; in of the cosets in the & decomposition of N ðH1 Þ : D transform G1 to Gm. The lemma is proven by coset decomposition of N ðH1 Þ relative to D. If H < G is a t-subgroup, for the translation groups T ðN ðGÞÞ and T ðN ðHÞÞ of the normalizers T ðN ðGÞÞ T ðN ðHÞÞ always holds (Wondratschek & Aroyo, 2001). Therefore, for t-supergroups there may be translations of T ðN ðHÞÞ which transform the space group G > H into another one, Gj > H. Transformation of G and H by an element of D will map G as well as H onto itself. Transformation of G and H by an element ðW ; wÞ 2 N ðHÞ but ðW ; wÞ 2 = D will map H onto itself but will map the supergroup G onto another supergroup Gj. For applications it is transparent to split the index in into the index iL of the translation lattices and iP of the point-group parts, in ¼ iL iP . For the application of Lemma 2.1.7.4.1, the kind of the normalizers N A ðHÞ and N A ðGÞ and in particular the index in ¼

Fig. 2.1.7.1. In the left-hand diagram, there are i conjugate subgroups Hk < G of G if [i] is the index of H in G; in the right-hand diagram, H / G is a normal subgroup of G. The group D ¼ N ðHÞ \ N ðGÞ is the intersection of the normalizers N ðHÞ and N ðGÞ. The group X is the group generated by the groups D and G.

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2.1. GUIDE TO THE SUBGROUP TABLES AND GRAPHS iL iP are decisive. The afﬁne normalizers of the plane groups and space groups are listed in the tables of Chapter 15.2 of IT A. Their point-group parts are:

type of afﬁne normalizers corresponds to the type of the highest-symmetry Euclidean normalizers belonging to that space (plane)-group type.] The afﬁne normalizers of the supergroups Pmmm, No. 47, and Pnnn, No. 48, are the same as that of P222. The index in ¼ 1 ¼ 1 1, there is only one supergroup with the same origin (origin choice 1 for Pnnn) of each of these types. The afﬁne normalizers of G ¼ Pccm, No. 49, and G ¼ Pban, No. 50, are N A ðGÞ ¼ Pð1=2; 1=2; 1=2Þ4=mmm ¼ D such that the index in ¼ 3 ¼ 1 3. Coset decomposition of N A ðP222Þ relative to D reveals the coset representatives 3111 and 31 111 so that from Pccm (origin shift 0; 0; 1=4) the space groups (in unconventional HM symbols) Pmaa and Pbmb are generated with the origin shifts 1=4; 0; 0 and 0; 1=4; 0; from Pban (no origin shift for origin choice 1) one obtains Pncb and Pcna with no origin shifts. To all these orthorhombic supergroups there are no translationally equivalent supergroups. The tetragonal supergroups P422, No. 89, P42 22, No. 93, P42m, No. 111, and P42c, No. 112, are space groups if one of the conditions a = b or b = c or c = a holds for the lattice parameters, with the tetragonal axes perpendicular to the tetragonal plane. Otherwise the tetragonal supergroups are afﬁne groups which are only isomorphic to space groups. They all have afﬁne normalizer Cð1; 1; 1=2Þ4=mmm, such that the index in ¼ 6 ¼ 2 3. The supergroups P422 form three pairs; one pair with its tetragonal axes parallel to c and with the origins either coinciding with that of P222 or shifted by 12 a (or equivalently by 12 b). The other two pairs point with their tetragonal axes parallel to a and to b, with one origin at the origin of P222 and the other origin shifted by 12 b (or equivalently c) and 12 a (or equivalently c). The same holds for the six supergroups of type P42 22 and for the six supergroups of type P42m. The six supergroups of type P42c again form three pairs with their tetragonal axes along c or a or b but their origins are shifted against that of P222 by 1=4 along the tetragonal axes because of the entry p ¼ 0; 0; 1=4 in the subgroup table of P42c, see also Example 2.1.7.3.1. For the supergroups with the symbol P23, No. 195, which is a space group if a = b = c, the afﬁne normalizer is N A ðP23Þ ¼ Ið1; 1; 1Þm3m with point-group index iP ¼ 1 but translation index iL ¼ 4. Thus, there are four such (translationally equivalent) supergroups with their origins at 0; 0; 0; 1=2; 0; 0; 0; 1=2; 0; and 0; 0; 1=2.

(1) inﬁnite groups of integral matrices W with detðW Þ ¼ 1 for triclinic and monoclinic space groups; (2) (ﬁnite) crystallographic point groups or afﬁne groups isomorphic to crystallographic point groups for orthorhombic, tetragonal, trigonal, hexagonal and cubic space groups. The translation parts of the normalizers are continuous or partly continuous groups for polar space groups and lattices for non-polar space groups. The following cases may be distinguished in the use of Lemma 2.1.7.4.1: (1) Both H and G are triclinic: only for the pair P1–P1 can G be a minimal translationengleiche supergroup of H, cf. Example 2.1.6.1.2. The point-group index iP ¼ 1, the translation index iL is uncountably inﬁnite. Such translation indices can be represented by regions of the unit cell for the possible coordinates of the origin, here 0 x0 ; y0 ; z0 < 1=2. In any case, continuous ranges of parameters due to inﬁnite indices iL do not cause difﬁculties. (2) H is triclinic, G is monoclinic: index iP ¼ 1. Such relations cannot be treated by the procedure described earlier in this section; monoclinic minimal t-supergroups of triclinic space groups are not accessible by the procedure using Lemma 2.1.7.4.1. Other approved procedures are not known to the authors; they are being developed and tested but are not yet available for this edition of IT A1. (3) Both H and G are monoclinic: for minimal t-supergroups G, inversions 1 have to be added to the symmetry of the space group H. The procedure using Lemma 2.1.7.4.1 needs to be worked out because of the inﬁnity of the linear parts of the normalizers. Another procedure is being tested at the time of writing of this guide. Both methods and their results cannot be presented here. (4) H is triclinic or monoclinic, G is orthorhombic, trigonal or hexagonal: as in (2), iP ¼ 1 and Lemma 2.1.7.4.1 cannot be applied. No solution can be offered at present. (5) Both H and G are orthorhombic or higher symmetry: the ﬁnite index iP ¼ q allows the application of Lemma 2.1.7.4.1. The index iL ¼ r may be either ﬁnite or iL ¼ 1, there are either r translationally equivalent supergroups of the same space-group type or an inﬁnite number, described by a continuous region of parameters, e.g. of coordinates of the origin. One has to take care to select from these supergroups those which are space groups and those which are afﬁne groups isomorphic to space groups (Koch, 1984).

Example 2.1.7.4.3 Application of the normalizers to the t-supergroups of H ¼ P21 21 2, No. 18. The normalizer N ðP21 21 2Þ ¼ Pð1=2; 1=2; 1=2Þ4=mmm is the same as those for the minimal t-supergroups Pbam, No. 55, Pccn, No. 56, Pnnm, No. 58, and Pmmn, No. 59. There is one supergroup for each of these types with the origin shifts 0; 0; 0; 1=4; 1=4; 1=4; 0; 0; 1=4; and 0; 0; 0 (origin choice 1), respectively, according to the p values listed in the subgroup tables of the supergroups. This can also be concluded from the spacegroup diagrams of IT A. The listed HM symbols Pbcm, No. 57, and Pbcn, No. 60, do not refer to supergroups of P21 21 2, but Pbma and Pnca do. This can be seen either from the full HM symbols, cf. Example 2.1.6.2.1, or by applying the (P, p) data of the supergroups for the subgroup P21 21 2, from which the origin shifts may also be taken, cf. Example 2.1.7.3.2. Both supergroups have normalizer N A ¼ Pð1=2; 1=2; 1=2Þmmm with index in ¼ 2 ¼ 1 2. There are two supergroups of each type, the second one, Pmab

The application of Lemma 2.1.7.4.1 will be described by three examples. Example 2.1.7.4.2 is the continuation of Example 2.1.7.3.1, Example 2.1.7.4.3 is the continuation of Example 2.1.7.3.2 and Example 2.1.7.4.4 deals with the minimal t-supergroups of space group H ¼ Pmm2, No. 25. Example 2.1.7.4.2 Application of the normalizers to the minimal t-supergroups of H ¼ P222, No. 16; continuation of Example 2.1.7.3.1. The afﬁne normalizer N A ðP222Þ ¼ Pð1=2; 1=2; 1=2Þm3m, cf. IT A, Table 15.2.1.3. [In the header of this table, only the words Euclidean normalizer are found. It is mentioned in Section 15.2.2, Afﬁne normalizers of plane and space groups, that the

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2. MAXIMAL SUBGROUPS OF THE PLANE GROUPS AND SPACE GROUPS Suppose Gs > H is such a supergroup. According to the theorem of Bieberbach for space groups, isomorphism and afﬁne equivalence result in the same classiﬁcation of the space groups, cf. Section 8.2.2 of IT A. Therefore, there exists an afﬁne mapping as 2 A in the group A of all afﬁne mappings which transforms G onto the space group Gs ¼ a1 s G as and does not belong to = N A ðHÞ. Let Hs ¼ as H a1 N A ðHÞ, as 2 s be obtained from H by the inverse of the transformation which transforms G to Gs. Then the group Hs is a subgroup of G if and only if H is a subgroup of Gs . Therefore, if the space group G has a subgroup Hs < G in addition to H < G, then the space group H has an additional supergroup Gs > H which can be found using the transformation of H to Hs. This transformation is effective only if it does not belong to the normalizer N A ðGÞ, otherwise it would transform the space group G onto itself. Therefore, only those subgroups Hs < G have to be taken into consideration which are not conjugate to H under the normalizer N A ðGÞ. An example of the application of this procedure is given in Section 1.7.1 and Example 1.7.3.2.2. It refers to minimal k-supergroups; an example for minimal t-subgroups is not known to the authors. In Sections 2.1.7.4 and 2.1.7.5 only minimal t-supergroups are dealt with. The same considerations can be applied to minimal k-supergroups with corresponding results. It was not necessary to mention the minimal k-supergroups here, as because of the more detailed data in the tables of this volume, the simpler procedure of Sections 2.1.7.1 and 2.1.7.2 could be used to determine the minimal k-supergroups.

and Pcnb, generated from that already listed by the fourfold rotation in the normalizer of P21 21 2. The tetragonal supergroups (they are space groups if for their lattice parameters the equation a = b holds) P421 2, No. 90, P42 21 2, No. 94, P421 m, No. 113, and P421 c, No. 114, all have normalizer Cð1; 1; 1=2Þ4=mmm, such that the index in ¼ 2 ¼ 2 1. There are two translationally equivalent supergroups in each case, one with origin at 0; 0; 0; 0; 0; 1=4; 0; 0; 0; and 0; 0; 1=4, respectively, the other shifted against the ﬁrst one by the translation t ð1=2; 0; 0Þ or (equivalently) by t ð0; 1=2; 0Þ. The procedure just described works ﬁne if the symmetry of the group H is higher than triclinic or monoclinic. The following example shows that an inﬁnite lattice index is acceptable. Example 2.1.7.4.4 Application of the normalizers to the t-supergroups of H ¼ Pmm2, No. 25. The afﬁne normalizer is N A ðPmm2Þ ¼ Pð1=2; 1=2; "Þ4=mmm. The list of t-supergroups of Pmm2 starts with Pmmm, No. 47. The afﬁne normalizer N A ðPmmmÞ ¼ Pð1=2; 1=2; 1=2Þm3m; its point group m3m is a supergroup of the point group 4/mmm of N A ðPmm2Þ such that the intersection of the point groups is 4/mmm and iP ¼ 1. The index iL ¼ 1, leading to a continuous sequence of supergroups with origins at 0; 0; z0 ; 0 z0 < 1=2. The t-supergroup Pmma, No. 51, follows. N A ðPmmaÞ ¼ Pð1=2; 1=2; 1=2ÞPmmm such that D ¼ Pð1=2; 1=2; 1=2ÞPmmm with iP ¼ 2 and iL ¼ 1. There is a continuous set of supergroups Pmma. Because of the origin shift p ¼ 1=4; 0; 0 for Pmm2 in the subgroup table of Pmma, the origins of these supergroups are placed at 1=4; 0; z0 with 0 z0 < 1=2. Because iP ¼ 2, there is a second set of supergroups, rotated by 90 relative to the ﬁrst set, such that its (unconventional) HM symbol is Pmmb and its origins are placed at 0; 1=4; z0 , 0 z0 < 1=2. The normalizer N A ¼ Pð1=2; 1=2; 1=2Þ4=mmm of the last orthorhombic t-supergroup Pmmn, No. 59, has the same pointgroup part as that of Pmm2 and its translation part differs only in z ¼ 1=2 instead of z ¼ ". There are no transformation data for Pmm2 in the subgroup table of Pmmn, origin choice 1. There is one set of supergroups Pmmn with the origins at 0; 0; z0 for 0 z0 < 1=2. The tetragonal minimal supergroups P4mm, No. 99, P42 mc, No. 105, and P4m2, No. 115, are space groups if a = b holds for the lattice parameters of Pmm2. The afﬁne and Euclidean normalizers of tetragonal space groups are the same. The normalizers Cð1; 1; "Þ4=mmm of P4mm and P42 mc differ from N A ðPmm2Þ only in the translation part such that iP ¼ 1 and iL ¼ 2 and the additional translations of N A ðPmm2Þ relative to D may be represented by t ð1=2; 0; 0Þ. There are two supergroups each: P4mm and P42 mc, with their origins at 0; 0; 0 and 1=2; 0; 0 of Pmm2. Finally, the normalizer N ðP4m2Þ ¼ Cð1; 1; 1=2ÞP4=mmm and there are two continuous sets of supergroups P4m2 with their origins at 0; 0; z0 and 1=2; 0; z0 ; 0 z0 < 1=2.

2.1.8. The subgroup graphs 2.1.8.1. General remarks The group–subgroup relations between the space groups may also be described by graphs. This way is chosen in Chapters 2.4 and 2.5. Graphs for the group–subgroup relations between crystallographic point groups have been published, for example, in Internationale Tabellen zur Bestimmung von Kristallstrukturen (1935) and in IT A (2005), Figs. 10.1.3.2 and 10.1.4.3. Three kinds of graphs for subgroups of space groups have been constructed and can be found in the literature: (1) Graphs for t-subgroups, such as the graphs of Ascher (1968). (2) Graphs for k-subgroups, such as the graphs for cubic space groups of Neubu¨ser & Wondratschek (1966). (3) Mixed graphs, combining t- and k-subgroups. These are used, for example, when relations between existing or suspected crystal structures are to be displayed. Examples are the ‘family trees’ after Ba¨rnighausen (1980), as shown in Chapter 1.6, now called Ba¨rnighausen trees. A complete collection of graphs of the ﬁrst two kinds is presented in this volume: in Chapter 2.4 those displaying the translationengleiche or t-subgroup relations and in Chapter 2.5 those for the klassengleiche or k-subgroup relations. Neither type of graph is restricted to maximal subgroups but both contain t- or k-subgroups of higher indices, with the exception of isomorphic subgroups, cf. Section 2.1.8.3 below. The group–subgroup relations are direct relations between the space groups themselves, not between their types. However, each such relation is valid for a pair of space groups, one from each of the types, and for each space group of a given type there exists a corresponding relation. In this sense, one can speak of a

2.1.7.5. Derivation of the remaining minimal t-supergroups By the procedure discussed in Section 2.1.7.4, from a supergroup G > H other supergroups Gj > H could be found which are isomorphic to G with the same index. The question arises as to whether there exist further minimal supergroups Gq > H isomorphic to G and of the same index which can not be obtained by consideration of the normalizers.

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2.1. GUIDE TO THE SUBGROUP TABLES AND GRAPHS relation between the space-group types, keeping in mind the difference between space groups and space-group types, cf. Section 1.2.5.3. The space groups in the graphs are denoted by the standard HM symbols and the space-group numbers. In each graph, each space-group type is displayed at most once. Such graphs are called contracted graphs here. Without this contraction, the more complex graphs would be much too large for the page size of this volume. The symbol of a space group G is connected by uninterrupted straight lines with the symbols of all its maximal non-isomorphic subgroups or minimal non-isomorphic supergroups. In general, the maximal subgroups of G are drawn on a lower level than G; in the same way, the minimal supergroups of G are mostly drawn on a higher level than G. For exceptions see Section 2.1.8.3. Multiple lines may occur in the graphs for t-subgroups. They are explained in Section 2.1.8.2. No indices are attached to the lines. They can be taken from the corresponding subgroup tables of Chapter 2.3, and are also provided by the general formulae of Section 1.2.8. For the k-subgroup graphs, they are further speciﬁed at the end of Section 2.1.8.3.

Example 2.1.8.2.1 Compare the t-subgroup graphs in Figs. 2.4.4.2, 2.4.4.3 and 2.4.4.8 of Pnna, No. 52, Pmna, No. 53, and Cmce, No. 64, respectively. The complete (uncontracted) graphs would have the shape of the graph of the point group mmm with mmm at the top (ﬁrst level), seven point groups7 (222, mm2, m2m, 2mm, 112=m, 12=m1 and 2=m11) in the second level, seven point groups (112, 121, 211, 11m, 1m1, m11 and 1) in the third level and the point group 1 at the bottom (fourth level). The group mmm is connected to each of the seven subgroups at the second level by one line. Each of the groups of the second level is connected with three groups of the third level by one line. All seven groups of the third level are connected by one line each with the point group 1 at the bottom. The contracted graph of the point group mmm would have mmm at the top, three point-group types (222, mm2 and 2=m) at the second level and three point-group types (2, m and 1) at the third level. The point group 1 at the bottom would not be displayed (no fourth level). Single lines would connect mmm with 222, mm2 with 2, 2=m with 2, 2=m with m and 2=m with 1; a double line would connect mm2 with m; triple lines would connect mmm with mm2, mmm with 2=m and 222 with 2. The number of ﬁelds in a contracted t-subgroup graph is between the numbers of ﬁelds in the complete and in the contracted point-group graphs. The graph in Fig. 2.4.4.2 of Pnna, No. 52, has six space-group types at the second level and four space-group types at the third level. For the graph in Fig. 2.4.4.3 of Pmna, No. 53, these numbers are seven and ﬁve and for the graph in Fig. 2.4.4.8 of Cmce, No. 64 (formerly Cmca), the numbers are seven and six. However, in all these graphs the number of connections is always seven from top to the second level and three from each ﬁeld of the second level downwards to the ground level, independent of the amount of contraction and of the local multiplicity of lines.

2.1.8.2. Graphs for translationengleiche subgroups Let G be a space group and T (G) the normal subgroup of all its translations. Owing to the isomorphism between the factor group G=T ðGÞ and the point group P G, see Section 1.2.5.4, according to the ﬁrst isomorphism theorem, Ledermann & Weir (1996), t-subgroup graphs are the same (up to the symbols) as the corresponding graphs between point groups. However, in this volume, the graphs are not complete but are contracted, displaying each space-group type at most once. This contraction may cause the graphs to look different from the point-group graphs and also different for different space groups of the same point group, cf. Example 2.1.8.2.1. One can indicate the connections between a space group G and its maximal subgroups in different ways. In the contracted t-subgroup graphs one line is drawn for each conjugacy class of maximal subgroups of G. Thus, a line represents the connection to an individual subgroup only if this is a normal maximal subgroup of G, otherwise it represents the connection to more than one subgroup. The conjugacy relations are not necessarily transferable to non-maximal subgroups, cf. Example 2.1.8.2.2. On the other hand, multiple lines are possible, see the examples. Although it is not in general possible to reconstruct the complete graph from the contracted one, the content of information of such a graph is higher than that of a graph which is drawn with simple lines only. The graph for the space group at its top also contains the contracted graphs for all subgroups which occur in it, see Example 2.1.8.2.3. Owing to lack of space for the large graphs, in all graphs of t-subgroups the group P1, No. 1, and its connections have been omitted. Therefore, to obtain the full graph one has to supplement the graphs by P1 at the bottom and to connect P1 by one line to each of the symbols that have no connection downwards. Within the same graph, symbols on the same level indicate subgroups of the same index relative to the group at the top. The distance between the levels indicates the size of the index. For a more detailed discussion, see Example 2.1.8.2.2. For the sequence and the numbers of the graphs, see the paragraph after Example 2.1.8.2.2.

Example 2.1.8.2.2 Compare the t-subgroup graphs shown in Fig. 2.4.1.1 for Pm3m, No. 221, and Fig. 2.4.1.5, Fm3m, No. 225. These graphs are contracted from the point-group graph m3m. There are altogether nine levels (without the lowest level of P1). The indices relative to the top space groups Pm3m and Fm3m are 1, 2, 3, 4, 6, 8, 12, 16 and 24, corresponding to the point-group orders 48, 24, 16, 12, 8, 6, 4, 3 and 2, respectively. The height of the levels in the graphs reﬂects the index; the distances between the levels are proportional to the logarithms of the indices but are slightly distorted here in order to adapt to the density of the lines. From the top space-group symbol there are ﬁve lines to the symbols of maximal subgroups: The three symbols at the level of index 2 are those of cubic normal subgroups, the one (tetragonal) symbol at the level of index 3 represents a conjugacy class of three, the symbol R3m, No. 166, at the level of index 4 represents a conjugacy class of four subgroups. The graphs differ in the levels of the indices 12 and 24 (orthorhombic, monoclinic and triclinic subgroups) by the number of symbols (nine and seven for index 12, ﬁve and three for index 24). The number of lines between neighbouring connected levels depends only on the number and kind of symbols in the upper level. 7 The HM symbols used here are nonconventional. They display the setting of the point group and follow the rules of IT A, Section 2.2.4.

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2. MAXIMAL SUBGROUPS OF THE PLANE GROUPS AND SPACE GROUPS However, for non-maximal subgroups the conjugacy relations may not hold. For example, in Fig. 2.4.1.1, the space group P222 has three normal maximal subgroups of type P2 and is thus connected to its symbol by a triple line, although these subgroups are conjugate subgroups of the non-minimal supergroup Pm3m.

For the index of a maximal t-subgroup, Lemma 1.2.8.2.3 is repeated: the index of a maximal non-isomorphic subgroup H is always 2 for oblique, rectangular and square plane groups and for triclinic, monoclinic, orthorhombic and tetragonal space groups G. The index is 2 or 3 for hexagonal plane groups and for trigonal and hexagonal space groups G. The index is 2, 3 or 4 for cubic space groups G.

Example 2.1.8.2.3 The t-subgroup graphs in Figs. 2.4.1.1 and 2.4.1.5 contain the t-subgroups of the summits Pm3m and Fm3m and their relations. In addition, the t-subgroup graph of Pm3m includes the t-subgroup graphs of the cubic summits P432, P43m, Pm3 and P23, those of the tetragonal summits P4=mmm, P42m, P4m2, P4mm, P422, P4=m, P4 and P4, those of the rhombohedral summits R3m, R3m, R32, R3 and R3 etc., as well as the t-subgroup graphs of several orthorhombic and monoclinic summits. The graph of the summit Fm3m includes analogously the graphs of the cubic summits F432, F43m, Fm3 and F23, of the tetragonal summits I4=mmm, I4m2 etc., also the graphs of rhombohedral summits R3m etc. Thus, many other graphs are included in the two basic graphs and can be extracted from them. The same holds for the other graphs displayed in Figs. 2.4.1.2 to 2.4.4.8: each of them includes the contracted graphs of all its subgroups as summits. For this reason one does not need 229 or 218 different graphs to cover all t-subgroup graphs of the 229 space-group types but only 37 (P1 can be excluded as trivial).

2.1.8.3. Graphs for klassengleiche subgroups There are 29 graphs for klassengleiche or k-subgroups, one for each crystal class with the exception of the crystal classes 1, 1 and 6 with only one space-group type each: P1, No. 1, P1, No. 2, and P6, No. 174, respectively. The sequence of the graphs is determined by the sequence of the point groups in IT A, Table 2.1.2.1, fourth column. The graphs of 4, 3 and 6=m are nearly trivial, because to these crystal classes only two space-group types belong. The graphs of mm2 with 22, of mmm with 28 and of 4=mmm with 20 space-group types are the most complicated ones. Isomorphic subgroups are special cases of k-subgroups. With the exception of both partners of the enantiomorphic spacegroup types, isomorphic subgroups are not displayed in the graphs. The explicit display of the isomorphic subgroups would add an inﬁnite number of lines from each ﬁeld for a space group back to this ﬁeld, or at least one line (e.g. a circle) implicitly representing the inﬁnite number of isomorphic subgroups, see the tables of maximal subgroups of Chapter 2.3.8 Such a line would have to be attached to every space-group symbol. Thus, there would be no more information. Nevertheless, connections between isomorphic space groups are included indirectly if the group–subgroup chain encloses a space group of another type. In this case, a space group X may be a subgroup of a space group Y and Y 0 a subgroup of X, where Y and Y 0 belong to the same space-group type. The subgroup chain is then Y – X – Y 0 . The two space groups Y and Y 0 are not identical but isomorphic. Whereas in general the label for the subgroup is positioned at a lower level than that for the original space group, for such relations the symbols for X and Y can only be drawn on the same level, connected by horizontal lines. If this happens at the top of a graph, the top level is occupied by more than one symbol (the number of symbols in the top level is the same as the number of symmorphic space-group types of the crystal class). Horizontal lines are drawn as left or right arrows depending on the kind of relation. The arrow is always directed from the supergroup to the subgroup. If the relation is two-sided, as is always the case for enantiomorphic space-group types, then the relation is displayed by a pair of horizontal lines, one of them formed by a right and the other by a left arrow. In the graph in Fig. 2.5.1.5 for crystal class mm2, the connections of Pmm2 with Cmm2 and with Amm2 are displayed by double-headed arrows instead. Furthermore, some arrows in Fig. 2.5.1.5, crystal class mm2, and Fig. 2.5.1.6, mmm, are dashed or dotted in order to better distinguish the different lines and to increase clarity. The different kinds of relations are demonstrated in the following examples.

The preceding Example 2.1.8.2.3 suggests that one should choose the graphs in such a way that their number can be kept small. It is natural to display the ‘big’ graphs ﬁrst and later those smaller graphs that are still missing. This procedure is behind the sequence of the t-subgroup graphs in this volume. (1) The ten graphs of Pm3m, No. 221, to Ia3d, No. 230, form the ﬁrst set of graphs in Figs. 2.4.1.1 to 2.4.1.10. (2) There are a few cubic space groups left which do not appear in the ﬁrst set. They are covered by the graphs of P41 32, No. 213, P43 32, No. 212, and Pa3, No. 205. These graphs have large parts in common so that they can be united in Fig. 2.4.1.11. (3) No cubic space group is left now, but only eight tetragonal space groups of crystal class 4=mmm have appeared up to now. Among them are all graphs for 4=mmm space groups with an I lattice which are contained in Figs. 2.4.1.5 to 2.4.1.8 of the F-centred cubic space groups. The next 12 graphs, Figs. 2.4.2.1 to 2.4.2.12, are those for the missing space groups of the crystal class 4=mmm with lattice symbol P. They start with P4=mcc, No. 124, and end with P42 =ncm, No. 138. (4) Four (enantiomorphic) tetragonal space-group types are left which are compiled in Fig. 2.4.2.13. (5) The next set is formed by the four graphs in Figs. 2.4.3.1 to 2.4.3.4 of the hexagonal space groups P6=mmm, No. 191, to P63 =mmc, No. 194. The hexagonal and trigonal enantiomorphic space groups do not appear in these graphs. They are combined in Fig. 2.4.3.5, the last one of hexagonal origin. (6) Several orthorhombic space groups are still left. They are treated in the eight graphs in Figs. 2.4.4.1 to 2.4.4.8, from Pmma, No. 51, to Cmce, No. 64 (formerly Cmca). (7) For each space group, the contracted graph of all its t-subgroups is provided in at least one of these 37 graphs.

Example 2.1.8.3.1 In the graph in Fig. 2.5.1.1, crystal class 2, a space group P2 may be a subgroup of index 2 of a space group C2 by ‘Loss of 8 One could contemplate adding one line for each series of maximal isomorphic subgroups. However, the number of series depends on the rules that deﬁne the distribution of the isomorphic subgroups into the series and is thus not constant.

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2.1. GUIDE TO THE SUBGROUP TABLES AND GRAPHS centring translations’. On the other hand, subgroups of P2 in the block ‘Enlarged unit cell’, a0 ¼ 2a; b0 ¼ 2b; C2 ð3Þ belong to the type C2, see the tables of maximal subgroups in Chapter 2.3. Therefore, both symbols are drawn at the same level and are connected by a pair of arrows pointing in opposite directions. Thus, the top level is occupied twice. In the graph in Fig. 2.5.1.2 of crystal class m, both the top level and the bottom level are each occupied by the symbols of two space-group types.

and R3m–R3c with space-group Nos. between 156 and 167. They have index 2. (3) A non-isomorphic maximal k-subgroup H of a hexagonal space group has index 2 or 3. (4) A non-isomorphic maximal k-subgroup H of a cubic space group G has either index 2 or index 4. The index is 2 if G has an I lattice and H a P lattice or if G has a P lattice and H an F lattice. The index is 4 if G has an F lattice and H a P lattice or if G has a P lattice and H an I lattice.

Example 2.1.8.3.2 There are four symbols at the top level of the graph in Fig. 2.5.1.4, crystal class 222. Their relations are rather complicated. Whereas one can go (by index 2) from P222 directly to a subgroup of type C222 and vice versa, the connection from F222 directly to C222 is one-way. One always has to pass C222 on the way from F222 to a subgroup of the types P222 or I222. Thus, the only maximal subgroup of F222 among these groups is C222. One can go directly from P222 to F222 but not to I222 etc.

2.1.8.4. Graphs for plane groups There are no graphs for plane groups in this volume. The four graphs for t-subgroups of plane groups are apart from the symbols the same as those for the corresponding space groups: p4mm–P4mm, p6mm–P6mm, p2mg–Pma2 and p2gg–Pba2, where the graphs for the space groups are included in the t-subgroup graphs in Figs. 2.4.1.1, 2.4.3.1, 2.4.2.1 and 2.4.2.3, respectively. The k-subgroup graphs are trivial for the plane groups p1, p2, p4, p3, p6 and p6mm because there is only one plane group in its crystal class. The graphs for the crystal classes 4mm and 3m consist of two plane groups each: p4mm and p4gm, p3m1 and p31m. Nevertheless, the graphs are different: the relation is onesided for the tetragonal plane-group pair as it is in the spacegroup pair P6=m ð175Þ–P63 =m ð176Þ and it is two-sided for the hexagonal plane-group pair as it is in the space-group pair P4 (81)–I4 (82). The graph for the three plane groups of the crystal class m corresponds to the space-group graph for the crystal class 2. Finally, the graph for the four plane groups of crystal class 2mm has no direct analogue among the k-subgroup graphs of the space groups. It can be obtained, however, from the graph in Fig. 2.5.1.3 of crystal class 2=m by removing the ﬁelds of C2=c, No. 15, and P21 =m, No. 11, with all their connections to the remaining ﬁelds. The replacements are then: C2=m, No. 12, by c2mm, No. 9, P2=m, No. 10, by p2mm, No. 6, P2=c, No. 13, by p2mg, No. 7, and P21 =c, No. 14, by p2gg, No. 8.

Because of the horizontal connecting arrows, it is clear that there cannot be much correspondence between the level in the graphs and the subgroup index. However, in no graph is a subgroup positioned at a higher level than the supergroup. Example 2.1.8.3.3 Consider the graph in Fig. 2.5.1.6 for crystal class mmm. To the space group Cmmm, No. 65, belong maximal non-isomorphic subgroups of the 11 space-group types (from left to right) Ibam, No. 72, Cmcm, No. 63, Imma, No. 74, Pmmn, No. 59, Pbam, No. 55, Pban, No. 50, Pmma, No. 51, Pmna, No. 53, Cccm, No. 66, Pmmm, No. 47, and Immm, No. 71. Although all of them have index 2, their symbols are positioned at very different levels of the graph. The table for the subgroups of Cmmm in Chapter 2.3 lists 22 non-isomorphic k-subgroups of index 2, because some of the space-group types mentioned above are represented by two or four different subgroups. This multiplicity cannot be displayed by multiple lines because the density of the lines in some of the k-subgroup graphs does not permit this kind of presentation, e.g. for mmm. The multiplicity may be taken from the subgroup tables in Chapter 2.3, where each non-isomorphic subgroup is listed individually. Consider the connections from Cmmm, No. 65, to Pbam, No. 55. There are among others: the direct connection of index 2, the connection of index 4 over Ibam, No. 72, the connection of index 8 over Imma, No. 74, and Pmma, No. 51. Thus, starting from the same space group of type Cmmm one arrives at different space groups of the type Pbam with different unit cells but all belonging to the same space-group type and thus represented by the same ﬁeld of the graph.

2.1.8.5. Application of the graphs If a subgroup is not maximal then there must be a group– subgroup chain G–H of maximal subgroups with more than two members which connects G with H. There are three possibilities: H may be a t-subgroup or a k-subgroup or a general subgroup of G. In the ﬁrst two cases, the application of the graphs is straightforward because at least one of the graphs will permit one to ﬁnd the possible chains directly. If H is a k-subgroup of G, isomorphic subgroups have to be included if necessary. If H is a general subgroup of G one has to combine t- and k-subgroup graphs. There is, however, a severe shortcoming to using contracted graphs for the analysis of group–subgroup relations, and great care has to be taken in such investigations. All subgroups Hj with the same space-group type are represented by the same ﬁeld of the graph, but these different non-maximal subgroups may permit different routes to a common original (super)group.

The index of a k-subgroup is restricted by Lemma 1.2.8.2.3 and by additional conditions. For the following statements one has to note that enantiomorphic space groups are isomorphic. (1) A non-isomorphic maximal k-subgroup of an oblique, rectangular or tetragonal plane group or of a triclinic, monoclinic, orthorhombic or tetragonal space group always has index 2. (2) In general, a non-isomorphic maximal k-subgroup H of a trigonal space group G has index 3. Exceptions are the pairs P3m1–P3c1, P31m–P31c, R3m–R3c, P31m–P31c, P3m1–P3c1

Example 2.1.8.5.1 An example for translationengleiche subgroups is provided by the group–subgroup chain Fm3m ð225Þ–C2=m ð12Þ of index 12. The contracted graph may be drawn by the program Subgroupgraph from the Bilbao Crystallographic Server,

93

2. MAXIMAL SUBGROUPS OF THE PLANE GROUPS AND SPACE GROUPS http://www.cryst.ehu.es/. It is shown in Fig. 2.1.8.1; each ﬁeld represents all occurring subgroups of a space-group type: I4=mmm, No. 139, represents three subgroups, R3m, No. 166, represents four subgroups, . . . and C2=m, No. 12, represents nine subgroups belonging to two conjugacy classes. Fig. 2.1.8.1 is part of the contracted total graph of the translationengleiche subgroups of the space group Fm3m, which is displayed in Fig. 2.4.1.5. With Subgroupgraph one can also obtain the complete graph between Fm3m and the set of all nine subgroups of the type C2=m. It is too large to be reproduced here. More instructive are the complete graphs for different single subgroups of the type C2=m of Fm3m. They can also be obtained with the program Subgroupgraph with the exception of the direction indices. In Fig. 2.1.8.2 such a ‘complete’ graph is displayed for one of the six subgroups of type C2=m of index 12 whose monoclinic axes point in the h110i directions of Fm3m. Similarly, in Fig. 2.1.8.3 the complete graph is drawn for one of the three subgroups of C2=m of index 12 whose monoclinic axes point in the h001i directions of Fm3m. It differs markedly from the contracted graph and from the ﬁrst complete graph. It is easily seen that it may be very misleading to use the contracted graph or the wrong individual complete graph instead of the right individual complete graph.

Fig. 2.1.8.1. Contracted graph of the group–subgroup chains from Fm3m (225) to one of those subgroups with index 12 which belong to the spacegroup type C2=m (12). The graph forms part of the total contracted graph of t-subgroups of Fm3m (Fig. 2.4.1.5).

In a contracted graph, no basis transformations and origin shifts can be included because they are often ambiguous. In the complete graphs the basis transformations and origin shifts should be listed if these graphs display structural information and not just group–subgroup relations. The group–subgroup relations do not depend on the coordinate systems relative to which the groups are described. On the other hand, the coordinate system is decisive for the coordinates of the atoms of the crystal structures displayed and connected in a Ba¨rnighausen tree. Therefore, for the description of structural relations in a Ba¨rnighausen tree knowledge of the transformations (matrix and column) is essential and great care has to be taken to list them correctly, see Chapter 1.6 and Example 2.1.8.5.4. If one wants to list the transformations in subgroup graphs, one can use the transformations which are presented in the subgroup tables. The use of the graphs of Chapters 2.4 and 2.5 is advantageous if general subgroups, in particular those of higher indices, are sought. As stated by Hermann’s theorem, Lemma 1.2.8.1.2, a Hermann group M always exists and it is uniquely determined for any speciﬁc group–subgroup pair G > H. If the subgroup relation is general, the group M divides the chain G > H into two subchains, the chain between the translationengleiche space groups G > M and that between the klassengleiche space groups M > H. Thus, however long and complicated the real chain may be, there is always a chain for which only two graphs are needed: a t-subgroup graph for the relation between G and M and a k-subgroup graph for the relation between M and H. For a given pair of space-group types G > H and a given index [i], however, there could exist several Hermann groups of different space-group types. The graphs of this volume are very helpful in their determination. The index [i] is the product of the index ½iP , which is the ratio of the crystal class orders of G and H, and the index ½iL of the lattice reduction from G to H, ½i ¼ ½iP ½iL . The graphs of t-subgroups (Chapter 2.4) are used to ﬁnd the types of subgroups Z of G with index ½iP , belonging to the crystal class of H. From the k-graphs (Chapter 2.5) it can be seen whether Z can be a supergroup of H with index ½iL and thus a possible Hermann group M. The following two examples

Fig. 2.1.8.2. Complete graph of the group–subgroup chains from Fm3m (225) to one representative belonging to those six C2=m (12) subgroups with index 12 whose monoclinic axes are along the h110i directions of Fm3m. The direction symbols given as subscripts refer to the basis of Fm3m.

Fig. 2.1.8.3. Complete graph of the group–subgroup chains from Fm3m (225) to A112=m, which is one representative of those three C2=m (12) subgroups with index 12 whose monoclinic axes are along the h001i directions of Fm3m.

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2.1. GUIDE TO THE SUBGROUP TABLES AND GRAPHS illustrate this two-step procedure for the determination of the space-group types of Hermann groups.

but different origin shifts. There can be no other subgroups of type I4/mcm because P4/mmm is the only possible Hermann group M. Are there other chains from Pm3m to the subgroups I4=mcm? Such a new chain of subgroups must have two steps. The ﬁrst one leads from Pm3m to a k-subgroup. In the graph for k-subgroups of Pm3m one ﬁnds two subgroups of index 2, namely Fm3m and Fm3c. One ﬁnds from the corresponding graphs of t-subgroups or from the subgroup tables that only Fm3c has subgroups of space-group type I4/mcm. The subgroup tables of Pm3m and Fm3c show that there are two nonconjugate subgroups of type Fm3c which each have one conjugacy class of three subgroups of type I4/mcm. For the preferred direction c only one of the conjugate subgroups is relevant. Therefore, there are two subgroups I4/mcm of index 2 belonging to chains passing Fm3c. It follows that two of the four subgroups obtained from Hermann’s group are also subgroups of Fm3c and two are not. The two common subgroups are found by comparing their origin shifts from Pm3m, which must be the same for both ways. The use of (4 4) matrices is convenient. The relevant equations for the bases and origin shifts are:

Example 2.1.8.5.2 Consider a pair of the space-group types Pm3m > P21 =m with index [i] = 24. The factorization of the index [i] into ½iP ¼ 12 and ½iL ¼ 2 follows from the crystal classes of G and H. From the graph of t-subgroups of Pm3m (Fig. 2.4.1.1) one ﬁnds that there are two space-group types, namely P2/m and C2/m, of the relevant crystal class (2/m) and index ½iP ¼ 12. Checking the graph (Fig. 2.5.1.3) of the k-subgroups of the space groups of the crystal class 2/m conﬁrms that space groups of both spacegroup types P2/m and C2/m have (maximal) P21 =m subgroups, i.e. space groups of both space-group types are Hermann groups for the pair Pm3m > P21 =m of index [i] = 24, depending on the individual space group P21 =m. Example 2.1.8.5.3 The determination of the space-group types of the Hermann groups for the pair Im3m (No. 229) > Cmcm (No. 63) of index [i] = 12 follows the same procedure as in the previous example. The index [i] = 12 is factorized into ½iP ¼ 6 and ½iL ¼ 2 taking into account the orders of the point groups of G and H. The graph of t-subgroups of Im3m (Fig. 2.4.1.9) shows that the subgroups of the space-group types Fmmm and Immm are candidates for Hermann groups. Reference to the graph of k-subgroups of the crystal class mmm (Fig. 2.5.1.6) indicates that Immm has no maximal subgroups of Cmcm type, i.e. only space groups of Fmmm type can be Hermann groups for the pair Im3m > Cmcm of index [i] = 12.

0

2 0

B B0 2 B P1 ¼ B B0 0 @ 0 0

0 0 2 0

0

1 0

1 2

1 2

0

C B 1 1 B 0C C B2 2 0 C B B 0C A @ 0 0 1 1 0 0 0

1 4

1

0

1 1

B B 1 1 B C¼B B 0C A @ 0 0 C

1C 4C

1

1

0

1 2

0

1C 2C

C

2

C; 0C A

0

1

1 0

1

0 0

leading to an origin shift of ð12 ; 12 ; 0Þ and

Apart from the chains G M H that can be found by the above considerations, other chains may exist. In some relatively simple cases, the graphs of this volume may be helpful to ﬁnd such chains. However, one has to take into account that the tabulated graphs are contracted ones. In particular this means that they contain nothing about the numbers of subgroups of a certain kind and on their relations, for example conjugacy relations. The following practical example may display the situation. It is based on the combination of the graphs of Chapters 2.4 and 2.5 with the subgroup tables of Chapters 2.2 and 2.3.

0

2 0 B B0 2 B P2 ¼ B B0 0 @

0 0

0 0

0

2

1 2

1 0

1 1 0 2 2 C B 1C B 1 1 0 2C B 2 2 C B 1C B 0 1 2A @ 0 1 0 0 0

0 1 C B 1C B 1 4C B C¼B B 0C A @ 0 1 0 1 4

1

1 0 0 2 0 0

1

C 1C C C; 1C 2A 1

leading to an origin shift of ð1; 1; 12Þ, which is equivalent to ð0; 0; 12Þ because an integer origin shift means only the choice of another conventional origin. The result is Fig. 2.1.8.4, which is a complete graph, i.e. each ﬁeld of the graph represents exactly one space group. The names of the substances belonging to the different subgroups show that the occurrence of such unexpected relations is not unrealistic. The crystal structures of KCuF3 and LT-SrTiO3 both belong to the space-group type I4/mcm. They can be derived by different distortions from the same ideal perovskite ABX3 structure, space group Pm3m (Figs. 2.1.8.4 and 2.1.8.5). The subgroup realized by LT-SrTiO3 is not a subgroup of Fm3c. The other subgroup, which is realized by KCuF3, is a subgroup of Fm3c. This cannot be concluded from the contracted graphs, but can be seen from the combination of the graphs with the tables or from the complete graph (Billiet, 1981; Koch, 1984; Wondratschek & Aroyo, 2001). The remaining two subgroups do not form symmetries of distorted perovskites. The listed orbits of the Wyckoff positions, 4a, 4b, 4c, 4d and 8e, are all extraordinary orbits, i.e. they have more translations than the lattice of I4/mcm, see Engel et al. (1984). In Strukturbericht 1 (1931) the possible space groups for the cubic perovskites are listed on p. 301; there are ﬁve possible space groups from P23 to Pm3m. The latter space-group symbol is framed and is considered to be the true

Example 2.1.8.5.4 Let G be a space group of type Pm3m, No. 221. What are its subgroups Hi of type I4/mcm, No. 140, and index 6? As the order of the crystal class is reduced from cubic (48) to tetragonal (16) by index 3, the reduction of the translation subgroup must have index 2. To ﬁnd the Hermann group M, we look in the subgroup table of Pm3m for tetragonal t-subgroups and ﬁnd one class of three conjugate maximal t-subgroups of crystal class 4/mmm: P4/mmm, No. 123. By each of the three conjugate subgroups one of the axes a, b or c is distinguished. As this distinguished direction is kept in the other steps, one can take one of the conjugates as the representative and can continue with the consideration of only this representative. For the representative direction we choose the c axis, because this is the standard setting of P4/mmm. The relations of the other conjugates can then be obtained by replacing c by a or b. The coordinate systems of Pm3m and P4/mmm are the same, but c 6¼ a; b is possible for P4/mmm. In the subgroup table of P4/mmm one looks for subgroups of type I4/mcm and index 2 and ﬁnds four non-conjugate subgroups with the same basis

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2. MAXIMAL SUBGROUPS OF THE PLANE GROUPS AND SPACE GROUPS

Fig. 2.1.8.4. Complete graph of the group–subgroup chains from perovskite, Pm3m, here high-temperature SrTiO3, to the four subgroups of type I4/mcm with their tetragonal axes in the c direction. Two of them correspond to KCuF3 and low-temperature SrTiO3. The transformations and origin shifts given in the connecting lines specify the basis vectors and origins of the maximal subgroups in terms of the bases of the preceding space groups (Ba¨rnighausen tree as explained in Section 1.6.3).

original (only tetragonally distorted) lattice and not I4/mcm.9 For the two (empty) subgroups I4/mcm a distorted variant of the perovskite structure does not exist. Other special positions or the general position of the original cubic space group have to be occupied if the space group I4/mcm shall be realized. This example shows clearly the difference between the subgroup graphs of group theory and the Ba¨rnighausen trees of crystal chemistry. References Ascher, E. (1968). Lattices of Equi-Translation Subgroups of the Space Groups. Geneva: Battelle Institute. Ba¨rnighausen, H. (1980). Group–subgroup relations between space groups: a useful tool in crystal chemistry. MATCH Commun. Math. Chem. 9, 139–175. Billiet, Y. (1981). Des dangers d’utiliser sans pre´caution des tables de sous-groupes maximum; retour au the´ore`me d’Hermann. Acta Cryst. A37, 649–652. Engel, P., Matsumoto, T., Steinmann, G. & Wondratschek, H (1984). The non-characteristic orbits of the space groups. Z. Kristallogr. Supplement issue No. 1. International Tables for Crystallography (2005). Vol. A, SpaceGroup Symmetry, edited by Th. Hahn, 5th ed. Heidelberg: Springer. International Tables for X-ray Crystallography (1952, 1965, 1969). Vol. I, Symmetry Groups, edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press. Internationale Tabellen zur Bestimmung von Kristallstrukturen (1935). 1. Bd. Edited by C. Hermann. Berlin: Borntraeger. (In German, English and French.) Koch, E. (1984). The implications of normalizers on group–subgroup relations between space groups. Acta Cryst. A40, 593–600. Ledermann, W. & Weir, A. J. (1996). Introduction to Group Theory, 2nd ed. Harlow: Addison Wesley Longman. Neubu¨ser, J. & Wondratschek, H. (1966). Untergruppen der Raumgruppen. Krist. Tech. 1, 529–543. Strukturbericht 1913–1928 (1931). Edited by P. P. Ewald & C. Hermann. Leipzig: Akademische Verlagsgesellschaft. Continued by Structure Reports. Wondratschek, H. & Aroyo, M. I. (2001). The application of Hermann’s group M in group–subgroup relations between space groups. Acta Cryst. A57, 311–320.

Fig. 2.1.8.5. Two different subgroups of P4/mmm, both of type I4/mcm, correspond to two kinds of distortions of the coordination octahedra of the perovskite structure. The difference is due to the origin shift by 1 1 2 ; 2 ; 0 on the left side, resulting in a different selection of the fourfold rotation axes that are retained in the subgroups.

symmetry of cubic perovskites. The other space groups are t-subgroups of Pm3m. They would be taken if for some reason the site symmetries of the orbits would contradict the site symmetries of Pm3m. Similarly, the true symmetry of these tetragonal perovskite derivatives would be P4/mmm with the 9 The restriction to t-subgroups in the space-group tables of Strukturbericht 1 is probably a consequence of the fact that only these subgroups of space groups were known in 1931. No tetragonal perovskites are described in that volume. In the later volumes, Strukturbericht 2 ff., the listing of the space-group tables has been discontinued.

96

references

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of plane group 1, p. 98.

p1

No. 1

p1

Generators selected (1); t(1, 0); t(0, 1) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

1

a

1

(1) x, y

I Maximal translationengleiche subgroups

none

II Maximal klassengleiche subgroups

• Enlarged unit cell

[2] a = 2a p1 (1) [2] b = 2b p1 (1) [2] a = 2a, b = 2b c1 (1, p1) [3] a = 3a p1 (1) [3] a = 3a, b = −a + b p1 (1) [3] a = 3a, b = −2a + b p1 (1) [3] b = 3b p1 (1)

1

2a, b

1

a, 2b

1

2a, −a + b

1

3a, b

1

3a, −a + b

1

3a, −2a + b

1

a, 3b

• Series of maximal isomorphic subgroups [p] a = pa, b = −qa + b p1 (1) 1 p prime; 0 ≤ q < p no conjugate subgroups [p] b = pb p1 (1) 1 p prime no conjugate subgroups

pa, −qa + b

a, pb

I Minimal translationengleiche supergroups [2] p2 (2); [2] pm (3); [2] pg (4); [2] cm (5); [3] p3 (13) II Minimal non-isomorphic klassengleiche supergroups

Copyright © 2011 International Union of Crystallography

none

98

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of plane group 2, p. 99.

p2

p2

No. 2

Generators selected (1); t(1, 0); t(0, 1); (2) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

2

e

1

(1) x, y

(2) x, ¯ y¯

I Maximal translationengleiche subgroups [2] p1 (1) 1 II Maximal klassengleiche subgroups

• Enlarged unit cell [2] a = 2a p2 (2) p2 (2) [2] b = 2b p2 (2) p2 (2) [2] a = 2a, b = 2b c2 (2, p2) c2 (2, p2) [3] a = 3a p2 (2) p2 (2) p2 (2) [3] a = 3a, b = −a + b p2 (2) p2 (2) p2 (2) [3] a = 3a, b = −2a + b p2 (2) p2 (2) p2 (2) [3] b = 3b p2 (2) p2 (2) p2 (2)

2 2 + (1, 0)

2a, b 2a, b

1/2, 0

2 2 + (0, 1)

a, 2b a, 2b

0, 1/2

2 2 + (1, 0)

2a, −a + b 2a, −a + b

1/2, 0

2 2 + (2, 0) 2 + (4, 0)

3a, b 3a, b 3a, b

1, 0 2, 0

2 2 + (2, 0) 2 + (4, 0)

3a, −a + b 3a, −a + b 3a, −a + b

1, 0 2, 0

2 2 + (2, 0) 2 + (4, 0)

3a, −2a + b 3a, −2a + b 3a, −2a + b

1, 0 2, 0

2 2 + (0, 2) 2 + (0, 4)

a, 3b a, 3b a, 3b

0, 1 0, 2

pa, −qa + b

u, 0

a, pb

0, u

• Series of maximal isomorphic subgroups [p] a = pa, b = −qa + b p2 (2) 2 + (2u, 0) prime p > 2; 0 ≤ q < p; 0 ≤ u < p p conjugate subgroups for each pair of q and p [p] b = pb p2 (2) 2 + (0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups I Minimal translationengleiche supergroups [2] p2mm (6); [2] p2mg (7); [2] p2gg (8); [2] c2mm (9); [2] p4 (10); [2] p6 (16) II Minimal non-isomorphic klassengleiche supergroups

Copyright © 2011 International Union of Crystallography

none

99

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of plane group 3, p. 100.

pm

No. 3

p1m1

Generators selected (1); t(1, 0); t(0, 1); (2) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

2

c

1

(1) x, y

(2) x, ¯y

I Maximal translationengleiche subgroups [2] p1 (1) 1 II Maximal klassengleiche subgroups

• Enlarged unit cell [2] a = 2a pm (3) pm (3) [2] b = 2b pg (4) pm (3) [2] a = 2a, b = 2b cm (5) cm (5) [3] a = 3a pm (3) pm (3) pm (3) [3] b = 3b pm (3)

2 2 + (1, 0)

2a, b 2a, b

2 + (0, 1) 2

a, 2b a, 2b

2 2 + (1, 0)

2a, 2b 2a, 2b

1/2, 0

2 2 + (2, 0) 2 + (4, 0)

3a, b 3a, b 3a, b

1, 0 2, 0

2

a, 3b

• Series of maximal isomorphic subgroups [p] a = pa pm (3) 2 + (2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] b = pb pm (3) 2 p prime no conjugate subgroups

pa, b

a, pb

I Minimal translationengleiche supergroups [2] p2mm (6); [2] p2mg (7) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] cm (5)

• Decreased unit cell

Copyright © 2011 International Union of Crystallography

none

100

1/2, 0

u, 0

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of plane group 4, p. 101.

p1g1

pg

No. 4

Generators selected (1); t(1, 0); t(0, 1); (2) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

2

a

1

(1) x, y

(2) x, ¯ y + 12

I Maximal translationengleiche subgroups [2] p1 (1) 1 II Maximal klassengleiche subgroups

• Enlarged unit cell

[2] a = 2a pg (4) pg (4) [3] a = 3a pg (4) pg (4) pg (4) [3] b = 3b pg (4)

2 2 + (1, 0)

2a, b 2a, b

1/2, 0

2 2 + (2, 0) 2 + (4, 0)

3a, b 3a, b 3a, b

1, 0 2, 0

2 + (0, 1)

a, 3b

• Series of maximal isomorphic subgroups [p] a = pa pg (4) 2 + (2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] b = pb pg (4) 2 + (0, 2p − 12 ) prime p > 2 no conjugate subgroups

pa, b

a, pb

I Minimal translationengleiche supergroups [2] p2mg (7); [2] p2gg (8) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] cm (5)

• Decreased unit cell

[2] b = 12 b pm (3)

Copyright © 2011 International Union of Crystallography

101

u, 0

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of plane group 5, p. 102.

cm

No. 5

c1m1

Generators selected (1); t(1, 0); t(0, 1); t( 21 , 12 ); (2) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

(0, 0)+ ( 21 , 12 )+

4

(1) x, y

b

1

I Maximal translationengleiche subgroups [2] c1 (1, p1) 1+

(2) x, ¯y

1/2(a − b), 1/2(a + b)

II Maximal klassengleiche subgroups

• Loss of centring translations [2] pg (4) [2] pm (3)

1; 2 + ( 21 , 12 ) 1; 2

• Enlarged unit cell [3] a = 3a cm (5) cm (5) cm (5) [3] b = 3b cm (5)

2 2 + (2, 0) 2 + (4, 0)

3a, b 3a, b 3a, b

2

a, 3b

1/4, 0

• Series of maximal isomorphic subgroups [p] a = pa cm (5) 2 + (2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] b = pb cm (5) 2 prime p > 2 no conjugate subgroups

pa, b

a, pb

I Minimal translationengleiche supergroups [2] c2mm (9); [3] p3m1 (14); [3] p31m (15) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations

none

• Decreased unit cell

[2] a = 12 a, b = 12 b pm (3)

Copyright © 2011 International Union of Crystallography

102

1, 0 2, 0

u, 0

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of plane group 6, p. 103.

p2mm

p2mm

No. 6

Generators selected (1); t(1, 0); t(0, 1); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

4

i

1

(1) x, y

(2) x, ¯ y¯

I Maximal translationengleiche subgroups [2] p1m1 (3, pm) 1; 3 [2] p11m (3, pm) 1; 4 [2] p211 (2, p2) 1; 2

(3) x, ¯y

(4) x, y¯

b, −a

II Maximal klassengleiche subgroups

• Enlarged unit cell

[2] a = 2a p2mg (7) p2mg (7) p2mm (6) p2mm (6) [2] b = 2b p2gm (7, p2mg) p2gm (7, p2mg) p2mm (6) p2mm (6) [2] a = 2a, b = 2b c2mm (9) c2mm (9) c2mm (9) c2mm (9) [3] a = 3a p2mm (6) p2mm (6) p2mm (6) [3] b = 3b p2mm (6) p2mm (6) p2mm (6)

2; 3 + (1, 0) 3; 2 + (1, 0) 2; 3 (2; 3) + (1, 0)

2a, b 2a, b 2a, b 2a, b

2; 3 + (0, 1) (2; 3) + (0, 1) 2; 3 3; 2 + (0, 1)

2b, −a 2b, −a a, 2b a, 2b

2; 3 3; 2 + (0, 1) (2; 3) + (1, 0) 2 + (1, 1); 3 + (1, 0)

2a, 2b 2a, 2b 2a, 2b 2a, 2b

0, 1/2 1/2, 0 1/2, 1/2

2; 3 (2; 3) + (2, 0) (2; 3) + (4, 0)

3a, b 3a, b 3a, b

1, 0 2, 0

2; 3 3; 2 + (0, 2) 3; 2 + (0, 4)

a, 3b a, 3b a, 3b

0, 1 0, 2

pa, b

u, 0

a, pb

0, u

• Series of maximal isomorphic subgroups [p] a = pa p2mm (6) (2; 3) + (2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] b = pb p2mm (6) 3; 2 + (0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups I Minimal translationengleiche supergroups [2] p4mm (11) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] c2mm (9)

• Decreased unit cell

Copyright © 2011 International Union of Crystallography

none

103

1/2, 0 1/2, 0

0, 1/2 0, 1/2

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of plane group 7, p. 104.

p2mg

No. 7

p2mg

Generators selected (1); t(1, 0); t(0, 1); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

4

d

1

(1) x, y

(2) x, ¯ y¯

I Maximal translationengleiche subgroups [2] p11g (4, pg) 1; 4 [2] p1m1 (3, pm) 1; 3 [2] p211 (2, p2) 1; 2

(3) x¯ + 12 , y

(4) x + 12 , y¯

−b, a 1/4, 0

II Maximal klassengleiche subgroups

• Enlarged unit cell [2] b = 2b p2gg (8) p2gg (8) p2mg (7) p2mg (7) [3] a = 3a p2mg (7) p2mg (7) p2mg (7) [3] b = 3b p2mg (7) p2mg (7) p2mg (7)

2; 3 + (0, 1) (2; 3) + (0, 1) 2; 3 3; 2 + (0, 1)

a, 2b a, 2b a, 2b a, 2b

2; 3 + (1, 0) 2 + (2, 0); 3 + (3, 0) 2 + (4, 0); 3 + (5, 0)

3a, b 3a, b 3a, b

1, 0 2, 0

2; 3 3; 2 + (0, 2) 3; 2 + (0, 4)

a, 3b a, 3b a, 3b

0, 1 0, 2

2 + (2u, 0); 3 + ( 2p − 12 + 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

pa, b

u, 0

3; 2 + (0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, pb

0, u

0, 1/2 0, 1/2

• Series of maximal isomorphic subgroups

[p] a = pa p2mg (7)

[p] b = pb p2mg (7)

I Minimal translationengleiche supergroups

none

II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] c2mm (9) • Decreased unit cell

[2] a = 12 a p2mm (6)

Copyright © 2011 International Union of Crystallography

104

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of plane group 8, p. 105.

p2gg

p2gg

No. 8

Generators selected (1); t(1, 0); t(0, 1); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

4

c

1

(1) x, y

(2) x, ¯ y¯

(3) x¯ + 12 , y + 12

I Maximal translationengleiche subgroups [2] p1g1 (4, pg) 1; 3 [2] p11g (4, pg) 1; 4 [2] p211 (2, p2) 1; 2

(4) x + 12 , y¯ + 12

−b, a

1/4, 0 0, 1/4

2; 3 + (1, 0) 2 + (2, 0); 3 + (3, 0) 2 + (4, 0); 3 + (5, 0)

3a, b 3a, b 3a, b

1, 0 2, 0

2; 3 + (0, 1) 2 + (0, 2); 3 + (0, 1) 2 + (0, 4); 3 + (0, 1)

a, 3b a, 3b a, 3b

0, 1 0, 2

2 + (2u, 0); 3 + ( 2p − 12 + 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

pa, b

u, 0

2 + (0, 2u); 3 + (0, 2p − 12 ) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, pb

0, u

II Maximal klassengleiche subgroups

• Enlarged unit cell [3] a = 3a p2gg (8) p2gg (8) p2gg (8) [3] b = 3b p2gg (8) p2gg (8) p2gg (8)

• Series of maximal isomorphic subgroups

[p] a = pa p2gg (8)

[p] b = pb p2gg (8)

I Minimal translationengleiche supergroups [2] p4gm (12) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] c2mm (9)

• Decreased unit cell

[2] a = 12 a p2mg (7); [2] b = 12 b p2gm (7, p2mg)

Copyright © 2011 International Union of Crystallography

105

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of plane group 9, p. 106.

c2mm

No. 9

c2mm

Generators selected (1); t(1, 0); t(0, 1); t( 21 , 12 ); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

f

(0, 0)+ ( 21 , 12 )+

1

(1) x, y

(2) x, ¯ y¯

I Maximal translationengleiche subgroups [2] c1m1 (5, cm) (1; 3)+ [2] c11m (5, cm) (1; 4)+ [2] c211 (2, p2) (1; 2)+

(3) x, ¯y

(4) x, y¯

b, −a 1/2(a − b), 1/2(a + b)

II Maximal klassengleiche subgroups

• Loss of centring translations [2] [2] [2] [2]

p2gg (8) p2gm (7, p2mg) p2mg (7) p2mm (6)

• Enlarged unit cell [3] a = 3a c2mm (9) c2mm (9) c2mm (9) [3] b = 3b c2mm (9) c2mm (9) c2mm (9)

1; 1; 1; 1;

2; 4; 3; 2;

(3; 4) + ( 21 , 12 ) (2; 3) + ( 21 , 12 ) (2; 4) + ( 21 , 12 ) 3; 4

b, −a

1/4, 1/4 1/4, 1/4

2; 3 (2; 3) + (2, 0) (2; 3) + (4, 0)

3a, b 3a, b 3a, b

1, 0 2, 0

2; 3 3; 2 + (0, 2) 3; 2 + (0, 4)

a, 3b a, 3b a, 3b

0, 1 0, 2

(2; 3) + (2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

pa, b

u, 0

3; 2 + (0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, pb

0, u

• Series of maximal isomorphic subgroups

[p] a = pa c2mm (9)

[p] b = pb c2mm (9)

I Minimal translationengleiche supergroups [2] p4mm (11); [2] p4gm (12); [3] p6mm (17) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations

none

• Decreased unit cell [2] a = 12 a, b = 12 b p2mm (6)

Copyright © 2011 International Union of Crystallography

106

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of plane group 10, p. 107.

p4

p4

No. 10

Generators selected (1); t(1, 0); t(0, 1); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

4

d

1

(1) x, y

(2) x, ¯ y¯

(3) y, ¯x

(4) y, x¯

I Maximal translationengleiche subgroups [2] p2 (2) 1; 2 II Maximal klassengleiche subgroups

• Enlarged unit cell [2] a = 2a, b = 2b c4 (10, p4) c4 (10, p4)

2; 3 2 + (1, 1); 3 + (1, 0)

a − b, a + b a − b, a + b

1/2, 1/2

pa, pb

u, v

qa − rb, ra + qb

u, 0

• Series of maximal isomorphic subgroups

[p2 ] a = pa, b = pb p4 (10)

2 + (2u, 2v); 3 + (u + v, −u + v) prime p > 2; 0 ≤ u < p; 0 ≤ v < p p2 conjugate subgroups for p = 4n − 1 [p = q2 + r2 ] a = qa − rb, b = ra + qb p4 (10) 2 + (2u, 0); 3 + (u, −u) prime p = 4n + 1; q > 0; r > 0; 0 ≤ u < p p conjugate subgroups for each pair of q and r

I Minimal translationengleiche supergroups [2] p4mm (11); [2] p4gm (12) II Minimal non-isomorphic klassengleiche supergroups

Copyright © 2011 International Union of Crystallography

none

107

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of plane group 11, p. 108.

p4mm

No. 11

p4mm

Generators selected (1); t(1, 0); t(0, 1); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

g

1

(1) x, y (5) x, ¯y

(2) x, ¯ y¯ (6) x, y¯

(3) y, ¯x (7) y, x

I Maximal translationengleiche subgroups [2] p411 (10, p4) 1; 2; 3; 4 [2] p21m (9, c2mm) 1; 2; 7; 8 [2] p2m1 (6, p2mm) 1; 2; 5; 6

(4) y, x¯ (8) y, ¯ x¯

a − b, a + b

II Maximal klassengleiche subgroups

• Enlarged unit cell [2] a = 2a, b = 2b c4mg (12, p4gm) c4mg (12, p4gm) c4mm (11, p4mm) c4mm (11, p4mm)

2 + (1, 1); 3 + (1, 0); 5 + (2, 0) 2; 3; 5 + (1, 0) 2; 3; 5 2 + (1, 1); (3; 5) + (1, 0)

a − b, a + b a − b, a + b a − b, a + b a − b, a + b

1/2, 1/2

• Series of maximal isomorphic subgroups [p2 ] a = pa, b = pb p4mm (11) 2 + (2u, 2v); 3 + (u + v, −u + v); 5 + (2u, 0) prime p > 2; 0 ≤ u < p; 0 ≤ v < p p2 conjugate subgroups

pa, pb

u, v

I Minimal translationengleiche supergroups

none

II Minimal non-isomorphic klassengleiche supergroups

none

Copyright © 2011 International Union of Crystallography

108

1/2, 1/2

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of plane group 12, p. 109.

p4gm

p4gm

No. 12

Generators selected (1); t(1, 0); t(0, 1); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

d

1

(1) x, y (5) x¯ + 12 , y + 12

(2) x, ¯ y¯ (6) x + 12 , y¯ + 12

I Maximal translationengleiche subgroups [2] p411 (10, p4) 1; 2; 3; 4 [2] p21m (9, c2mm) 1; 2; 7; 8 [2] p2g1 (8, p2gg) 1; 2; 5; 6

(3) y, ¯x (7) y + 12 , x + 12

(4) y, x¯ (8) y¯ + 12 , x¯ + 12

a − b, a + b

0, 1/2

II Maximal klassengleiche subgroups

• Enlarged unit cell

none

• Series of maximal isomorphic subgroups [p2 ] a = pa, b = pb p4gm (12) 2 + (2u, 2v); 3 + (u + v, −u + v); 5 + ( 2p − 12 + 2u, 2p − 12 ) prime p > 2; 0 ≤ u < p; 0 ≤ v < p p2 conjugate subgroups

pa, pb

I Minimal translationengleiche supergroups

none

II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] c4gm (11, p4mm)

• Decreased unit cell

Copyright © 2011 International Union of Crystallography

none

109

u, v

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of plane group 13, p. 110.

p3

No. 13

p3

Generators selected (1); t(1, 0); t(0, 1); (2) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

3

d

1

(1) x, y

(2) y, ¯ x−y

(3) x¯ + y, x¯

I Maximal translationengleiche subgroups [3] p1 (1) 1 II Maximal klassengleiche subgroups

• Enlarged unit cell [3] a = 3a, b = 3b h3 (13, p3) h3 (13, p3) h3 (13, p3) [4] a = 2a, b = 2b ⎧ p3 (13) ⎪ ⎨ p3 (13) ⎪ ⎩ p3 (13) p3 (13)

2 2 + (1, 0) 2 + (1, 1)

a − b, a + 2b a − b, a + 2b a − b, a + 2b

2/3, 1/3 1/3, 2/3

2 2 + (1, −1) 2 + (1, 2) 2 + (2, 1)

2a, 2b 2a, 2b 2a, 2b 2a, 2b

1, 0 0, 1 1, 1

pa, pb

u, v

qa − rb, ra + (q + r)b

u, 0

• Series of maximal isomorphic subgroups

[p2 ] a = pa, b = pb p3 (13)

2 + (u + v, −u + 2v) p prime; 0 ≤ u < p; 0 ≤ v < p p2 conjugate subgroups for p = 2 or p = 6n − 1 [p = q2 + r2 + qr] a = qa − rb, b = ra + (q + r)b p3 (13) 2 + (u, −u) prime p = 6n + 1; q > 0; r > 0; 0 ≤ u < p p conjugate subgroups for each pair of q and r

I Minimal translationengleiche supergroups [2] p3m1 (14); [2] p31m (15); [2] p6 (16) II Minimal non-isomorphic klassengleiche supergroups

Copyright © 2011 International Union of Crystallography

none

110

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of plane group 14, p. 111.

p3m1

p3m1

No. 14

Generators selected (1); t(1, 0); t(0, 1); (2); (4) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

6

e

1

(1) x, y (4) y, ¯ x¯

(2) y, ¯ x−y (5) x¯ + y, y

I Maximal translationengleiche subgroups [2] p311 (13, p3) 1; 2; 3 [3] p1m1 (5, cm) 1; 4 [3] p1m1 (5, cm) 1; 5 [3] p1m1 (5, cm) 1; 6

(3) x¯ + y, x¯ (6) x, x − y

a + b, −a + b −a, −a − 2b −b, 2a + b

II Maximal klassengleiche subgroups

• Enlarged unit cell [3] a = 3a, b = 3b h3m1 (15, p31m) h3m1 (15, p31m) h3m1 (15, p31m) h3m1 (15, p31m) h3m1 (15, p31m) h3m1 (15, p31m) h3m1 (15, p31m) h3m1 (15, p31m) h3m1 (15, p31m) [4] a = 2a, b = 2b ⎧ p3m1 (14) ⎪ ⎨ p3m1 (14) ⎪ ⎩ p3m1 (14) p3m1 (14)

2; 4 2 + (1, −1); 4 + (1, 1) 2 + (2, −2); 4 + (2, 2) 4; 2 + (0, 1) 2 + (1, 0); 4 + (1, 1) 2 + (2, −1); 4 + (2, 2) 4; 2 + (0, 2) (2; 4) + (1, 1) 2 + (2, 0); 4 + (2, 2)

a − b, a + 2b a − b, a + 2b a − b, a + 2b a − b, a + 2b a − b, a + 2b a − b, a + 2b a − b, a + 2b a − b, a + 2b a − b, a + 2b

1, 0 2, 0 −1/3, 1/3 2/3, 1/3 5/3, 1/3 −2/3, 2/3 1/3, 2/3 4/3, 2/3

2; 4 2 + (1, −1); 4 + (1, 1) 2 + (1, 2); 4 + (1, 1) 2 + (2, 1); 4 + (2, 2)

2a, 2b 2a, 2b 2a, 2b 2a, 2b

1, 0 0, 1 1, 1

pa, pb

u, v

• Series of maximal isomorphic subgroups [p2 ] a = pa, b = pb p3m1 (14) 2 + (u + v, −u + 2v); 4 + (u + v, u + v) p prime; p = 3; 0 ≤ u < p; 0 ≤ v < p p2 conjugate subgroups I Minimal translationengleiche supergroups [2] p6mm (17) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [3] h3m1 (15, p31m)

• Decreased unit cell

Copyright © 2011 International Union of Crystallography

none

111

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of plane group 15, p. 112.

p31m

No. 15

p31m

Generators selected (1); t(1, 0); t(0, 1); (2); (4) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

6

d

1

(1) x, y (4) y, x

(2) y, ¯ x−y (5) x − y, y¯

I Maximal translationengleiche subgroups [2] p311 (13, p3) 1; 2; 3 [3] p11m (5, cm) 1; 4 [3] p11m (5, cm) 1; 5 [3] p11m (5, cm) 1; 6

(3) x¯ + y, x¯ (6) x, ¯ x¯ + y

a − b, a + b a + 2b, −a −2a − b, −b

II Maximal klassengleiche subgroups

• Enlarged unit cell [3] a = 3a, b = 3b h31m (14, p3m1) [4] a = 2a, b = 2b ⎧ p31m (15) ⎪ ⎨ p31m (15) ⎪ ⎩ p31m (15) p31m (15)

2; 4

a − b, a + 2b

2; 4 (2; 4) + (1, −1) 2 + (1, 2); 4 + (−1, 1) 4; 2 + (2, 1)

2a, 2b 2a, 2b 2a, 2b 2a, 2b

1, 0 0, 1 1, 1

pa, pb

u, v

• Series of maximal isomorphic subgroups [p2 ] a = pa, b = pb p31m (15) 2 + (u + v, −u + 2v); 4 + (u − v, −u + v) p prime; p = 3; 0 ≤ u < p; 0 ≤ v < p p2 conjugate subgroups I Minimal translationengleiche supergroups [2] p6mm (17) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [3] h31m (14, p3m1) • Decreased unit cell

Copyright © 2011 International Union of Crystallography

none

112

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of plane group 16, p. 113.

p6

p6

No. 16

Generators selected (1); t(1, 0); t(0, 1); (2); (4) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

6

d

1

(1) x, y (4) x, ¯ y¯

(2) y, ¯ x−y (5) y, x¯ + y

(3) x¯ + y, x¯ (6) x − y, x

I Maximal translationengleiche subgroups [2] p3 (13) 1; 2; 3 [3] p2 (2) 1; 4 II Maximal klassengleiche subgroups

• Enlarged unit cell

[3] a = 3a, b = 3b h6 (16, p6) h6 (16, p6) h6 (16, p6) [4] a = 2a, b = 2b ⎧ p6 (16) ⎪ ⎨ p6 (16) ⎪ ⎩ p6 (16) p6 (16)

2; 4 2 + (1, −1); 4 + (2, 0) 2 + (2, −2); 4 + (4, 0)

a − b, a + 2b a − b, a + 2b a − b, a + 2b

1, 0 2, 0

2; 4 2 + (1, −1); 4 + (2, 0) 2 + (1, 2); 4 + (0, 2) 2 + (2, 1); 4 + (2, 2)

2a, 2b 2a, 2b 2a, 2b 2a, 2b

1, 0 0, 1 1, 1

• Series of maximal isomorphic subgroups

[p2 ] a = pa, b = pb p6 (16)

pa, pb 2 + (u + v, −u + 2v); 4 + (2u, 2v) p prime; 0 ≤ u < p; 0 ≤ v < p p2 conjugate subgroups for p = 2 or p = 6n − 1 [p = q2 + r2 + qr] a = qa − rb, b = ra + (q + r)b qa − rb, ra + (q + r)b p6 (16) 2 + (u, −u); 4 + (2u, 0) prime p = 3 or p = 6n + 1; q > 0; r > 0; 0 ≤ u < p p conjugate subgroups for each pair of q and r

I Minimal translationengleiche supergroups [2] p6mm (17) II Minimal non-isomorphic klassengleiche supergroups

Copyright © 2011 International Union of Crystallography

none

113

u, v

u, 0

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of plane group 17, p. 114.

p6mm

No. 17

p6mm

Generators selected (1); t(1, 0); t(0, 1); (2); (4); (7) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

12

f

1

(1) (4) (7) (10)

x, y x, ¯ y¯ y, ¯ x¯ y, x

(2) (5) (8) (11)

I Maximal translationengleiche subgroups [2] p611 (16, p6) 1; 2; 3; 4; 5; 6 [2] p31m (15) 1; 2; 3; 10; 11; 12 [2] p3m1 (14) 1; 2; 3; 7; 8; 9 [3] p2mm (9, c2mm) 1; 4; 7; 10 [3] p2mm (9, c2mm) 1; 4; 8; 11 [3] p2mm (9, c2mm) 1; 4; 9; 12

y, ¯ x−y y, x¯ + y x¯ + y, y x − y, y¯

(3) (6) (9) (12)

x¯ + y, x¯ x − y, x x, x − y x, ¯ x¯ + y

a − b, a + b a + 2b, −a 2a + b, b

II Maximal klassengleiche subgroups

• Enlarged unit cell [3] a = 3a, b = 3b h6mm (17, p6mm) h6mm (17, p6mm) h6mm (17, p6mm) [4] a = 2a, b = 2b ⎧ p6mm (17) ⎪ ⎨ p6mm (17) ⎪ ⎩ p6mm (17) p6mm (17)

2; 4; 7 2 + (1, −1); 4 + (2, 0); 7 + (1, 1) 2 + (2, −2); 4 + (4, 0); 7 + (2, 2)

a − b, a + 2b a − b, a + 2b a − b, a + 2b

1, 0 2, 0

2; 4; 7 2 + (1, −1); 4 + (2, 0); 7 + (1, 1) 2 + (1, 2); 4 + (0, 2); 7 + (1, 1) 2 + (2, 1); (4; 7) + (2, 2)

2a, 2b 2a, 2b 2a, 2b 2a, 2b

1, 0 0, 1 1, 1

pa, pb

u, v

• Series of maximal isomorphic subgroups

[p2 ] a = pa, b = pb p6mm (17)

2 + (u + v, −u + 2v); 4 + (2u, 2v); 7 + (u + v, u + v) p prime; p = 3; 0 ≤ u < p; 0 ≤ v < p p2 conjugate subgroups

I Minimal translationengleiche supergroups

none

II Minimal non-isomorphic klassengleiche supergroups

none

Copyright © 2011 International Union of Crystallography

114

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 1, pp. 116–117.

P1

No. 1

P1

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1) General position Multiplicity, Wyckoff letter, Site symmetry

1

a

Coordinates

1

(1) x, y, z

I Maximal translationengleiche subgroups

none

II Maximal klassengleiche subgroups

• Enlarged unit cell

[2] a = 2a P1 (1) 1 [2] b = 2b P1 (1) 1 [2] c = 2c P1 (1) 1 [2] b = 2b, c = 2c A1 (1, P1) 1 [2] a = 2a, c = 2c B1 (1, P1) 1 [2] a = 2a, b = 2b C1 (1, P1) 1 [2] a = 2a, b = 2b, c = 2c F1 (1, P1) 1 [3] a = 3a P1 (1) 1 [3] a = 3a, b = a + b P1 (1) 1 [3] a = 3a, b = 2a + b P1 (1) 1 [3] a = 3a, c = a + c P1 (1) 1 [3] a = 3a, c = 2a + c P1 (1) 1 [3] a = 3a, b = a + b, c = a + c P1 (1) 1 [3] a = 3a, b = 2a + b, c = a + c P1 (1) 1 [3] a = 3a, b = a + b, c = 2a + c P1 (1) 1 [3] a = 3a, b = 2a + b, c = 2a + c P1 (1) 1 [3] b = 3b P1 (1) 1 [3] b = 3b, c = b + c P1 (1) 1 [3] b = 3b, c = 2b + c P1 (1) 1 [3] c = 3c P1 (1) 1

Copyright © 2011 International Union of Crystallography

2a, b, c a, 2b, c a, b, 2c a, 2b, b + c 2a, b, a + c 2a, a + b, c 2a, a + b, a + c 3a, b, c 3a, a + b, c 3a, 2a + b, c 3a, b, a + c 3a, b, 2a + c 3a, a + b, a + c 3a, 2a + b, a + c 3a, a + b, 2a + c 3a, 2a + b, 2a + c a, 3b, c a, 3b, b + c a, 3b, 2b + c a, b, 3c

116

C11

CONTINUED

P1

No. 1

• Series of maximal isomorphic subgroups [p] a = pa, b = qa + b, c = ra + c P1 (1) 1 p prime; 0 ≤ q < p; 0 ≤ r < p no conjugate subgroups [p] b = pb, c = qb + c P1 (1) 1 p prime; 0 ≤ q < p no conjugate subgroups [p] c = pc P1 (1) 1 p prime no conjugate subgroups

pa, qa + b, ra + c

a, pb, qb + c

a, b, pc

I Minimal translationengleiche supergroups [2] P1¯ (2); [2] P121 (3); [2] P112 (3); [2] P121 1 (4); [2] P1121 (4); [2] C121 (5); [2] A112 (5); [2] P1m1 (6); [2] P11m (6); [2] P1c1 (7); [2] P11a (7); [2] C1m1 (8); [2] A11m (8); [2] C1c1 (9); [2] A11a (9); [3] P3 (143); [3] P31 (144); [3] P32 (145); [3] R3 (146) II Minimal non-isomorphic klassengleiche supergroups

none

117

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 2, pp. 118–119.

P 1¯

Ci1

P 1¯

No. 2

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2) General position Multiplicity, Wyckoff letter, Site symmetry

2

i

1

Coordinates

(1) x, y, z

(2) x, ¯ y, ¯ z¯

I Maximal translationengleiche subgroups [2] P1 (1) 1 II Maximal klassengleiche subgroups

• Enlarged unit cell [2] a = 2a P1¯ (2) 2 P1¯ (2) 2 + (1, 0, 0) [2] b = 2b P1¯ (2) 2 P1¯ (2) 2 + (0, 1, 0) [2] c = 2c P1¯ (2) 2 P1¯ (2) 2 + (0, 0, 1) [2] b = 2b, c = 2c ¯ A1¯ (2, P1) 2 ¯ ¯ A1 (2, P1) 2 + (0, 1, 0) [2] a = 2a, c = 2c ¯ B1¯ (2, P1) 2 ¯ B1¯ (2, P1) 2 + (1, 0, 0) [2] a = 2a, b = 2b ¯ C1¯ (2, P1) 2 ¯ C1¯ (2, P1) 2 + (1, 0, 0) [2] a = 2a, b = 2b, c = 2c ¯ F 1¯ (2, P1) 2 ¯ ¯ F 1 (2, P1) 2 + (1, 0, 0) [3] ⎧ a = 3a 2 ⎨ P1¯ (2) ¯ P1 (2) 2 + (2, 0, 0) ⎩ ¯ P1 (2) 2 + (4, 0, 0) = 3a, b = a + b [3] a ⎧ 2 ⎨ P1¯ (2) P1¯ (2) 2 + (2, 0, 0) ⎩ ¯ P1 (2) 2 + (4, 0, 0) = 3a, b = 2a + b [3] a ⎧ 2 ⎨ P1¯ (2) P1¯ (2) 2 + (2, 0, 0) ⎩ ¯ P1 (2) 2 + (4, 0, 0) = 3a, c = a + c [3] a ⎧ 2 ⎨ P1¯ (2) P1¯ (2) 2 + (2, 0, 0) ⎩ ¯ P1 (2) 2 + (4, 0, 0) = 3a, c = 2a + c [3] a ⎧ 2 ⎨ P1¯ (2) P1¯ (2) 2 + (2, 0, 0) ⎩ ¯ P1 (2) 2 + (4, 0, 0) = 3a, b = a + b, c = a + c [3] a ⎧ 2 ⎨ P1¯ (2) P1¯ (2) 2 + (2, 0, 0) ⎩ ¯ P1 (2) 2 + (4, 0, 0) = 3a, b = 2a + b, c = a + c [3] a ⎧ 2 ⎨ P1¯ (2) P1¯ (2) 2 + (2, 0, 0) ⎩ ¯ P1 (2) 2 + (4, 0, 0) Copyright © 2011 International Union of Crystallography

118

2a, b, c 2a, b, c

1/2, 0, 0

a, 2b, c a, 2b, c

0, 1/2, 0

a, b, 2c a, b, 2c

0, 0, 1/2

a, 2b, b + c a, 2b, b + c

0, 1/2, 0

2a, b, a + c 2a, b, a + c

1/2, 0, 0

2a, a + b, c 2a, a + b, c

1/2, 0, 0

2a, a + b, a + c 2a, a + b, a + c

1/2, 0, 0

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

3a, a + b, c 3a, a + b, c 3a, a + b, c

1, 0, 0 2, 0, 0

3a, 2a + b, c 3a, 2a + b, c 3a, 2a + b, c

1, 0, 0 2, 0, 0

3a, b, a + c 3a, b, a + c 3a, b, a + c

1, 0, 0 2, 0, 0

3a, b, 2a + c 3a, b, 2a + c 3a, b, 2a + c

1, 0, 0 2, 0, 0

3a, a + b, a + c 3a, a + b, a + c 3a, a + b, a + c

1, 0, 0 2, 0, 0

3a, 2a + b, a + c 3a, 2a + b, a + c 3a, 2a + b, a + c

1, 0, 0 2, 0, 0

CONTINUED

P 1¯

No. 2

[3] ⎧ a = 3a, b = a + b, c = 2a + c ¯ 2 ⎨ P1 (2) P1¯ (2) 2 + (2, 0, 0) ⎩ ¯ P1 (2) 2 + (4, 0, 0) = 3a, b = 2a + b, c = 2a + c [3] a ⎧ 2 ⎨ P1¯ (2) P1¯ (2) 2 + (2, 0, 0) ⎩ ¯ P1 (2) 2 + (4, 0, 0) = 3b [3] b ⎧ 2 ⎨ P1¯ (2) P1¯ (2) 2 + (0, 2, 0) ⎩ ¯ P1 (2) 2 + (0, 4, 0) = 3b, c = b + c [3] b ⎧ 2 ⎨ P1¯ (2) P1¯ (2) 2 + (0, 2, 0) ⎩ ¯ P1 (2) 2 + (0, 4, 0) = 3b, c = 2b + c [3] b ⎧ 2 ⎨ P1¯ (2) P1¯ (2) 2 + (0, 2, 0) ⎩ ¯ P1 (2) 2 + (0, 4, 0) = 3c [3] c ⎧ 2 ⎨ P1¯ (2) P1¯ (2) 2 + (0, 0, 2) ⎩ ¯ P1 (2) 2 + (0, 0, 4)

• Series of maximal isomorphic subgroups [p] a = pa, b = qa + b, c = ra + c P1¯ (2) 2 + (2u, 0, 0) prime p > 2; 0 ≤ q < p; 0 ≤ r < p; 0 ≤ u < p p conjugate subgroups for each triplet of q, r, and p [p] b = pb, c = qb + c P1¯ (2) 2 + (0, 2u, 0) prime p > 2; 0 ≤ q < p; 0 ≤ u < p p conjugate subgroups for each pair of q and p [p] c = pc P1¯ (2) 2 + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups

3a, a + b, 2a + c 3a, a + b, 2a + c 3a, a + b, 2a + c

1, 0, 0 2, 0, 0

3a, 2a + b, 2a + c 3a, 2a + b, 2a + c 3a, 2a + b, 2a + c

1, 0, 0 2, 0, 0

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

a, 3b, b + c a, 3b, b + c a, 3b, b + c

0, 1, 0 0, 2, 0

a, 3b, 2b + c a, 3b, 2b + c a, 3b, 2b + c

0, 1, 0 0, 2, 0

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

pa, qa + b, ra + c

u, 0, 0

a, pb, qb + c

0, u, 0

a, b, pc

0, 0, u

I Minimal translationengleiche supergroups [2] P12/m1 (10); [2] P112/m (10); [2] P121 /m1 (11); [2] P1121 /m (11); [2] C12/m1 (12); [2] A112/m (12); [2] P12/c1 (13); [2] P112/a (13); [2] P121 /c1 (14); [2] P1121 /a (14); [2] C12/c1 (15); [2] A112/a (15); [3] P3¯ (147); [3] R3¯ (148) II Minimal non-isomorphic klassengleiche supergroups

none

119

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 3, pp. 120–122.

P2

No. 3

UNIQUE AXIS

C21

P121

b

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

2

e

1

(1) x, y, z

(2) x, ¯ y, z¯

I Maximal translationengleiche subgroups [2] P1 (1) 1 II Maximal klassengleiche subgroups

• Enlarged unit cell

[2] b = 2b P121 1 (4) P121 (3) [2] c = 2c P121 (3) P121 (3) [2] a = 2a P121 (3) P121 (3) [2] a = 2a, c = 2c B121 (3, P121) B121 (3, P121) [2] a = 2a, b = 2b C121 (5) C121 (5) [2] b = 2b, c = 2c A121 (5, C121) A121 (5, C121) [2] a = 2a, b = 2b, c = 2c F121 (5, C121) F121 (5, C121) [3] b = 3b P121 (3) [3] c = 3c P121 (3) P121 (3) P121 (3) [3] a = a − c, c = 3c P121 (3) P121 (3) P121 (3) [3] a = a − 2c, c = 3c P121 (3) P121 (3) P121 (3) [3] a = 3a P121 (3) P121 (3) P121 (3)

2 + (0, 1, 0) 2

a, 2b, c a, 2b, c

2 2 + (0, 0, 1)

a, b, 2c a, b, 2c

0, 0, 1/2

2 2 + (1, 0, 0)

2a, b, c 2a, b, c

1/2, 0, 0

2 2 + (0, 0, 1)

a − c, b, 2c a − c, b, 2c

0, 0, 1/2

2 2 + (1, 0, 0)

2a, 2b, c 2a, 2b, c

1/2, 0, 0

2 2 + (0, 0, 1)

2c, 2b, −a − 2c 2c, 2b, −a − 2c

0, 0, 1/2

2 2 + (1, 0, 0)

2a, 2b, −a + c 2a, 2b, −a + c

1/2, 0, 0

2

a, 3b, c

2 2 + (0, 0, 2) 2 + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

2 2 + (0, 0, 2) 2 + (0, 0, 4)

a − c, b, 3c a − c, b, 3c a − c, b, 3c

0, 0, 1 0, 0, 2

2 2 + (0, 0, 2) 2 + (0, 0, 4)

a − 2c, b, 3c a − 2c, b, 3c a − 2c, b, 3c

0, 0, 1 0, 0, 2

2 2 + (2, 0, 0) 2 + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

Copyright © 2011 International Union of Crystallography

120

No. 3

CONTINUED

UNIQUE AXIS

• Series of maximal isomorphic subgroups [p] b = pb P121 (3) 2 p prime no conjugate subgroups [p] a = a − qc, c = pc P121 (3) 2 + (0, 0, 2u) prime p > 2; 0 ≤ q < p; 0 ≤ u < p p conjugate subgroups for each pair of q and p [p] a = pa P121 (3) 2 + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

b

P2

a, pb, c

a − qc, b, pc

0, 0, u

pa, b, c

u, 0, 0

I Minimal translationengleiche supergroups [2] P12/m1 (10); [2] P12/c1 (13); [2] P222 (16); [2] P2221 (17); [2] P21 21 2 (18); [2] C222 (21); [2] Pmm2 (25); [2] Pcc2 (27); [2] Pma2 (28); [2] Pnc2 (30); [2] Pba2 (32); [2] Pnn2 (34); [2] Cmm2 (35); [2] Ccc2 (37); [2] P4 (75); [2] P42 (77); [2] P4¯ (81); [3] P6 (168); [3] P62 (171); [3] P64 (172) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] C121 (5); [2] A121 (5, C121); [2] I121 (5, C121)

• Decreased unit cell

none

(Continued from the following page)

No. 3

UNIQUE AXIS

c

• Series of maximal isomorphic subgroups

[p] c = pc P112 (3)

[p] a = pa, b = −qa + b P112 (3) [p] b = pb P112 (3)

2 p prime no conjugate subgroups

a, b, pc

2 + (2u, 0, 0) prime p > 2; 0 ≤ q < p; 0 ≤ u < p p conjugate subgroups for each pair of q and p

pa, −qa + b, c

u, 0, 0

2 + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, pb, c

0, u, 0

I Minimal translationengleiche supergroups [2] P112/m (10); [2] P112/a (13); [2] P222 (16); [2] P2221 (17); [2] P21 21 2 (18); [2] C222 (21); [2] Pmm2 (25); [2] Pcc2 (27); [2] Pma2 (28); [2] Pnc2 (30); [2] Pba2 (32); [2] Pnn2 (34); [2] Cmm2 (35); [2] Ccc2 (37); [2] P4 (75); [2] P42 (77); [2] P4¯ (81); [3] P6 (168); [3] P62 (171); [3] P64 (172) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] A112 (5); [2] B112 (5, A112); [2] I112 (5, A112) • Decreased unit cell

none

121

P2

P2

No. 3

UNIQUE AXIS

C21

P112

c

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

2

e

1

(1) x, y, z

(2) x, ¯ y, ¯z

I Maximal translationengleiche subgroups [2] P1 (1) 1 II Maximal klassengleiche subgroups

• Enlarged unit cell [2] c = 2c P1121 (4) P112 (3) [2] a = 2a P112 (3) P112 (3) [2] b = 2b P112 (3) P112 (3) [2] a = 2a, b = 2b C112 (3, P112) C112 (3, P112) [2] b = 2b, c = 2c A112 (5) A112 (5) [2] a = 2a, c = 2c B112 (5, A112) B112 (5, A112) [2] a = 2a, b = 2b, c = 2c F112 (5, A112) F112 (5, A112) [3] c = 3c P112 (3) [3] a = 3a P112 (3) P112 (3) P112 (3) [3] a = 3a, b = −a + b P112 (3) P112 (3) P112 (3) [3] a = 3a, b = −2a + b P112 (3) P112 (3) P112 (3) [3] b = 3b P112 (3) P112 (3) P112 (3)

2 + (0, 0, 1) 2

a, b, 2c a, b, 2c

2 2 + (1, 0, 0)

2a, b, c 2a, b, c

1/2, 0, 0

2 2 + (0, 1, 0)

a, 2b, c a, 2b, c

0, 1/2, 0

2 2 + (1, 0, 0)

2a, −a + b, c 2a, −a + b, c

1/2, 0, 0

2 2 + (0, 1, 0)

a, 2b, 2c a, 2b, 2c

0, 1/2, 0

2 2 + (1, 0, 0)

−2a − b, 2a, 2c −2a − b, 2a, 2c

1/2, 0, 0

2 2 + (0, 1, 0)

a − b, 2b, 2c a − b, 2b, 2c

0, 1/2, 0

2

a, b, 3c

2 2 + (2, 0, 0) 2 + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2 2 + (2, 0, 0) 2 + (4, 0, 0)

3a, −a + b, c 3a, −a + b, c 3a, −a + b, c

1, 0, 0 2, 0, 0

2 2 + (2, 0, 0) 2 + (4, 0, 0)

3a, −2a + b, c 3a, −2a + b, c 3a, −2a + b, c

1, 0, 0 2, 0, 0

2 2 + (0, 2, 0) 2 + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

(Continued on the preceding page)

122

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 4, pp. 123–124.

C22

P 1 21 1

UNIQUE AXIS

P 21

No. 4

b

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

2

a

1

(1) x, y, z

(2) x, ¯ y + 12 , z¯

I Maximal translationengleiche subgroups [2] P1 (1) 1 II Maximal klassengleiche subgroups

• Enlarged unit cell [2] c = 2c P121 1 (4) P121 1 (4) [2] a = 2a P121 1 (4) P121 1 (4) [2] a = 2a, c = 2c B121 1 (4, P121 1) B121 1 (4, P121 1) [3] b = 3b P121 1 (4) [3] c = 3c P121 1 (4) P121 1 (4) P121 1 (4) [3] a = a − c, c = 3c P121 1 (4) P121 1 (4) P121 1 (4) [3] a = a − 2c, c = 3c P121 1 (4) P121 1 (4) P121 1 (4) [3] a = 3a P121 1 (4) P121 1 (4) P121 1 (4)

2 2 + (0, 0, 1)

a, b, 2c a, b, 2c

0, 0, 1/2

2 2 + (1, 0, 0)

2a, b, c 2a, b, c

1/2, 0, 0

2 2 + (0, 0, 1)

a − c, b, 2c a − c, b, 2c

0, 0, 1/2

2 + (0, 1, 0)

a, 3b, c

2 2 + (0, 0, 2) 2 + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

2 2 + (0, 0, 2) 2 + (0, 0, 4)

a − c, b, 3c a − c, b, 3c a − c, b, 3c

0, 0, 1 0, 0, 2

2 2 + (0, 0, 2) 2 + (0, 0, 4)

a − 2c, b, 3c a − 2c, b, 3c a − 2c, b, 3c

0, 0, 1 0, 0, 2

2 2 + (2, 0, 0) 2 + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

• Series of maximal isomorphic subgroups [p] b = pb 2 + (0, 2p − 12 , 0) P121 1 (4) prime p > 2 no conjugate subgroups [p] a = a − qc, c = pc 2 + (0, 0, 2u) P121 1 (4) prime p > 2; 0 ≤ q < p; 0 ≤ u < p p conjugate subgroups for each pair of q and p [p] a = pa 2 + (2u, 0, 0) P121 1 (4) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, pb, c

a − qc, b, pc

0, 0, u

pa, b, c

u, 0, 0

I Minimal translationengleiche supergroups [2] P121 /m1 (11); [2] P121 /c1 (14); [2] P2221 (17); [2] P21 21 2 (18); [2] P21 21 21 (19); [2] C2221 (20); [2] Pmc21 (26); [2] Pca21 (29); [2] Pmn21 (31); [2] Pna21 (33); [2] Cmc21 (36); [2] P41 (76); [2] P43 (78); [3] P61 (169); [3] P65 (170); [3] P63 (173) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] C121 (5); [2] A121 (5, C121); [2] I121 (5, C121) • Decreased unit cell [2] b = 12 b P121 (3) Copyright © 2011 International Union of Crystallography

123

P 21

No. 4

UNIQUE AXIS

C22

P 1 1 21

c

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

2

a

1

(1) x, y, z

(2) x, ¯ y, ¯ z + 12

I Maximal translationengleiche subgroups [2] P1 (1) 1 II Maximal klassengleiche subgroups

• Enlarged unit cell

[2] a = 2a P1121 (4) P1121 (4) [2] b = 2b P1121 (4) P1121 (4) [2] a = 2a, b = 2b C1121 (4, P1121 ) C1121 (4, P1121 ) [3] c = 3c P1121 (4) [3] a = 3a P1121 (4) P1121 (4) P1121 (4) [3] a = 3a, b = −a + b P1121 (4) P1121 (4) P1121 (4) [3] a = 3a, b = −2a + b P1121 (4) P1121 (4) P1121 (4) [3] b = 3b P1121 (4) P1121 (4) P1121 (4)

2 2 + (1, 0, 0)

2a, b, c 2a, b, c

1/2, 0, 0

2 2 + (0, 1, 0)

a, 2b, c a, 2b, c

0, 1/2, 0

2 2 + (1, 0, 0)

2a, −a + b, c 2a, −a + b, c

1/2, 0, 0

2 + (0, 0, 1)

a, b, 3c

2 2 + (2, 0, 0) 2 + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2 2 + (2, 0, 0) 2 + (4, 0, 0)

3a, −a + b, c 3a, −a + b, c 3a, −a + b, c

1, 0, 0 2, 0, 0

2 2 + (2, 0, 0) 2 + (4, 0, 0)

3a, −2a + b, c 3a, −2a + b, c 3a, −2a + b, c

1, 0, 0 2, 0, 0

2 2 + (0, 2, 0) 2 + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

• Series of maximal isomorphic subgroups

[p] c = pc P1121 (4)

[p] a = pa, b = −qa + b P1121 (4) [p] b = pb P1121 (4)

2 + (0, 0, 2p − 12 ) prime p > 2 no conjugate subgroups

a, b, pc

2 + (2u, 0, 0) prime p > 2; 0 ≤ q < p; 0 ≤ u < p p conjugate subgroups for each pair of q and p

pa, −qa + b, c

u, 0, 0

2 + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, pb, c

0, u, 0

I Minimal translationengleiche supergroups [2] P1121 /m (11); [2] P1121 /a (14); [2] P2221 (17); [2] P21 21 2 (18); [2] P21 21 21 (19); [2] C2221 (20); [2] Pmc21 (26); [2] Pca21 (29); [2] Pmn21 (31); [2] Pna21 (33); [2] Cmc21 (36); [2] P41 (76); [2] P43 (78); [3] P61 (169); [3] P65 (170); [3] P63 (173) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] A112 (5); [2] B112 (5, A112); [2] I112 (5, A112) • Decreased unit cell [2] c = 12 c P112 (3) 124

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 5, pp. 125–127.

C23

C121

UNIQUE AXIS

C2

No. 5

b, CELL CHOICE 1

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t( 21 , 12 , 0); (2) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

4

c

(0, 0, 0)+ ( 21 , 12 , 0)+

1

(1) x, y, z

I Maximal translationengleiche subgroups [2] C1 (1, P1) 1+

(2) x, ¯ y, z¯

1/2(a − b), 1/2(a + b), c

II Maximal klassengleiche subgroups

• Loss of centring translations [2] P121 1 (4) 1; 2 + ( 21 , 12 , 0) [2] P121 (3) 1; 2 • Enlarged unit cell [2] c = 2c C121 (5) C121 (5) I121 (5, C121) I121 (5, C121) [3] b = 3b C121 (5) [3] c = 3c C121 (5) C121 (5) C121 (5) [3] a = a − 2c, c = 3c C121 (5) C121 (5) C121 (5) [3] a = a − 4c, c = 3c C121 (5) C121 (5) C121 (5) [3] a = 3a C121 (5) C121 (5) C121 (5)

1/4, 0, 0

2 2 + (0, 0, 1) 2 2 + (0, 0, 1)

a, b, 2c a, b, 2c a − 2c, b, 2c a − 2c, b, 2c

2

a, 3b, c

2 2 + (0, 0, 2) 2 + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

2 2 + (0, 0, 2) 2 + (0, 0, 4)

a − 2c, b, 3c a − 2c, b, 3c a − 2c, b, 3c

0, 0, 1 0, 0, 2

2 2 + (0, 0, 2) 2 + (0, 0, 4)

a − 4c, b, 3c a − 4c, b, 3c a − 4c, b, 3c

0, 0, 1 0, 0, 2

2 2 + (2, 0, 0) 2 + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

• Series of maximal isomorphic subgroups [p] b = pb C121 (5) 2 prime p > 2 no conjugate subgroups [p] a = a − 2qc, c = pc C121 (5) 2 + (0, 0, 2u) prime p > 2; 0 ≤ q < p; 0 ≤ u < p p conjugate subgroups for each pair of q and p [p] a = pa C121 (5) 2 + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

Copyright © 2011 International Union of Crystallography

125

0, 0, 1/2 0, 0, 1/2

a, pb, c

a − 2qc, b, pc

0, 0, u

pa, b, c

u, 0, 0

C2

UNIQUE AXIS

b

No. 5

CONTINUED

I Minimal translationengleiche supergroups [2] C12/m1 (12); [2] C12/c1 (15); [2] C2221 (20); [2] C222 (21); [2] F222 (22); [2] I222 (23); [2] I21 21 21 (24); [2] Amm2 (38); [2] Aem2 (39); [2] Ama2 (40); [2] Aea2 (41); [2] Fmm2 (42); [2] Fdd2 (43); [2] Imm2 (44); [2] Iba2 (45); [2] Ima2 (46); [2] I4 (79); [2] I41 (80); [2] I 4¯ (82); [3] P312 (149); [3] P321 (150); [3] P31 12 (151); [3] P31 21 (152); [3] P32 12 (153); [3] P32 21 (154); [3] R32 (155) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations

none

• Decreased unit cell

[2] a = 12 a, b = 12 b P121 (3)

C2

UNIQUE AXIS

c

No. 5

(Continued from the facing page)

I Minimal translationengleiche supergroups [2] A112/m (12); [2] A112/a (15); [2] C2221 (20); [2] C222 (21); [2] F222 (22); [2] I222 (23); [2] I21 21 21 (24); [2] Amm2 (38); [2] Aem2 (39); [2] Ama2 (40); [2] Aea2 (41); [2] Fmm2 (42); [2] Fdd2 (43); [2] Imm2 (44); [2] Iba2 (45); [2] Ima2 (46); [2] I4 (79); [2] I41 (80); [2] I 4¯ (82); [3] P312 (149); [3] P321 (150); [3] P31 12 (151); [3] P31 21 (152); [3] P32 12 (153); [3] P32 21 (154); [3] R32 (155) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations

none

• Decreased unit cell

[2] b = 12 b, c = 12 c P112 (3)

126

C23

A112

UNIQUE AXIS

C2

No. 5

c, CELL CHOICE 1

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t(0, 12 , 12 ); (2) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

4

c

(0, 0, 0)+ (0, 12 , 12 )+

1

(1) x, y, z

I Maximal translationengleiche subgroups [2] A1 (1, P1) 1+

(2) x, ¯ y, ¯z

a, 1/2(b − c), 1/2(b + c)

II Maximal klassengleiche subgroups

• Loss of centring translations [2] P1121 (4) 1; 2 + (0, 12 , 12 ) [2] P112 (3) 1; 2

0, 1/4, 0

• Enlarged unit cell

[2] a = 2a A112 (5) A112 (5) I112 (5, A112) I112 (5, A112) [3] c = 3c A112 (5) [3] a = 3a A112 (5) A112 (5) A112 (5) [3] a = 3a, b = −2a + b A112 (5) A112 (5) A112 (5) [3] a = 3a, b = −4a + b A112 (5) A112 (5) A112 (5) [3] b = 3b A112 (5) A112 (5) A112 (5)

2 2 + (1, 0, 0) 2 2 + (1, 0, 0)

2a, b, c 2a, b, c 2a, −2a + b, c 2a, −2a + b, c

2

a, b, 3c

2 2 + (2, 0, 0) 2 + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2 2 + (2, 0, 0) 2 + (4, 0, 0)

3a, −2a + b, c 3a, −2a + b, c 3a, −2a + b, c

1, 0, 0 2, 0, 0

2 2 + (2, 0, 0) 2 + (4, 0, 0)

3a, −4a + b, c 3a, −4a + b, c 3a, −4a + b, c

1, 0, 0 2, 0, 0

2 2 + (0, 2, 0) 2 + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

1/2, 0, 0 1/2, 0, 0

• Series of maximal isomorphic subgroups

[p] c = pc A112 (5)

[p] a = pa, b = −2qa + b A112 (5) [p] b = pb A112 (5)

2 prime p > 2 no conjugate subgroups

a, b, pc

2 + (2u, 0, 0) prime p > 2; 0 ≤ q < p; 0 ≤ u < p p conjugate subgroups for each pair of q and p

pa, −2qa + b, c

u, 0, 0

2 + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, pb, c

0, u, 0

(Continued on the facing page)

127

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 6, pp. 128–130.

Pm

No. 6

UNIQUE AXIS

Cs1

P1m1

b

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

2

c

1

(1) x, y, z

(2) x, y, ¯z

I Maximal translationengleiche subgroups [2] P1 (1) 1 II Maximal klassengleiche subgroups

• Enlarged unit cell [2] b = 2b P1m1 (6) P1m1 (6) [2] c = 2c P1c1 (7) P1m1 (6) [2] a = 2a P1a1 (7, P1c1) P1m1 (6) [2] a = 2a, c = 2c B1e1 (7, P1c1) B1m1 (6, P1m1) [2] a = 2a, b = 2b C1m1 (8) C1m1 (8) [2] b = 2b, c = 2c A1m1 (8, C1m1) A1m1 (8, C1m1) [2] a = 2a, b = 2b, c = 2c F1m1 (8, C1m1) F1m1 (8, C1m1) [3] b = 3b P1m1 (6) P1m1 (6) P1m1 (6) [3] c = 3c P1m1 (6) [3] a = a − c, c = 3c P1m1 (6) [3] a = a − 2c, c = 3c P1m1 (6) [3] a = 3a P1m1 (6)

2 2 + (0, 1, 0)

a, 2b, c a, 2b, c

2 + (0, 0, 1) 2

a, b, 2c a, b, 2c

2 + (1, 0, 0) 2

−2a − c, b, 2a 2a, b, c

2 + (0, 0, 1) 2

a − c, b, 2c a − c, b, 2c

2 2 + (0, 1, 0)

2a, 2b, c 2a, 2b, c

0, 1/2, 0

2 2 + (0, 1, 0)

2c, 2b, −a − 2c 2c, 2b, −a − 2c

0, 1/2, 0

2 2 + (0, 1, 0)

2a, 2b, −a + c 2a, 2b, −a + c

0, 1/2, 0

2 2 + (0, 2, 0) 2 + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2

a, b, 3c

2

a − c, b, 3c

2

a − 2c, b, 3c

2

3a, b, c

0, 1/2, 0

• Series of maximal isomorphic subgroups

[p] b = pb P1m1 (6)

[p] a = a − qc, c = pc P1m1 (6) [p] a = pa P1m1 (6)

2 + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, pb, c

2 p prime; 0 ≤ q < p no conjugate subgroups

a − qc, b, pc

2 p prime no conjugate subgroups

pa, b, c

Copyright © 2011 International Union of Crystallography

128

0, u, 0

CONTINUED

No. 6

UNIQUE AXIS

b

Pm

I Minimal translationengleiche supergroups [2] P12/m1 (10); [2] P121 /m1 (11); [2] Pmm2 (25); [2] Pmc21 (26); [2] Pma2 (28); [2] Pmn21 (31); [2] Amm2 (38); [2] Ama2 (40); [3] P6¯ (174) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] C1m1 (8); [2] A1m1 (8, C1m1); [2] I1m1 (8, C1m1) • Decreased unit cell

(Continued from the following page)

none

No. 6

UNIQUE AXIS

c

Pm

I Minimal translationengleiche supergroups [2] P112/m (10); [2] P1121 /m (11); [2] Pmm2 (25); [2] Pmc21 (26); [2] Pma2 (28); [2] Pmn21 (31); [2] Amm2 (38); [2] Ama2 (40); [3] P6¯ (174) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] A11m (8); [2] B11m (8, A11m); [2] I11m (8, A11m)

• Decreased unit cell

none

129

Pm

No. 6

UNIQUE AXIS

Cs1

P11m

c

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

2

c

1

(1) x, y, z

(2) x, y, z¯

I Maximal translationengleiche subgroups [2] P1 (1) 1 II Maximal klassengleiche subgroups

• Enlarged unit cell

[2] c = 2c P11m (6) P11m (6) [2] a = 2a P11a (7) P11m (6) [2] b = 2b P11b (7, P11a) P11m (6) [2] a = 2a, b = 2b C11e (7, P11a) C11m (6, P11m) [2] b = 2b, c = 2c A11m (8) A11m (8) [2] a = 2a, c = 2c B11m (8, A11m) B11m (8, A11m) [2] a = 2a, b = 2b, c = 2c F11m (8, A11m) F11m (8, A11m) [3] c = 3c P11m (6) P11m (6) P11m (6) [3] a = 3a P11m (6) [3] a = 3a, b = −a + b P11m (6) [3] a = 3a, b = −2a + b P11m (6) [3] b = 3b P11m (6)

2 2 + (0, 0, 1)

a, b, 2c a, b, 2c

2 + (1, 0, 0) 2

2a, b, c 2a, b, c

2 + (0, 1, 0) 2

2b, −a − 2b, c a, 2b, c

2 + (1, 0, 0) 2

2a, −a + b, c 2a, −a + b, c

2 2 + (0, 0, 1)

a, 2b, 2c a, 2b, 2c

0, 0, 1/2

2 2 + (0, 0, 1)

−2a − b, 2a, 2c −2a − b, 2a, 2c

0, 0, 1/2

2 2 + (0, 0, 1)

a − b, 2b, 2c a − b, 2b, 2c

0, 0, 1/2

2 2 + (0, 0, 2) 2 + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

2

3a, b, c

2

3a, −a + b, c

2

3a, −2a + b, c

2

a, 3b, c

• Series of maximal isomorphic subgroups [p] c = pc P11m (6) 2 + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] a = pa, b = −qa + b P11m (6) 2 p prime; 0 ≤ q < p no conjugate subgroups [p] b = pb P11m (6) 2 p prime no conjugate subgroups (Continued on the preceding page)

a, b, pc

pa, −qa + b, c

a, pb, c

130

0, 0, 1/2

0, 0, u

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 7, pp. 131–132.

Cs2

P1c1

UNIQUE AXIS

Pc

No. 7

b, CELL CHOICE 1

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

2

a

1

(1) x, y, z

(2) x, y, ¯ z + 12

I Maximal translationengleiche subgroups [2] P1 (1) 1 II Maximal klassengleiche subgroups

• Enlarged unit cell

[2] b = 2b P1c1 (7) P1c1 (7) [2] a = 2a P1c1 (7) P1n1 (7, P1c1) [2] a = 2a, b = 2b C1c1 (9) C1c1 (9) [3] b = 3b P1c1 (7) P1c1 (7) P1c1 (7) [3] c = 3c P1c1 (7) [3] a = 3a P1c1 (7) [3] a = 3a, c = −2a + c P1c1 (7) [3] a = 3a, c = −4a + c P1c1 (7)

2 2 + (0, 1, 0)

a, 2b, c a, 2b, c

2 2 + (1, 0, 0)

2a, b, c 2a, b, −2a + c

2 2 + (0, 1, 0)

2a, 2b, c 2a, 2b, c

0, 1/2, 0

2 2 + (0, 2, 0) 2 + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2 + (0, 0, 1)

a, b, 3c

2

3a, b, c

2 + (−1, 0, 0)

3a, b, −2a + c

2 + (−2, 0, 0)

3a, b, −4a + c

0, 1/2, 0

• Series of maximal isomorphic subgroups

[p] b = pb P1c1 (7) [p] c = pc P1c1 (7)

[p] a = pa, c = −2qa + c P1c1 (7)

2 + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, pb, c

2 + (0, 0, 2p − 12 ) prime p > 2 no conjugate subgroups

a, b, pc

2 + (−q, 0, 0) p prime; 0 ≤ q < p no conjugate subgroups

pa, b, −2qa + c

0, u, 0

I Minimal translationengleiche supergroups [2] P12/c1 (13); [2] P121 /c1 (14); [2] Pmc21 (26); [2] Pcc2 (27); [2] Pma2 (28); [2] Pca21 (29); [2] Pnc2 (30); [2] Pmn21 (31); [2] Pba2 (32); [2] Pna21 (33); [2] Pnn2 (34); [2] Aem2 (39); [2] Aea2 (41) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] A1m1 (8, C1m1); [2] C1c1 (9); [2] I1c1 (9, C1c1) • Decreased unit cell [2] c = 12 c P1m1 (6) Copyright © 2011 International Union of Crystallography

131

Pc

No. 7

UNIQUE AXIS

Cs2

P11a

c, CELL CHOICE 1

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

2

a

1

(1) x, y, z

(2) x + 12 , y, z¯

I Maximal translationengleiche subgroups [2] P1 (1) 1 II Maximal klassengleiche subgroups

• Enlarged unit cell

[2] c = 2c P11a (7) P11a (7) [2] b = 2b P11a (7) P11n (7, P11a) [2] b = 2b, c = 2c A11a (9) A11a (9) [3] c = 3c P11a (7) P11a (7) P11a (7) [3] a = 3a P11a (7) [3] b = 3b P11a (7) [3] a = a − 2b, b = 3b P11a (7) [3] a = a − 4b, b = 3b P11a (7)

2 2 + (0, 0, 1)

a, b, 2c a, b, 2c

2 2 + (0, 1, 0)

a, 2b, c a − 2b, 2b, c

2 2 + (0, 0, 1)

a, 2b, 2c a, 2b, 2c

0, 0, 1/2

2 2 + (0, 0, 2) 2 + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

2 + (1, 0, 0)

3a, b, c

2

a, 3b, c

2 + (0, −1, 0)

a − 2b, 3b, c

2 + (0, −2, 0)

a − 4b, 3b, c

0, 0, 1/2

• Series of maximal isomorphic subgroups

[p] c = pc P11a (7)

[p] a = pa P11a (7) [p] a = a − 2qb, b = pb P11a (7)

2 + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, b, pc

2 + ( 2p − 12 ), 0, 0 prime p > 2 no conjugate subgroups

pa, b, c

2 + (0, −q, 0) p prime; 0 ≤ q < p no conjugate subgroups

a − 2qb, pb, c

0, 0, u

I Minimal translationengleiche supergroups [2] P112/a (13); [2] P1121 /a (14); [2] Pmc21 (26); [2] Pcc2 (27); [2] Pma2 (28); [2] Pca21 (29); [2] Pnc2 (30); [2] Pmn21 (31); [2] Pba2 (32); [2] Pna21 (33); [2] Pnn2 (34); [2] Aem2 (39); [2] Aea2 (41) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] B11m (8, A11m); [2] A11a (9); [2] I11a (9, A11a) • Decreased unit cell

[2] a = 12 a P11m (6)

132

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 8, pp. 133–134.

Cs3

C1m1

UNIQUE AXIS

Cm

No. 8

b, CELL CHOICE 1

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t( 21 , 12 , 0); (2) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

4

b

(0, 0, 0)+ ( 21 , 12 , 0)+

1

(1) x, y, z

I Maximal translationengleiche subgroups [2] C1 (1, P1) 1+

(2) x, y, ¯z

1/2(a − b), 1/2(a + b), c

II Maximal klassengleiche subgroups

• Loss of centring translations [2] P1a1 (7, P1c1) 1; 2 + ( 21 , 12 , 0) [2] P1m1 (6) 1; 2 • Enlarged unit cell [2] c = 2c C1c1 (9) I1c1 (9, C1c1) C1m1 (8) I1m1 (8, C1m1) [3] b = 3b C1m1 (8) C1m1 (8) C1m1 (8) [3] c = 3c C1m1 (8) [3] a = a − 2c, c = 3c C1m1 (8) [3] a = a − 4c, c = 3c C1m1 (8) [3] a = 3a C1m1 (8)

−a − c, b, a

2 + (0, 0, 1) 2 + (0, 0, 1) 2 2

a, b, 2c a − 2c, b, 2c a, b, 2c a − 2c, b, 2c

2 2 + (0, 2, 0) 2 + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

2

a, b, 3c

2

a − 2c, b, 3c

2

a − 4c, b, 3c

2

3a, b, c

0, 1/4, 0

0, 1, 0 0, 2, 0

• Series of maximal isomorphic subgroups

[p] b = pb C1m1 (8)

[p] a = a − 2qc, c = pc C1m1 (8) [p] a = pa C1m1 (8)

2 + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, pb, c

2 p prime; 0 ≤ q < p no conjugate subgroups

a − 2qc, b, pc

2 prime p > 2 no conjugate subgroups

pa, b, c

0, u, 0

I Minimal translationengleiche supergroups [2] C12/m1 (12); [2] Cmm2 (35); [2] Cmc21 (36); [2] Amm2 (38); [2] Aem2 (39); [2] Fmm2 (42); [2] Imm2 (44); [2] Ima2 (46); [3] P3m1 (156); [3] P31m (157); [3] R3m (160) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations

none

• Decreased unit cell [2] a = 12 a, b = 12 b P1m1 (6) Copyright © 2011 International Union of Crystallography

133

Cm

No. 8

UNIQUE AXIS

Cs3

A11m

c, CELL CHOICE 1

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t(0, 12 , 12 ); (2) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

4

b

(0, 0, 0)+ (0, 12 , 12 )+

1

(1) x, y, z

I Maximal translationengleiche subgroups [2] A1 (1, P1) 1+

(2) x, y, z¯

a, 1/2(b − c), 1/2(b + c)

II Maximal klassengleiche subgroups

• Loss of centring translations [2] P11b (7, P11a) 1; 2 + (0, 12 , 12 ) [2] P11m (6) 1; 2 • Enlarged unit cell [2] a = 2a A11a (9) I11a (9, A11a) A11m (8) I11m (8, A11m) [3] c = 3c A11m (8) A11m (8) A11m (8) [3] a = 3a A11m (8) [3] a = 3a, b = −2a + b A11m (8) [3] a = 3a, b = −4a + b A11m (8) [3] b = 3b A11m (8)

b, −a − b, c

2 + (1, 0, 0) 2 + (1, 0, 0) 2 2

2a, b, c 2a, −2a + b, c 2a, b, c 2a, −2a + b, c

2 2 + (0, 0, 2) 2 + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

2

3a, b, c

2

3a, −2a + b, c

2

3a, −4a + b, c

2

a, 3b, c

0, 0, 1/4

0, 0, 1 0, 0, 2

• Series of maximal isomorphic subgroups

[p] c = pc A11m (8)

[p] a = pa, b = −2qa + b A11m (8) [p] b = pb A11m (8)

2 + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, b, pc

2 p prime; 0 ≤ q < p no conjugate subgroups

pa, −2qa + b, c

2 prime p > 2 no conjugate subgroups

a, pb, c

0, 0, u

I Minimal translationengleiche supergroups [2] A112/m (12); [2] Cmm2 (35); [2] Cmc21 (36); [2] Amm2 (38); [2] Aem2 (39); [2] Fmm2 (42); [2] Imm2 (44); [2] Ima2 (46); [3] P3m1 (156); [3] P31m (157); [3] R3m (160) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations

none

• Decreased unit cell [2] b = 12 b, c = 12 c P11m (6) 134

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 9, pp. 135–136.

Cs4

C1c1

UNIQUE AXIS

Cc

No. 9

b, CELL CHOICE 1

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t( 21 , 12 , 0); (2) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

4

a

(0, 0, 0)+ ( 21 , 12 , 0)+

1

(1) x, y, z

I Maximal translationengleiche subgroups [2] C1 (1, P1) 1+

(2) x, y, ¯ z + 12

1/2(a − b), 1/2(a + b), c

II Maximal klassengleiche subgroups

• Loss of centring translations [2] P1c1 (7) [2] P1n1 (7, P1c1)

• Enlarged unit cell [3] b = 3b C1c1 (9) C1c1 (9) C1c1 (9) [3] c = 3c C1c1 (9) [3] a = a − 2c, c = 3c C1c1 (9) [3] a = a − 4c, c = 3c C1c1 (9) [3] a = 3a C1c1 (9)

1; 2 1; 2 + ( 21 , 12 , 0)

a, b, −a + c

0, 1/4, 0

2 2 + (0, 2, 0) 2 + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2 + (0, 0, 1)

a, b, 3c

2 + (0, 0, 1)

a − 2c, b, 3c

2 + (0, 0, 1)

a − 4c, b, 3c

2

3a, b, c

• Series of maximal isomorphic subgroups [p] b = pb C1c1 (9) 2 + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] a = a − 2qc, c = pc C1c1 (9) 2 + (0, 0, 2p − 12 ) prime p > 2; 0 ≤ q < p no conjugate subgroups [p] a = pa C1c1 (9) 2 prime p > 2 no conjugate subgroups

a, pb, c

0, u, 0

a − 2qc, b, pc

pa, b, c

I Minimal translationengleiche supergroups [2] C12/c1 (15); [2] Cmc21 (36); [2] Ccc2 (37); [2] Ama2 (40); [2] Aea2 (41); [2] Fdd2 (43); [2] Iba2 (45); [2] Ima2 (46); [3] P3c1 (158); [3] P31c (159); [3] R3c (161) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] F1m1 (8, C1m1)

• Decreased unit cell [2] c = 12 c C1m1 (8); [2] a = 12 a, b = 12 b P1c1 (7)

Copyright © 2011 International Union of Crystallography

135

Cc

No. 9

UNIQUE AXIS

Cs4

A11a

c, CELL CHOICE 1

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t(0, 12 , 12 ); (2) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

4

a

(0, 0, 0)+ (0, 12 , 12 )+

1

(1) x, y, z

I Maximal translationengleiche subgroups [2] A1 (1, P1) 1+

(2) x + 12 , y, z¯

a, 1/2(b − c), 1/2(b + c)

II Maximal klassengleiche subgroups

• Loss of centring translations [2] P11a (7) [2] P11n (7, P11a)

• Enlarged unit cell [3] c = 3c A11a (9) A11a (9) A11a (9) [3] a = 3a A11a (9) [3] a = 3a, b = −2a + b A11a (9) [3] a = 3a, b = −4a + b A11a (9) [3] b = 3b A11a (9)

1; 2 1; 2 + (0, 12 , 12 )

a − b, b, c

0, 0, 1/4

2 2 + (0, 0, 2) 2 + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

2 + (1, 0, 0)

3a, b, c

2 + (1, 0, 0)

3a, −2a + b, c

2 + (1, 0, 0)

3a, −4a + b, c

2

a, 3b, c

• Series of maximal isomorphic subgroups [p] c = pc A11a (9) 2 + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] a = pa, b = −2qa + b A11a (9) 2 + ( 2p − 12 , 0, 0) prime p > 2; 0 ≤ q < p no conjugate subgroups [p] b = pb A11a (9) 2 prime p > 2 no conjugate subgroups

a, b, pc

0, 0, u

pa, −2qa + b, c

a, pb, c

I Minimal translationengleiche supergroups [2] A112/a (15); [2] Cmc21 (36); [2] Ccc2 (37); [2] Ama2 (40); [2] Aea2 (41); [2] Fdd2 (43); [2] Iba2 (45); [2] Ima2 (46); [3] P3c1 (158); [3] P31c (159); [3] R3c (161) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] F11m (8, A11m)

• Decreased unit cell [2] a = 12 a A11m (8); [2] b = 12 b, c = 12 c P11a (7)

136

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 10, pp. 137–140.

C2h1

P 1 2/m 1

UNIQUE AXIS

No. 10

P 2/m

b

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

4

o

1

(1) x, y, z

(2) x, ¯ y, z¯

(3) x, ¯ y, ¯ z¯

(4) x, y, ¯z

I Maximal translationengleiche subgroups [2] P1m1 (6) 1; 4 [2] P121 (3) 1; 2 [2] P1¯ (2) 1; 3 II Maximal klassengleiche subgroups

• Enlarged unit cell

[2] b = 2b P121 /m1 (11) P121 /m1 (11) P12/m1 (10) P12/m1 (10) [2] c = 2c P12/c1 (13) P12/c1 (13) P12/m1 (10) P12/m1 (10) [2] a = 2a P12/a1 (13, P12/c1) P12/a1 (13, P12/c1) P12/m1 (10) P12/m1 (10) [2] a = 2a, c = 2c B12/e1 (13, P12/c1) B12/e1 (13, P12/c1) B12/m1 (10, P12/m1) B12/m1 (10, P12/m1) [2] a = 2a, b = 2b C12/m1 (12) C12/m1 (12) C12/m1 (12) C12/m1 (12) [2] b = 2b, c = 2c A12/m1 (12, C12/m1) A12/m1 (12, C12/m1) A12/m1 (12, C12/m1) A12/m1 (12, C12/m1) [2] a = 2a, b = 2b, c = 2c F12/m1 (12, C12/m1) F12/m1 (12, C12/m1) F12/m1 (12, C12/m1) F12/m1 (12, C12/m1) [3] b = 3b P12/m1 (10) P12/m1 (10) P12/m1 (10) [3] c = 3c P12/m1 (10) P12/m1 (10) P12/m1 (10) [3] a = a − c, c = 3c P12/m1 (10) P12/m1 (10) P12/m1 (10)

3; 2 + (0, 1, 0) (2; 3) + (0, 1, 0) 2; 3 2; 3 + (0, 1, 0)

a, 2b, c a, 2b, c a, 2b, c a, 2b, c

3; 2 + (0, 0, 1) 2; 3 + (0, 0, 1) 2; 3 (2; 3) + (0, 0, 1)

a, b, 2c a, b, 2c a, b, 2c a, b, 2c

3; 2 + (1, 0, 0) 2; 3 + (1, 0, 0) 2; 3 (2; 3) + (1, 0, 0)

−2a − c, b, 2a −2a − c, b, 2a 2a, b, c 2a, b, c

3; 2 + (0, 0, 1) 2; 3 + (0, 0, 1) 2; 3 (2; 3) + (0, 0, 1)

a − c, b, 2c a − c, b, 2c a − c, b, 2c a − c, b, 2c

0, 0, 1/2

2; 3 2; 3 + (0, 1, 0) (2; 3) + (1, 0, 0) 2 + (1, 0, 0); 3 + (1, 1, 0)

2a, 2b, c 2a, 2b, c 2a, 2b, c 2a, 2b, c

0, 1/2, 0 1/2, 0, 0 1/2, 1/2, 0

2; 3 (2; 3) + (0, 0, 1) 2; 3 + (0, 1, 0) 2 + (0, 0, 1); 3 + (0, 1, 1)

2c, 2b, −a − 2c 2c, 2b, −a − 2c 2c, 2b, −a − 2c 2c, 2b, −a − 2c

0, 0, 1/2 0, 1/2, 0 0, 1/2, 1/2

2; 3 2; 3 + (0, 1, 0) (2; 3) + (1, 0, 0) 2 + (1, 0, 0); 3 + (1, 1, 0)

2a, 2b, −a + c 2a, 2b, −a + c 2a, 2b, −a + c 2a, 2b, −a + c

0, 1/2, 0 1/2, 0, 0 1/2, 1/2, 0

2; 3 2; 3 + (0, 2, 0) 2; 3 + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2; 3 (2; 3) + (0, 0, 2) (2; 3) + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

2; 3 (2; 3) + (0, 0, 2) (2; 3) + (0, 0, 4)

a − c, b, 3c a − c, b, 3c a − c, b, 3c

0, 0, 1 0, 0, 2

Copyright © 2011 International Union of Crystallography

137

0, 1/2, 0 0, 1/2, 0 0, 0, 1/2 0, 0, 1/2 1/2, 0, 0 1/2, 0, 0 0, 0, 1/2

P 2/m

UNIQUE AXIS

[3] a = a − 2c, c = 3c P12/m1 (10) P12/m1 (10) P12/m1 (10) [3] a = 3a P12/m1 (10) P12/m1 (10) P12/m1 (10)

b

No. 10

CONTINUED

2; 3 (2; 3) + (0, 0, 2) (2; 3) + (0, 0, 4)

a − 2c, b, 3c a − 2c, b, 3c a − 2c, b, 3c

0, 0, 1 0, 0, 2

2; 3 (2; 3) + (2, 0, 0) (2; 3) + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

a, pb, c

0, u, 0

a − qc, b, pc

0, 0, u

pa, b, c

u, 0, 0

• Series of maximal isomorphic subgroups [p] b = pb P12/m1 (10) 2; 3 + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] a = a − qc, c = pc P12/m1 (10) (2; 3) + (0, 0, 2u) prime p > 2; 0 ≤ q < p; 0 ≤ u < p p conjugate subgroups for each pair of q and p [p] a = pa P12/m1 (10) (2; 3) + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

I Minimal translationengleiche supergroups [2] Pmmm (47); [2] Pccm (49); [2] Pmma (51); [2] Pmna (53); [2] Pbam (55); [2] Pnnm (58); [2] Cmmm (65); [2] Cccm (66); [2] P4/m (83); [2] P42 /m (84); [3] P6/m (175) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] C12/m1 (12); [2] A12/m1 (12, C12/m1); [2] I12/m1 (12, C12/m1)

• Decreased unit cell

none

138

C2h1

P 1 1 2/m

UNIQUE AXIS

No. 10

P 2/m

c

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

4

o

1

(1) x, y, z

(2) x, ¯ y, ¯z

(3) x, ¯ y, ¯ z¯

(4) x, y, z¯

I Maximal translationengleiche subgroups [2] P11m (6) 1; 4 [2] P112 (3) 1; 2 [2] P1¯ (2) 1; 3 II Maximal klassengleiche subgroups

• Enlarged unit cell

[2] c = 2c P1121 /m (11) P1121 /m (11) P112/m (10) P112/m (10) [2] a = 2a P112/a (13) P112/a (13) P112/m (10) P112/m (10) [2] b = 2b P112/b (13, P112/a) P112/b (13, P112/a) P112/m (10) P112/m (10) [2] a = 2a, b = 2b C112/e (13, P112/a) C112/e (13, P112/a) C112/m (10, P112/m) C112/m (10, P112/m) [2] b = 2b, c = 2c A112/m (12) A112/m (12) A112/m (12) A112/m (12) [2] a = 2a, c = 2c B112/m (12, A112/m) B112/m (12, A112/m) B112/m (12, A112/m) B112/m (12, A112/m) [2] a = 2a, b = 2b, c = 2c F112/m (12, A112/m) F112/m (12, A112/m) F112/m (12, A112/m) F112/m (12, A112/m) [3] c = 3c P112/m (10) P112/m (10) P112/m (10) [3] a = 3a P112/m (10) P112/m (10) P112/m (10) [3] a = 3a, b = −a + b P112/m (10) P112/m (10) P112/m (10)

3; 2 + (0, 0, 1) (2; 3) + (0, 0, 1) 2; 3 2; 3 + (0, 0, 1)

a, b, 2c a, b, 2c a, b, 2c a, b, 2c

3; 2 + (1, 0, 0) 2; 3 + (1, 0, 0) 2; 3 (2; 3) + (1, 0, 0)

2a, b, c 2a, b, c 2a, b, c 2a, b, c

3; 2 + (0, 1, 0) 2; 3 + (0, 1, 0) 2; 3 (2; 3) + (0, 1, 0)

2b, −a − 2b, c 2b, −a − 2b, c a, 2b, c a, 2b, c

3; 2 + (1, 0, 0) 2; 3 + (1, 0, 0) 2; 3 (2; 3) + (1, 0, 0)

2a, −a + b, c 2a, −a + b, c 2a, −a + b, c 2a, −a + b, c

1/2, 0, 0

2; 3 2; 3 + (0, 0, 1) (2; 3) + (0, 1, 0) 2 + (0, 1, 0); 3 + (0, 1, 1)

a, 2b, 2c a, 2b, 2c a, 2b, 2c a, 2b, 2c

0, 0, 1/2 0, 1/2, 0 0, 1/2, 1/2

2; 3 (2; 3) + (1, 0, 0) 2; 3 + (0, 0, 1) 2 + (1, 0, 0); 3 + (1, 0, 1)

−2a − b, 2a, 2c −2a − b, 2a, 2c −2a − b, 2a, 2c −2a − b, 2a, 2c

1/2, 0, 0 0, 0, 1/2 1/2, 0, 1/2

2; 3 2; 3 + (0, 0, 1) (2; 3) + (0, 1, 0) 2 + (0, 1, 0); 3 + (0, 1, 1)

a − b, 2b, 2c a − b, 2b, 2c a − b, 2b, 2c a − b, 2b, 2c

0, 0, 1/2 0, 1/2, 0 0, 1/2, 1/2

2; 3 2; 3 + (0, 0, 2) 2; 3 + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

2; 3 (2; 3) + (2, 0, 0) (2; 3) + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 3 (2; 3) + (2, 0, 0) (2; 3) + (4, 0, 0)

3a, −a + b, c 3a, −a + b, c 3a, −a + b, c

1, 0, 0 2, 0, 0

139

0, 0, 1/2 0, 0, 1/2 1/2, 0, 0 1/2, 0, 0 0, 1/2, 0 0, 1/2, 0 1/2, 0, 0

P 2/m

UNIQUE AXIS

[3] a = 3a, b = −2a + b P112/m (10) P112/m (10) P112/m (10) [3] b = 3b P112/m (10) P112/m (10) P112/m (10)

c

No. 10

CONTINUED

2; 3 (2; 3) + (2, 0, 0) (2; 3) + (4, 0, 0)

3a, −2a + b, c 3a, −2a + b, c 3a, −2a + b, c

1, 0, 0 2, 0, 0

2; 3 (2; 3) + (0, 2, 0) (2; 3) + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

a, b, pc

0, 0, u

pa, −qa + b, c

u, 0, 0

a, pb, c

0, u, 0

• Series of maximal isomorphic subgroups [p] c = pc P112/m (10) 2; 3 + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] a = pa, b = −qa + b P112/m (10) (2; 3) + (2u, 0, 0) prime p > 2; 0 ≤ q < p; 0 ≤ u < p p conjugate subgroups for each pair of q and p [p] b = pb P112/m (10) (2; 3) + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

I Minimal translationengleiche supergroups [2] Pmmm (47); [2] Pccm (49); [2] Pmma (51); [2] Pmna (53); [2] Pbam (55); [2] Pnnm (58); [2] Cmmm (65); [2] Cccm (66); [2] P4/m (83); [2] P42 /m (84); [3] P6/m (175) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] A112/m (12); [2] B112/m (12, A112/m); [2] I112/m (12, A112/m)

• Decreased unit cell

none

140

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 11, pp. 141–143.

C2h2

P 1 21/m 1

UNIQUE AXIS

No. 11

P 21/m

b

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

4

f

1

(1) x, y, z

(2) x, ¯ y + 12 , z¯

(3) x, ¯ y, ¯ z¯

(4) x, y¯ + 12 , z

I Maximal translationengleiche subgroups [2] P1m1 (6) 1; 4 [2] P121 1 (4) 1; 2 [2] P1¯ (2) 1; 3

0, 1/4, 0

II Maximal klassengleiche subgroups

• Enlarged unit cell [2] c = 2c P121 /c1 (14) P121 /c1 (14) P121 /m1 (11) P121 /m1 (11) [2] a = 2a P121 /a1 (14, P121 /c1) P121 /a1 (14, P121 /c1) P121 /m1 (11) P121 /m1 (11) [2] a = 2a, c = 2c B121 /e1 (14, P121 /c1) B121 /e1 (14, P121 /c1) B121 /m1 (11, P121 /m1) B121 /m1 (11, P121 /m1) [3] b = 3b P121 /m1 (11) P121 /m1 (11) P121 /m1 (11) [3] c = 3c P121 /m1 (11) P121 /m1 (11) P121 /m1 (11) [3] a = a − c, c = 3c P121 /m1 (11) P121 /m1 (11) P121 /m1 (11) [3] a = a − 2c, c = 3c P121 /m1 (11) P121 /m1 (11) P121 /m1 (11) [3] a = 3a P121 /m1 (11) P121 /m1 (11) P121 /m1 (11)

3; 2 + (0, 0, 1) 2; 3 + (0, 0, 1) 2; 3 (2; 3) + (0, 0, 1)

a, b, 2c a, b, 2c a, b, 2c a, b, 2c

3; 2 + (1, 0, 0) 2; 3 + (1, 0, 0) 2; 3 (2; 3) + (1, 0, 0)

−2a − c, b, 2a −2a − c, b, 2a 2a, b, c 2a, b, c

1/2, 0, 0

3; 2 + (0, 0, 1) 2; 3 + (0, 0, 1) 2; 3 (2; 3) + (0, 0, 1)

a − c, b, 2c a − c, b, 2c a − c, b, 2c a − c, b, 2c

0, 0, 1/2

3; 2 + (0, 1, 0) 2 + (0, 1, 0); 3 + (0, 2, 0) 2 + (0, 1, 0); 3 + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2; 3 (2; 3) + (0, 0, 2) (2; 3) + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

2; 3 (2; 3) + (0, 0, 2) (2; 3) + (0, 0, 4)

a − c, b, 3c a − c, b, 3c a − c, b, 3c

0, 0, 1 0, 0, 2

2; 3 (2; 3) + (0, 0, 2) (2; 3) + (0, 0, 4)

a − 2c, b, 3c a − 2c, b, 3c a − 2c, b, 3c

0, 0, 1 0, 0, 2

2; 3 (2; 3) + (2, 0, 0) (2; 3) + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

a, pb, c

0, u, 0

a − qc, b, pc

0, 0, u

pa, b, c

u, 0, 0

• Series of maximal isomorphic subgroups [p] b = pb P121 /m1 (11) 2 + (0, 2p − 12 , 0); 3 + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] a = a − qc, c = pc P121 /m1 (11) (2; 3) + (0, 0, 2u) prime p > 2; 0 ≤ q < p; 0 ≤ u < p p conjugate subgroups for each pair of q and p [p] a = pa P121 /m1 (11) (2; 3) + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups Copyright © 2011 International Union of Crystallography

141

0, 0, 1/2 0, 0, 1/2

1/2, 0, 0

0, 0, 1/2

P 21/m

UNIQUE AXIS

b

No. 11

CONTINUED

I Minimal translationengleiche supergroups [2] Pmma (51); [2] Pbcm (57); [2] Pmmn (59); [2] Pnma (62); [2] Cmcm (63); [3] P63 /m (176) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] C12/m1 (12); [2] A12/m1 (12, C12/m1); [2] I12/m1 (12, C12/m1) • Decreased unit cell

[2] b = 12 b P12/m1 (10)

P 21/m

UNIQUE AXIS

c

No. 11

I Minimal translationengleiche supergroups [2] Pmma (51); [2] Pbcm (57); [2] Pmmn (59); [2] Pnma (62); [2] Cmcm (63); [3] P63 /m (176) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] A112/m (12); [2] B112/m (12, A112/m); [2] I112/m (12, A112/m)

• Decreased unit cell [2] c = 12 c P112/m (10)

142

(Continued from the facing page)

C2h2

P 1 1 21/m

UNIQUE AXIS

No. 11

P 21/m

c

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

4

f

1

(1) x, y, z

(2) x, ¯ y, ¯ z + 12

(3) x, ¯ y, ¯ z¯

(4) x, y, z¯ + 12

I Maximal translationengleiche subgroups [2] P11m (6) 1; 4 1; 2 [2] P1121 (4) [2] P1¯ (2) 1; 3

0, 0, 1/4

II Maximal klassengleiche subgroups

• Enlarged unit cell [2] a = 2a P1121 /a (14) P1121 /a (14) P1121 /m (11) P1121 /m (11) [2] b = 2b P1121 /b (14, P1121 /a) P1121 /b (14, P1121 /a) P1121 /m (11) P1121 /m (11) [2] a = 2a, b = 2b C1121 /e (14, P1121 /a) C1121 /e (14, P1121 /a) C1121 /m (11, P1121 /m) C1121 /m (11, P1121 /m) [3] c = 3c P1121 /m (11) P1121 /m (11) P1121 /m (11) [3] a = 3a P1121 /m (11) P1121 /m (11) P1121 /m (11) [3] a = 3a, b = −a + b P1121 /m (11) P1121 /m (11) P1121 /m (11) [3] a = 3a, b = −2a + b P1121 /m (11) P1121 /m (11) P1121 /m (11) [3] b = 3b P1121 /m (11) P1121 /m (11) P1121 /m (11)

3; 2 + (1, 0, 0) 2; 3 + (1, 0, 0) 2; 3 (2; 3) + (1, 0, 0)

2a, b, c 2a, b, c 2a, b, c 2a, b, c

3; 2 + (0, 1, 0) 2; 3 + (0, 1, 0) 2; 3 (2; 3) + (0, 1, 0)

2b, −a − 2b, c 2b, −a − 2b, c a, 2b, c a, 2b, c

0, 1/2, 0

3; 2 + (1, 0, 0) 2; 3 + (1, 0, 0) 2; 3 (2; 3) + (1, 0, 0)

2a, −a + b, c 2a, −a + b, c 2a, −a + b, c 2a, −a + b, c

1/2, 0, 0 1/2, 0, 0

3; 2 + (0, 0, 1) 2 + (0, 0, 1); 3 + (0, 0, 2) 2 + (0, 0, 1); 3 + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

2; 3 (2; 3) + (2, 0, 0) (2; 3) + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 3 (2; 3) + (2, 0, 0) (2; 3) + (4, 0, 0)

3a, −a + b, c 3a, −a + b, c 3a, −a + b, c

1, 0, 0 2, 0, 0

2; 3 (2; 3) + (2, 0, 0) (2; 3) + (4, 0, 0)

3a, −2a + b, c 3a, −2a + b, c 3a, −2a + b, c

1, 0, 0 2, 0, 0

2; 3 (2; 3) + (0, 2, 0) (2; 3) + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

a, b, pc

0, 0, u

pa, −qa + b, c

u, 0, 0

a, pb, c

0, u, 0

• Series of maximal isomorphic subgroups [p] c = pc P1121 /m (11) 2 + (0, 0, 2p − 12 ); 3 + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] a = pa, b = −qa + b P1121 /m (11) (2; 3) + (2u, 0, 0) prime p > 2; 0 ≤ q < p; 0 ≤ u < p p conjugate subgroups for each pair of q and p [p] b = pb P1121 /m (11) (2; 3) + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups (Continued on the facing page) 143

1/2, 0, 0 1/2, 0, 0

0, 1/2, 0

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 12, pp. 144–146.

C 2/m UNIQUE AXIS

No. 12

C2h3

C 1 2/m 1

b, CELL CHOICE 1

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t( 21 , 12 , 0); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

j

(0, 0, 0)+ ( 21 , 12 , 0)+

1

(1) x, y, z

(2) x, ¯ y, z¯

(3) x, ¯ y, ¯ z¯

I Maximal translationengleiche subgroups [2] C1m1 (8) (1; 4)+ [2] C121 (5) (1; 2)+ ¯ [2] C1¯ (2, P1) (1; 3)+

(4) x, y, ¯z

1/2(a − b), 1/2(a + b), c

II Maximal klassengleiche subgroups

• Loss of centring translations [2] [2] [2] [2]

P121 /a1 (14, P121 /c1) P12/a1 (13, P12/c1) P121 /m1 (11) P12/m1 (10)

• Enlarged unit cell [2] c = 2c C12/c1 (15) C12/c1 (15) I12/c1 (15, C12/c1) I12/c1 (15, C12/c1) C12/m1 (12) C12/m1 (12) I12/m1 (12, C12/m1) I12/m1 (12, C12/m1) [3] b = 3b C12/m1 (12) C12/m1 (12) C12/m1 (12) [3] c = 3c C12/m1 (12) C12/m1 (12) C12/m1 (12) [3] a = a − 2c, c = 3c C12/m1 (12) C12/m1 (12) C12/m1 (12) [3] a = a − 4c, c = 3c C12/m1 (12) C12/m1 (12) C12/m1 (12) [3] a = 3a C12/m1 (12) C12/m1 (12) C12/m1 (12)

1; 1; 1; 1;

3; 2; 4; 2;

(2; 4) + ( 21 , 12 , 0) (3; 4) + ( 21 , 12 , 0) (2; 3) + ( 21 , 12 , 0) 3; 4

c, b, −a c, b, −a c, b, −a

1/4, 1/4, 0 1/4, 1/4, 0

3; 2 + (0, 0, 1) 2; 3 + (0, 0, 1) 3; 2 + (0, 0, 1) 2; 3 + (0, 0, 1) 2; 3 (2; 3) + (0, 0, 1) 2; 3 (2; 3) + (0, 0, 1)

a, b, 2c a, b, 2c a − 2c, b, 2c a − 2c, b, 2c a, b, 2c a, b, 2c a − 2c, b, 2c a − 2c, b, 2c

0, 0, 1/2

2; 3 2; 3 + (0, 2, 0) 2; 3 + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2; 3 (2; 3) + (0, 0, 2) (2; 3) + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

2; 3 (2; 3) + (0, 0, 2) (2; 3) + (0, 0, 4)

a − 2c, b, 3c a − 2c, b, 3c a − 2c, b, 3c

0, 0, 1 0, 0, 2

2; 3 (2; 3) + (0, 0, 2) (2; 3) + (0, 0, 4)

a − 4c, b, 3c a − 4c, b, 3c a − 4c, b, 3c

0, 0, 1 0, 0, 2

2; 3 (2; 3) + (2, 0, 0) (2; 3) + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

a, pb, c

0, u, 0

a − 2qc, b, pc

0, 0, u

pa, b, c

u, 0, 0

• Series of maximal isomorphic subgroups [p] b = pb C12/m1 (12) 2; 3 + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] a = a − 2qc, c = pc C12/m1 (12) (2; 3) + (0, 0, 2u) prime p > 2; 0 ≤ q < p; 0 ≤ u < p p conjugate subgroups for each pair of q and p [p] a = pa C12/m1 (12) (2; 3) + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups Copyright © 2011 International Union of Crystallography

144

0, 0, 1/2 0, 0, 1/2 0, 0, 1/2

CONTINUED

No. 12

UNIQUE AXIS

b

C 2/m

I Minimal translationengleiche supergroups [2] Cmcm (63); [2] Cmce (64); [2] Cmmm (65); [2] Cmme (67); [2] Fmmm (69); [2] Immm (71); [2] Ibam (72); [2] Imma (74); ¯ ¯ ¯ ¯ ¯ ¯ [2] I4/m (87); [3] P312/m (162, P31m); [3] P32/m1 (164, P3m1); [3] R32/m (166, R3m) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations

none

• Decreased unit cell

[2] a = 12 a, b = 12 b P12/m1 (10)

(Continued from the following page)

No. 12

UNIQUE AXIS

c

C 2/m

I Minimal translationengleiche supergroups [2] Cmcm (63); [2] Cmce (64); [2] Cmmm (65); [2] Cmme (67); [2] Fmmm (69); [2] Immm (71); [2] Ibam (72); [2] Imma (74); ¯ ¯ ¯ ¯ ¯ ¯ [2] I4/m (87); [3] P312/m (162, P31m); [3] P32/m1 (164, P3m1); [3] R32/m (166, R3m) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations

none

• Decreased unit cell

[2] b = 12 b, c = 12 c P112/m (10)

145

C 2/m UNIQUE AXIS

No. 12

C2h3

A 1 1 2/m

c, CELL CHOICE 1

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t(0, 12 , 12 ); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

j

(0, 0, 0)+ (0, 12 , 12 )+

1

(1) x, y, z

(2) x, ¯ y, ¯z

I Maximal translationengleiche subgroups [2] A11m (8) (1; 4)+ [2] A112 (5) (1; 2)+ ¯ [2] A1¯ (2, P1) (1; 3)+

(3) x, ¯ y, ¯ z¯

(4) x, y, z¯

a, 1/2(b − c), 1/2(b + c)

II Maximal klassengleiche subgroups

• Loss of centring translations [2] [2] [2] [2]

P1121 /b (14, P1121 /a) P112/b (13, P112/a) P1121 /m (11) P112/m (10)

• Enlarged unit cell [2] a = 2a A112/a (15) A112/a (15) I112/a (15, A112/a) I112/a (15, A112/a) A112/m (12) A112/m (12) I112/m (12, A112/m) I112/m (12, A112/m) [3] c = 3c A112/m (12) A112/m (12) A112/m (12) [3] a = 3a A112/m (12) A112/m (12) A112/m (12) [3] a = 3a, b = −2a + b A112/m (12) A112/m (12) A112/m (12) [3] a = 3a, b = −4a + b A112/m (12) A112/m (12) A112/m (12) [3] b = 3b A112/m (12) A112/m (12) A112/m (12)

1; 1; 1; 1;

3; 2; 4; 2;

(2; 4) + (0, 12 , 12 ) (3; 4) + (0, 12 , 12 ) (2; 3) + (0, 12 , 12 ) 3; 4

−b, a, c −b, a, c −b, a, c

0, 1/4, 1/4 0, 1/4, 1/4

3; 2 + (1, 0, 0) 2; 3 + (1, 0, 0) 3; 2 + (1, 0, 0) 2; 3 + (1, 0, 0) 2; 3 (2; 3) + (1, 0, 0) 2; 3 (2; 3) + (1, 0, 0)

2a, b, c 2a, b, c 2a, −2a + b, c 2a, −2a + b, c 2a, b, c 2a, b, c 2a, −2a + b, c 2a, −2a + b, c

2; 3 2; 3 + (0, 0, 2) 2; 3 + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

2; 3 (2; 3) + (2, 0, 0) (2; 3) + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 3 (2; 3) + (2, 0, 0) (2; 3) + (4, 0, 0)

3a, −2a + b, c 3a, −2a + b, c 3a, −2a + b, c

1, 0, 0 2, 0, 0

2; 3 (2; 3) + (2, 0, 0) (2; 3) + (4, 0, 0)

3a, −4a + b, c 3a, −4a + b, c 3a, −4a + b, c

1, 0, 0 2, 0, 0

2; 3 (2; 3) + (0, 2, 0) (2; 3) + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

a, b, pc

0, 0, u

pa, −2qa + b, c

u, 0, 0

a, pb, c

0, u, 0

• Series of maximal isomorphic subgroups [p] c = pc A112/m (12) 2; 3 + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] a = pa, b = −2qa + b A112/m (12) (2; 3) + (2u, 0, 0) prime p > 2; 0 ≤ q < p; 0 ≤ u < p p conjugate subgroups for each pair of q and p [p] b = pb A112/m (12) (2; 3) + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups (Continued on the preceding page) 146

1/2, 0, 0 1/2, 0, 0 1/2, 0, 0 1/2, 0, 0

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 13, pp. 147–149.

C2h4

P 1 2/c 1

UNIQUE AXIS

No. 13

P 2/c

b, CELL CHOICE 1

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

4

g

1

(1) x, y, z

(2) x, ¯ y, z¯ + 12

(3) x, ¯ y, ¯ z¯

(4) x, y, ¯ z + 12

I Maximal translationengleiche subgroups [2] P1c1 (7) 1; 4 [2] P121 (3) 1; 2 [2] P1¯ (2) 1; 3

0, 0, 1/4

II Maximal klassengleiche subgroups

• Enlarged unit cell [2] b = 2b P121 /c1 (14) P121 /c1 (14) P12/c1 (13) P12/c1 (13) [2] a = 2a P12/c1 (13) P12/c1 (13) P12/n1 (13, P12/c1) P12/n1 (13, P12/c1) [2] a = 2a, b = 2b C12/c1 (15) C12/c1 (15) C12/c1 (15) C12/c1 (15) [3] b = 3b P12/c1 (13) P12/c1 (13) P12/c1 (13) [3] c = 3c P12/c1 (13) P12/c1 (13) P12/c1 (13) [3] a = 3a P12/c1 (13) P12/c1 (13) P12/c1 (13) [3] a = 3a, c = −2a + c P12/c1 (13) P12/c1 (13) P12/c1 (13) [3] a = 3a, c = −4a + c P12/c1 (13) P12/c1 (13) P12/c1 (13)

3; 2 + (0, 1, 0) (2; 3) + (0, 1, 0) 2; 3 2; 3 + (0, 1, 0)

a, 2b, c a, 2b, c a, 2b, c a, 2b, c

2; 3 (2; 3) + (1, 0, 0) 3; 2 + (1, 0, 0) 2 + (2, 0, 0); 3 + (1, 0, 0)

2a, b, c 2a, b, c 2a, b, −2a + c 2a, b, −2a + c

1/2, 0, 0

2; 3 2; 3 + (0, 1, 0) (2; 3) + (1, 0, 0) 2 + (1, 0, 0); 3 + (1, 1, 0)

2a, 2b, c 2a, 2b, c 2a, 2b, c 2a, 2b, c

0, 1/2, 0 1/2, 0, 0 1/2, 1/2, 0

2; 3 2; 3 + (0, 2, 0) 2; 3 + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

3; 2 + (0, 0, 1) 2 + (0, 0, 3); 3 + (0, 0, 2) 2 + (0, 0, 5); 3 + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

2; 3 (2; 3) + (2, 0, 0) (2; 3) + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

3; 2 + (−1, 0, 0) 2 + (1, 0, 0); 3 + (2, 0, 0) 2 + (3, 0, 0); 3 + (4, 0, 0)

3a, b, −2a + c 3a, b, −2a + c 3a, b, −2a + c

1, 0, 0 2, 0, 0

3; 2 + (−2, 0, 0) 2 + (0, 0, 0); 3 + (2, 0, 0) 2 + (2, 0, 0); 3 + (4, 0, 0)

3a, b, −4a + c 3a, b, −4a + c 3a, b, −4a + c

1, 0, 0 2, 0, 0

a, pb, c

0, u, 0

a, b, pc

0, 0, u

pa, b, −2qa + c

u, 0, 0

• Series of maximal isomorphic subgroups [p] b = pb P12/c1 (13) 2; 3 + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] c = pc P12/c1 (13) 2 + (0, 0, 2p − 12 + 2u); 3 + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] a = pa, c = −2qa + c P12/c1 (13) 2 + (−q + 2u, 0, 0); 3 + (2u, 0, 0) prime p > 2; 0 ≤ q < p; 0 ≤ u < p p conjugate subgroups for each pair of q and p Copyright © 2011 International Union of Crystallography

147

0, 1/2, 0 0, 1/2, 0 1/2, 0, 0

P 2/c

UNIQUE AXIS

b

No. 13

CONTINUED

I Minimal translationengleiche supergroups [2] Pnnn (48); [2] Pccm (49); [2] Pban (50); [2] Pmma (51); [2] Pnna (52); [2] Pmna (53); [2] Pcca (54); [2] Pccn (56); [2] Pbcm (57); [2] Pmmn (59); [2] Pbcn (60); [2] Cmme (67); [2] Ccce (68); [2] P4/n (85); [2] P42 /n (86) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] A12/m1 (12, C12/m1); [2] C12/c1 (15); [2] I12/c1 (15, C12/c1)

• Decreased unit cell

[2] c = 12 c P12/m1 (10)

P 2/c

UNIQUE AXIS

c

No. 13

(Continued from the facing page)

I Minimal translationengleiche supergroups [2] Pnnn (48); [2] Pccm (49); [2] Pban (50); [2] Pmma (51); [2] Pnna (52); [2] Pmna (53); [2] Pcca (54); [2] Pccn (56); [2] Pbcm (57); [2] Pmmn (59); [2] Pbcn (60); [2] Cmme (67); [2] Ccce (68); [2] P4/n (85); [2] P42 /n (86) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] A112/a (15); [2] B112/m (12, A112/m); [2] I112/a (15, A112/a) • Decreased unit cell [2] a = 12 a P112/m (10)

148

C2h4

P 1 1 2/a

UNIQUE AXIS

No. 13

P 2/c

c, CELL CHOICE 1

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

4

g

1

(1) x, y, z

(2) x¯ + 12 , y, ¯z

(3) x, ¯ y, ¯ z¯

(4) x + 12 , y, z¯

I Maximal translationengleiche subgroups [2] P11a (7) 1; 4 [2] P112 (3) 1; 2 [2] P1¯ (2) 1; 3

1/4, 0, 0

II Maximal klassengleiche subgroups

• Enlarged unit cell [2] c = 2c P1121 /a (14) P1121 /a (14) P112/a (13) P112/a (13) [2] b = 2b P112/a (13) P112/a (13) P112/n (13, P112/a) P112/n (13, P112/a) [2] b = 2b, c = 2c A112/a (15) A112/a (15) A112/a (15) A112/a (15) [3] c = 3c P112/a (13) P112/a (13) P112/a (13) [3] a = 3a P112/a (13) P112/a (13) P112/a (13) [3] b = 3b P112/a (13) P112/a (13) P112/a (13) [3] a = a − 2b, b = 3b P112/a (13) P112/a (13) P112/a (13) [3] a = a − 4b, b = 3b P112/a (13) P112/a (13) P112/a (13)

3; 2 + (0, 0, 1) (2; 3) + (0, 0, 1) 2; 3 2; 3 + (0, 0, 1)

a, b, 2c a, b, 2c a, b, 2c a, b, 2c

2; 3 (2; 3) + (0, 1, 0) 3; 2 + (0, 1, 0) 2 + (0, 2, 0); 3 + (0, 1, 0)

a, 2b, c a, 2b, c a − 2b, 2b, c a − 2b, 2b, c

0, 1/2, 0

2; 3 2; 3 + (0, 0, 1) (2; 3) + (0, 1, 0) 2 + (0, 1, 0); 3 + (0, 1, 1)

a, 2b, 2c a, 2b, 2c a, 2b, 2c a, 2b, 2c

0, 0, 1/2 0, 1/2, 0 0, 1/2, 1/2

2; 3 2; 3 + (0, 0, 2) 2; 3 + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

3; 2 + (1, 0, 0) 2 + (3, 0, 0); 3 + (2, 0, 0) 2 + (5, 0, 0); 3 + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 3 (2; 3) + (0, 2, 0) (2; 3) + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

3; 2 + (0, −1, 0) 2 + (0, 1, 0); 3 + (0, 2, 0) 2 + (0, 3, 0); 3 + (0, 4, 0)

a − 2b, 3b, c a − 2b, 3b, c a − 2b, 3b, c

0, 1, 0 0, 2, 0

3; 2 + (0, −2, 0) 2 + (0, 0, 0); 3 + (0, 2, 0) 2 + (0, 2, 0); 3 + (0, 4, 0)

a − 4b, 3b, c a − 4b, 3b, c a − 4b, 3b, c

0, 1, 0 0, 2, 0

a, b, pc

0, 0, u

pa, b, c

u, 0, 0

a − 2qb, pb, c

0, u, 0

• Series of maximal isomorphic subgroups [p] c = pc P112/a (13) 2; 3 + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] a = pa P112/a (13) 2 + ( 2p − 12 + 2u, 0, 0); 3 + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] a = a − 2qb, b = pb P112/a (13) 2 + (0, −q + 2u, 0); 3 + (0, 2u, 0) prime p > 2; 0 ≤ q < p; 0 ≤ u < p p conjugate subgroups for each pair of q and p (Continued on the facing page) 149

0, 0, 1/2 0, 0, 1/2 0, 1/2, 0

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 14, pp. 150–152.

P 21/c UNIQUE AXIS

C2h5

P 1 21/c 1

No. 14 b, CELL CHOICE 1

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

4

e

1

(1) x, y, z

(2) x, ¯ y + 12 , z¯ + 12

(3) x, ¯ y, ¯ z¯

(4) x, y¯ + 12 , z + 12

I Maximal translationengleiche subgroups [2] P1c1 (7) 1; 4 [2] P121 1 (4) 1; 2 [2] P1¯ (2) 1; 3

0, 1/4, 0 0, 0, 1/4

II Maximal klassengleiche subgroups

• Enlarged unit cell

[2] a = 2a P121 /c1 (14) P121 /c1 (14) P121 /n1 (14, P121 /c1) P121 /n1 (14, P121 /c1) [3] b = 3b P121 /c1 (14) P121 /c1 (14) P121 /c1 (14) [3] c = 3c P121 /c1 (14) P121 /c1 (14) P121 /c1 (14) [3] a = 3a P121 /c1 (14) P121 /c1 (14) P121 /c1 (14) [3] a = 3a, c = −2a + c P121 /c1 (14) P121 /c1 (14) P121 /c1 (14) [3] a = 3a, c = −4a + c P121 /c1 (14) P121 /c1 (14) P121 /c1 (14)

2; 3 (2; 3) + (1, 0, 0) 3; 2 + (1, 0, 0) 2 + (2, 0, 0); 3 + (1, 0, 0)

2a, b, c 2a, b, c 2a, b, −2a + c 2a, b, −2a + c

3; 2 + (0, 1, 0) 2 + (0, 1, 0); 3 + (0, 2, 0) 2 + (0, 1, 0); 3 + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

3; 2 + (0, 0, 1) 2 + (0, 0, 3); 3 + (0, 0, 2) 2 + (0, 0, 5); 3 + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

2; 3 (2; 3) + (2, 0, 0) (2; 3) + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

3; 2 + (−1, 0, 0) 2 + (1, 0, 0); 3 + (2, 0, 0) 2 + (3, 0, 0); 3 + (4, 0, 0)

3a, b, −2a + c 3a, b, −2a + c 3a, b, −2a + c

1, 0, 0 2, 0, 0

3; 2 + (−2, 0, 0) 2 + (0, 0, 0); 3 + (2, 0, 0) 2 + (2, 0, 0); 3 + (4, 0, 0)

3a, b, −4a + c 3a, b, −4a + c 3a, b, −4a + c

1, 0, 0 2, 0, 0

a, pb, c

0, u, 0

a, b, pc

0, 0, u

pa, b, −2qa + c

u, 0, 0

• Series of maximal isomorphic subgroups [p] b = pb 2 + (0, 2p − 12 , 0); 3 + (0, 2u, 0) P121 /c1 (14) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] c = pc P121 /c1 (14) 2 + (0, 0, 2p − 12 + 2u); 3 + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] a = pa, c = −2qa + c 2 + (−q + 2u, 0, 0); 3 + (2u, 0, 0) P121 /c1 (14) prime p > 2; 0 ≤ q < p; 0 ≤ u < p p conjugate subgroups for each pair of q and p

Copyright © 2011 International Union of Crystallography

150

1/2, 0, 0 1/2, 0, 0

CONTINUED

No. 14

UNIQUE AXIS

b

P 21/c

I Minimal translationengleiche supergroups [2] Pnna (52); [2] Pmna (53); [2] Pcca (54); [2] Pbam (55); [2] Pccn (56); [2] Pbcm (57); [2] Pnnm (58); [2] Pbcn (60); [2] Pbca (61); [2] Pnma (62); [2] Cmce (64) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] A12/m1 (12, C12/m1); [2] C12/c1 (15); [2] I12/c1 (15, C12/c1)

• Decreased unit cell

[2] c = 12 c P121 /m1 (11); [2] b = 12 b P12/c1 (13)

(Continued from the following page)

No. 14

UNIQUE AXIS

c

P 21/c

I Minimal translationengleiche supergroups [2] Pnna (52); [2] Pmna (53); [2] Pcca (54); [2] Pbam (55); [2] Pccn (56); [2] Pbcm (57); [2] Pnnm (58); [2] Pbcn (60); [2] Pbca (61); [2] Pnma (62); [2] Cmce (64) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] A112/a (15); [2] B112/m (12, A112/m); [2] I112/a (15, A112/a) • Decreased unit cell [2] a = 12 a P1121 /m (11); [2] c = 12 c P112/a (13)

151

P 21/c UNIQUE AXIS

C2h5

P 1 1 21/a

No. 14 c, CELL CHOICE 1

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

4

e

1

(1) x, y, z

(2) x¯ + 12 , y, ¯ z + 12

(3) x, ¯ y, ¯ z¯

(4) x + 12 , y, z¯ + 12

I Maximal translationengleiche subgroups [2] P11a (7) 1; 4 [2] P1121 (4) 1; 2 ¯ [2] P1 (2) 1; 3

0, 0, 1/4 1/4, 0, 0

II Maximal klassengleiche subgroups

• Enlarged unit cell [2] b = 2b P1121 /a (14) P1121 /a (14) P1121 /n (14, P1121 /a) P1121 /n (14, P1121 /a) [3] c = 3c P1121 /a (14) P1121 /a (14) P1121 /a (14) [3] a = 3a P1121 /a (14) P1121 /a (14) P1121 /a (14) [3] b = 3b P1121 /a (14) P1121 /a (14) P1121 /a (14) [3] a = a − 2b, b = 3b P1121 /a (14) P1121 /a (14) P1121 /a (14) [3] a = a − 4b, b = 3b P1121 /a (14) P1121 /a (14) P1121 /a (14)

2; 3 (2; 3) + (0, 1, 0) 3; 2 + (0, 1, 0) 2 + (0, 2, 0); 3 + (0, 1, 0)

a, 2b, c a, 2b, c a − 2b, 2b, c a − 2b, 2b, c

3; 2 + (0, 0, 1) 2 + (0, 0, 1); 3 + (0, 0, 2) 2 + (0, 0, 1); 3 + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

3; 2 + (1, 0, 0) 2 + (3, 0, 0); 3 + (2, 0, 0) 2 + (5, 0, 0); 3 + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 3 (2; 3) + (0, 2, 0) (2; 3) + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

3; 2 + (0, −1, 0) 2 + (0, 1, 0); 3 + (0, 2, 0) 2 + (0, 3, 0); 3 + (0, 4, 0)

a − 2b, 3b, c a − 2b, 3b, c a − 2b, 3b, c

0, 1, 0 0, 2, 0

3; 2 + (0, −2, 0) 2 + (0, 0, 0); 3 + (0, 2, 0) 2 + (0, 2, 0); 3 + (0, 4, 0)

a − 4b, 3b, c a − 4b, 3b, c a − 4b, 3b, c

0, 1, 0 0, 2, 0

a, b, pc

0, 0, u

pa, b, c

u, 0, 0

a − 2qb, pb, c

0, u, 0

• Series of maximal isomorphic subgroups [p] c = pc P1121 /a (14) 2 + (0, 0, 2p − 12 ); 3 + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] a = pa P1121 /a (14) 2 + ( 2p − 12 + 2u, 0, 0); 3 + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] a = a − 2qb, b = pb 2 + (0, −q + 2u, 0); 3 + (0, 2u, 0) P1121 /a (14) prime p > 2; 0 ≤ q < p; 0 ≤ u < p p conjugate subgroups for each pair of q and p (Continued on the preceding page)

152

0, 1/2, 0 0, 1/2, 0

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 15, pp. 153–155.

C2h6

C 1 2/c 1

UNIQUE AXIS

No. 15

C 2/c

b, CELL CHOICE 1

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t( 21 , 12 , 0); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

f

(0, 0, 0)+ ( 21 , 12 , 0)+

1

(1) x, y, z

(2) x, ¯ y, z¯ + 12

(3) x, ¯ y, ¯ z¯

I Maximal translationengleiche subgroups [2] C1c1 (9) (1; 4)+ [2] C121 (5) (1; 2)+ ¯ ¯ [2] C1 (2, P1) (1; 3)+

(4) x, y, ¯ z + 12

0, 0, 1/4 1/2(a − b), 1/2(a + b), c

II Maximal klassengleiche subgroups

• Loss of centring translations [2] [2] [2] [2]

P121 /n1 (14, P121 /c1) P121 /c1 (14) P12/c1 (13) P12/n1 (13, P12/c1)

1; 1; 1; 1;

3; 4; 2; 2;

(2; 4) + ( 21 , 12 , 0) (2; 3) + ( 21 , 12 , 0) 3; 4 (3; 4) + ( 21 , 12 , 0)

c, b, −a − c 1/4, 1/4, 0 c, b, −a − c

1/4, 1/4, 0

2; 3 2; 3 + (0, 2, 0) 2; 3 + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

3; 2 + (0, 0, 1) 2 + (0, 0, 3); 3 + (0, 0, 2) 2 + (0, 0, 5); 3 + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

3; 2 + (0, 0, 1) 2 + (0, 0, 3); 3 + (0, 0, 2) 2 + (0, 0, 5); 3 + (0, 0, 4)

a − 2c, b, 3c a − 2c, b, 3c a − 2c, b, 3c

0, 0, 1 0, 0, 2

3; 2 + (0, 0, 1) 2 + (0, 0, 3); 3 + (0, 0, 2) 2 + (0, 0, 5); 3 + (0, 0, 4)

a − 4c, b, 3c a − 4c, b, 3c a − 4c, b, 3c

0, 0, 1 0, 0, 2

2; 3 (2; 3) + (2, 0, 0) (2; 3) + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

a, pb, c

0, u, 0

a − 2qc, b, pc

0, 0, u

pa, b, c

u, 0, 0

• Enlarged unit cell

[3] b = 3b C12/c1 (15) C12/c1 (15) C12/c1 (15) [3] c = 3c C12/c1 (15) C12/c1 (15) C12/c1 (15) [3] a = a − 2c, c = 3c C12/c1 (15) C12/c1 (15) C12/c1 (15) [3] a = a − 4c, c = 3c C12/c1 (15) C12/c1 (15) C12/c1 (15) [3] a = 3a C12/c1 (15) C12/c1 (15) C12/c1 (15)

• Series of maximal isomorphic subgroups [p] b = pb C12/c1 (15) 2; 3 + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] a = a − 2qc, c = pc C12/c1 (15) 2 + (0, 0, 2p − 12 + 2u); 3 + (0, 0, 2u) prime p > 2; 0 ≤ q < p; 0 ≤ u < p p conjugate subgroups for each pair of q and p [p] a = pa C12/c1 (15) (2; 3) + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

Copyright © 2011 International Union of Crystallography

153

C 2/c

UNIQUE AXIS

b

No. 15

CONTINUED

I Minimal translationengleiche supergroups [2] Cmcm (63); [2] Cmce (64); [2] Cccm (66); [2] Ccce (68); [2] Fddd (70); [2] Ibam (72); [2] Ibca (73); [2] Imma (74); [2] I41 /a (88); ¯ ¯ ¯ ¯ ¯ ¯ [3] P312/c (163, P31c); [3] P32/c1 (165, P3c1); [3] R32/c (167, R3c) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] F12/m1 (12, C12/m1) • Decreased unit cell

[2] c = 12 c C12/m1 (12); [2] a = 12 a, b = 12 b P12/c1 (13)

C 2/c

UNIQUE AXIS

c

No. 15

(Continued from the facing page)

I Minimal translationengleiche supergroups [2] Cmcm (63); [2] Cmce (64); [2] Cccm (66); [2] Ccce (68); [2] Fddd (70); [2] Ibam (72); [2] Ibca (73); [2] Imma (74); [2] I41 /a (88); ¯ ¯ ¯ ¯ ¯ ¯ [3] P312/c (163, P31c); [3] P32/c1 (165, P3c1); [3] R32/c (167, R3c) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] F112/m (12, A112/m)

• Decreased unit cell

[2] a = 12 a A112/m (12); [2] b = 12 b, c = 12 c P112/a (13)

154

C2h6

A 1 1 2/a

UNIQUE AXIS

No. 15

C 2/c

c, CELL CHOICE 1

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t(0, 12 , 12 ); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

f

(0, 0, 0)+ (0, 12 , 12 )+

1

(1) x, y, z

¯z (2) x¯ + 12 , y,

I Maximal translationengleiche subgroups [2] A11a (9) (1; 4)+ [2] A112 (5) (1; 2)+ ¯ [2] A1¯ (2, P1) (1; 3)+

(3) x, ¯ y, ¯ z¯

(4) x + 12 , y, z¯

1/4, 0, 0 a, 1/2(b − c), 1/2(b + c)

II Maximal klassengleiche subgroups

• Loss of centring translations [2] [2] [2] [2]

P1121 /n (14, P1121 /a) P1121 /a (14) P112/a (13) P112/n (13, P112/a)

1; 1; 1; 1;

3; 4; 2; 2;

(2; 4) + (0, 12 , 12 ) (2; 3) + (0, 12 , 12 ) 3; 4 (3; 4) + (0, 12 , 12 )

−a − b, a, c 0, 1/4, 1/4 −a − b, a, c

0, 1/4, 1/4

• Enlarged unit cell

[3] c = 3c A112/a A112/a A112/a [3] a = 3a A112/a A112/a A112/a [3] a = 3a, A112/a A112/a A112/a [3] a = 3a, A112/a A112/a A112/a [3] b = 3b A112/a A112/a A112/a

(15) (15) (15)

2; 3 2; 3 + (0, 0, 2) 2; 3 + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

(15) (15) (15) b = −2a + b (15) (15) (15) b = −4a + b (15) (15) (15)

3; 2 + (1, 0, 0) 2 + (3, 0, 0); 3 + (2, 0, 0) 2 + (5, 0, 0); 3 + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

3; 2 + (1, 0, 0) 2 + (3, 0, 0); 3 + (2, 0, 0) 2 + (5, 0, 0); 3 + (4, 0, 0)

3a, −2a + b, c 3a, −2a + b, c 3a, −2a + b, c

1, 0, 0 2, 0, 0

3; 2 + (1, 0, 0) 2 + (3, 0, 0); 3 + (2, 0, 0) 2 + (5, 0, 0); 3 + (4, 0, 0)

3a, −4a + b, c 3a, −4a + b, c 3a, −4a + b, c

1, 0, 0 2, 0, 0

(15) (15) (15)

2; 3 (2; 3) + (0, 2, 0) (2; 3) + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

a, b, pc

0, 0, u

pa, −2qa + b, c

u, 0, 0

a, pb, c

0, u, 0

• Series of maximal isomorphic subgroups [p] c = pc A112/a (15) 2; 3 + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] a = pa, b = −2qa + b A112/a (15) 2 + ( 2p − 12 + 2u, 0, 0); 3 + (2u, 0, 0) prime p > 2; 0 ≤ q < p; 0 ≤ u < p p conjugate subgroups for each pair of q and p [p] b = pb A112/a (15) (2; 3) + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups (Continued on the facing page)

155

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 16, pp. 156–157.

P222

No. 16

D12

P222

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

4

u

1

(1) x, y, z

(2) x, ¯ y, ¯z

I Maximal translationengleiche subgroups [2] P112 (3) 1; 2 [2] P121 (3) 1; 3 [2] P211 (3, P121) 1; 4

(3) x, ¯ y, z¯

(4) x, y, ¯ z¯

c, a, b

II Maximal klassengleiche subgroups

• Enlarged unit cell

[2] a = 2a P21 22 (17, P2221 ) P21 22 (17, P2221 ) P222 (16) P222 (16) [2] b = 2b P221 2 (17, P2221 ) P221 2 (17, P2221 ) P222 (16) P222 (16) [2] c = 2c P2221 (17) P2221 (17) P222 (16) P222 (16) [2] b = 2b, c = 2c A222 (21, C222) A222 (21, C222) A222 (21, C222) A222 (21, C222) [2] a = 2a, c = 2c B222 (21, C222) B222 (21, C222) B222 (21, C222) B222 (21, C222) [2] a = 2a, b = 2b C222 (21) C222 (21) C222 (21) C222 (21) [2] a = 2a, b = 2b, c = 2c F222 (22) F222 (22) F222 (22) F222 (22) [3] a = 3a P222 (16) P222 (16) P222 (16) [3] b = 3b P222 (16) P222 (16) P222 (16) [3] c = 3c P222 (16) P222 (16) P222 (16)

3; 2 + (1, 0, 0) 2; 3 + (1, 0, 0) 2; 3 (2; 3) + (1, 0, 0)

b, c, 2a b, c, 2a 2a, b, c 2a, b, c

2; 3 + (0, 1, 0) (2; 3) + (0, 1, 0) 2; 3 3; 2 + (0, 1, 0)

c, a, 2b c, a, 2b a, 2b, c a, 2b, c

3; 2 + (0, 0, 1) (2; 3) + (0, 0, 1) 2; 3 2; 3 + (0, 0, 1)

a, b, 2c a, b, 2c a, b, 2c a, b, 2c

2; 3 3; 2 + (0, 1, 0) 2; 3 + (0, 0, 1) 2 + (0, 1, 0); 3 + (0, 0, 1)

2b, 2c, a 2b, 2c, a 2b, 2c, a 2b, 2c, a

0, 1/2, 0 0, 0, 1/2 0, 1/2, 1/2

2; 3 2; 3 + (0, 0, 1) (2; 3) + (1, 0, 0) 2 + (1, 0, 0); 3 + (1, 0, 1)

2c, 2a, b 2c, 2a, b 2c, 2a, b 2c, 2a, b

0, 0, 1/2 1/2, 0, 0 1/2, 0, 1/2

2; 3 (2; 3) + (1, 0, 0) 3; 2 + (0, 1, 0) 2 + (1, 1, 0); 3 + (1, 0, 0)

2a, 2b, c 2a, 2b, c 2a, 2b, c 2a, 2b, c

1/2, 0, 0 0, 1/2, 0 1/2, 1/2, 0

2; 3 3; 2 + (0, 1, 0) 2; 3 + (0, 0, 1) (2; 3) + (1, 0, 0)

2a, 2b, 2c 2a, 2b, 2c 2a, 2b, 2c 2a, 2b, 2c

0, 1/2, 0 0, 0, 1/2 1/2, 0, 0

2; 3 (2; 3) + (2, 0, 0) (2; 3) + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 3 3; 2 + (0, 2, 0) 3; 2 + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2; 3 2; 3 + (0, 0, 2) 2; 3 + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

Copyright © 2011 International Union of Crystallography

156

1/2, 0, 0 1/2, 0, 0

0, 1/2, 0 0, 1/2, 0 0, 0, 1/2 0, 0, 1/2

CONTINUED

P222

No. 16

• Series of maximal isomorphic subgroups [p] a = pa P222 (16) (2; 3) + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] b = pb P222 (16) 3; 2 + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] c = pc P222 (16) 2; 3 + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups

pa, b, c

u, 0, 0

a, pb, c

0, u, 0

a, b, pc

0, 0, u

I Minimal translationengleiche supergroups ¯ ¯ (112); [2] Pmmm (47); [2] Pnnn (48); [2] Pccm (49); [2] Pban (50); [2] P422 (89); [2] P42 22 (93); [2] P42m (111); [2] P42c [3] P23 (195) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] A222 (21, C222); [2] B222 (21, C222); [2] C222 (21); [2] I222 (23)

• Decreased unit cell

none

157

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 17, p. 158.

P 2 2 21

No. 17

D22

P 2 2 21

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

4

e

1

(1) x, y, z

(2) x, ¯ y, ¯ z + 12

I Maximal translationengleiche subgroups [2] P1121 (4) 1; 2 [2] P121 (3) 1; 3 [2] P211 (3, P121) 1; 4

(3) x, ¯ y, z¯ + 12

(4) x, y, ¯ z¯

0, 0, 1/4 c, a, b

II Maximal klassengleiche subgroups

• Enlarged unit cell [2] a = 2a P21 221 (18, P21 21 2) P21 221 (18, P21 21 2) P2221 (17) P2221 (17) [2] b = 2b P221 21 (18, P21 21 2) P221 21 (18, P21 21 2) P2221 (17) P2221 (17) [2] a = 2a, b = 2b C2221 (20) C2221 (20) C2221 (20) C2221 (20) [3] a = 3a P2221 (17) P2221 (17) P2221 (17) [3] b = 3b P2221 (17) P2221 (17) P2221 (17) [3] c = 3c P2221 (17) P2221 (17) P2221 (17)

2; 3 + (1, 0, 0) 3; 2 + (1, 0, 0) 2; 3 (2; 3) + (1, 0, 0)

c, 2a, b c, 2a, b 2a, b, c 2a, b, c

2; 3 + (0, 1, 0) (2; 3) + (0, 1, 0) 2; 3 3; 2 + (0, 1, 0)

2b, c, a 2b, c, a a, 2b, c a, 2b, c

0, 1/2, 0

2; 3 (2; 3) + (1, 0, 0) 3; 2 + (0, 1, 0) 2 + (1, 1, 0); 3 + (1, 0, 0)

2a, 2b, c 2a, 2b, c 2a, 2b, c 2a, 2b, c

1/2, 0, 0 0, 1/2, 0 1/2, 1/2, 0

2; 3 (2; 3) + (2, 0, 0) (2; 3) + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 3 3; 2 + (0, 2, 0) 3; 2 + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

(2; 3) + (0, 0, 1) 2 + (0, 0, 1); 3 + (0, 0, 3) 2 + (0, 0, 1); 3 + (0, 0, 5)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

(2; 3) + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

pa, b, c

u, 0, 0

3; 2 + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, pb, c

0, u, 0

2 + (0, 0, 2p − 12 ); 3 + (0, 0, 2p − 12 + 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, b, pc

0, 0, u

1/2, 0, 1/4 0, 0, 1/4 1/2, 0, 0 0, 1/2, 0

• Series of maximal isomorphic subgroups

[p] a = pa P2221 (17) [p] b = pb P2221 (17) [p] c = pc P2221 (17)

I Minimal translationengleiche supergroups [2] Pmma (51); [2] Pnna (52); [2] Pmna (53); [2] Pcca (54); [2] P41 22 (91); [2] P43 22 (95) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] C2221 (20); [2] A222 (21, C222); [2] B222 (21, C222); [2] I21 21 21 (24)

• Decreased unit cell [2] c = 12 c P222 (16)

Copyright © 2011 International Union of Crystallography

158

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 18, p. 159.

D32

P 21 21 2

P 21 21 2

No. 18

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

4

c

1

(1) x, y, z

(2) x, ¯ y, ¯z

(3) x¯ + 12 , y + 12 , z¯

I Maximal translationengleiche subgroups [2] P121 1 (4) 1; 3 1; 4 [2] P21 11 (4, P121 1) [2] P112 (3) 1; 2

(4) x + 12 , y¯ + 12 , z¯

c, a, b

1/4, 0, 0 0, 1/4, 0

II Maximal klassengleiche subgroups

• Enlarged unit cell [2] c = 2c P21 21 21 (19) P21 21 21 (19) P21 21 2 (18) P21 21 2 (18) [3] a = 3a P21 21 2 (18) P21 21 2 (18) P21 21 2 (18) [3] b = 3b P21 21 2 (18) P21 21 2 (18) P21 21 2 (18) [3] c = 3c P21 21 2 (18) P21 21 2 (18) P21 21 2 (18)

3; 2 + (0, 0, 1) (2; 3) + (0, 0, 1) 2; 3 2; 3 + (0, 0, 1)

a, b, 2c a, b, 2c a, b, 2c a, b, 2c

2; 3 + (1, 0, 0) 2 + (2, 0, 0); 3 + (3, 0, 0) 2 + (4, 0, 0); 3 + (5, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 3 + (0, 1, 0) 2 + (0, 2, 0); 3 + (0, 1, 0) 2 + (0, 4, 0); 3 + (0, 1, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2; 3 2; 3 + (0, 0, 2) 2; 3 + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

2 + (2u, 0, 0); 3 + ( 2p − 12 + 2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

pa, b, c

u, 0, 0

2 + (0, 2u, 0); 3 + (0, 2p − 12 , 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, pb, c

0, u, 0

2; 3 + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, b, pc

0, 0, u

1/4, 0, 1/2 1/4, 0, 0 0, 0, 1/2

• Series of maximal isomorphic subgroups

[p] a = pa P21 21 2 (18) [p] b = pb P21 21 2 (18) [p] c = pc P21 21 2 (18)

I Minimal translationengleiche supergroups [2] Pbam (55); [2] Pccn (56); [2] Pbcm (57); [2] Pnnm (58); [2] Pmmn (59); [2] Pbcn (60); [2] P421 2 (90); [2] P42 21 2 (94); ¯ 1 m (113); [2] P42 ¯ 1 c (114) [2] P42 II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] A21 22 (20, C2221 ); [2] B221 2 (20, C2221 ); [2] C222 (21); [2] I222 (23)

• Decreased unit cell [2] a = 12 a P221 2 (17, P2221 ); [2] b = 12 b P21 22 (17, P2221 )

Copyright © 2011 International Union of Crystallography

159

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 19, p. 160.

P 21 21 21

No. 19

D42

P 2 1 21 21

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

4

a

1

(1) x, y, z

(2) x¯ + 12 , y, ¯ z + 12

(3) x, ¯ y + 12 , z¯ + 12

I Maximal translationengleiche subgroups [2] P1121 (4) 1; 2 1; 3 [2] P121 1 (4) 1; 4 [2] P21 11 (4, P121 1)

(4) x + 12 , y¯ + 12 , z¯

c, a, b

1/4, 0, 0 0, 0, 1/4 0, 1/4, 0

3; 2 + (1, 0, 0) 2 + (3, 0, 0); 3 + (2, 0, 0) 2 + (5, 0, 0); 3 + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 3 + (0, 1, 0) 2 + (0, 2, 0); 3 + (0, 1, 0) 2 + (0, 4, 0); 3 + (0, 1, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

(2; 3) + (0, 0, 1) 2 + (0, 0, 1); 3 + (0, 0, 3) 2 + (0, 0, 1); 3 + (0, 0, 5)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

2 + ( 2p − 12 + 2u, 0, 0); 3 + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

pa, b, c

u, 0, 0

2 + (0, 2u, 0); 3 + (0, 2p − 12 , 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, pb, c

0, u, 0

2 + (0, 0, 2p − 12 ); 3 + (0, 0, 2p − 12 + 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, b, pc

0, 0, u

II Maximal klassengleiche subgroups

• Enlarged unit cell [3] a = 3a P21 21 21 (19) P21 21 21 (19) P21 21 21 (19) [3] b = 3b P21 21 21 (19) P21 21 21 (19) P21 21 21 (19) [3] c = 3c P21 21 21 (19) P21 21 21 (19) P21 21 21 (19)

• Series of maximal isomorphic subgroups

[p] a = pa P21 21 21 (19) [p] b = pb P21 21 21 (19) [p] c = pc P21 21 21 (19)

I Minimal translationengleiche supergroups [2] Pbca (61); [2] Pnma (62); [2] P41 21 2 (92); [2] P43 21 2 (96); [3] P21 3 (198) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] A21 22 (20, C2221 ); [2] B221 2 (20, C2221 ); [2] C2221 (20); [2] I21 21 21 (24) • Decreased unit cell [2] a = 12 a P221 21 (18, P21 21 2); [2] b = 12 b P21 221 (18, P21 21 2); [2] c = 12 c P21 21 2 (18)

Copyright © 2011 International Union of Crystallography

160

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 20, p. 161.

D52

C 2 2 21

C 2 2 21

No. 20

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t( 21 , 12 , 0); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

c

(0, 0, 0)+ ( 21 , 12 , 0)+

1

(1) x, y, z

(2) x, ¯ y, ¯ z + 12

I Maximal translationengleiche subgroups [2] C121 (5) (1; 3)+ [2] C211 (5, C121) (1; 4)+ [2] C1121 (4, P1121 ) (1; 2)+

(3) x, ¯ y, z¯ + 12

(4) x, y, ¯ z¯

−b, a, c 1/2(a − b), 1/2(a + b), c

0, 0, 1/4

II Maximal klassengleiche subgroups

• Loss of centring translations [2] [2] [2] [2]

P21 21 21 (19) P21 221 (18, P21 21 2) P221 21 (18, P21 21 2) P2221 (17)

1; 1; 1; 1;

2; 3; 4; 2;

(3; 4) + ( 21 , 12 , 0) (2; 4) + ( 21 , 12 , 0) (2; 3) + ( 21 , 12 , 0) 3; 4

c, a, b b, c, a

1/4, 0, 0 0, 1/4, 1/4 1/4, 0, 0

2; 3 (2; 3) + (2, 0, 0) (2; 3) + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 3 3; 2 + (0, 2, 0) 3; 2 + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

(2; 3) + (0, 0, 1) 2 + (0, 0, 1); 3 + (0, 0, 3) 2 + (0, 0, 1); 3 + (0, 0, 5)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

pa, b, c

u, 0, 0

a, pb, c

0, u, 0

a, b, pc

0, 0, u

• Enlarged unit cell

[3] a = 3a C2221 (20) C2221 (20) C2221 (20) [3] b = 3b C2221 (20) C2221 (20) C2221 (20) [3] c = 3c C2221 (20) C2221 (20) C2221 (20)

• Series of maximal isomorphic subgroups [p] a = pa (2; 3) + (2u, 0, 0) C2221 (20) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] b = pb 3; 2 + (0, 2u, 0) C2221 (20) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] c = pc 2 + (0, 0, 2p − 12 ); 3 + (0, 0, 2p − 12 + 2u) C2221 (20) prime p > 2; 0 ≤ u < p p conjugate subgroups

I Minimal translationengleiche supergroups [2] Cmcm (63); [2] Cmce (64); [2] P41 22 (91); [2] P41 21 2 (92); [2] P43 22 (95); [2] P43 21 2 (96); [3] P61 22 (178); [3] P65 22 (179); [3] P63 22 (182) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] F222 (22)

• Decreased unit cell

[2] a = 12 a, b = 12 b P2221 (17); [2] c = 12 c C222 (21)

Copyright © 2011 International Union of Crystallography

161

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 21, pp. 162–163.

C222

No. 21

D62

C222

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t( 21 , 12 , 0); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

l

(0, 0, 0)+ ( 21 , 12 , 0)+

1

(1) x, y, z

(2) x, ¯ y, ¯z

I Maximal translationengleiche subgroups [2] C121 (5) (1; 3)+ [2] C211 (5, C121) (1; 4)+ [2] C112 (3, P112) (1; 2)+

(3) x, ¯ y, z¯

(4) x, y, ¯ z¯

−b, a, c 1/2(a − b), 1/2(a + b), c

II Maximal klassengleiche subgroups

• Loss of centring translations [2] [2] [2] [2]

P21 21 2 (18) P21 22 (17, P2221 ) P221 2 (17, P2221 ) P222 (16)

• Enlarged unit cell [2] c = 2c I21 21 21 (24) I21 21 21 (24) I222 (23) I222 (23) C222 (21) C222 (21) C2221 (20) C2221 (20) [3] a = 3a C222 (21) C222 (21) C222 (21) [3] b = 3b C222 (21) C222 (21) C222 (21) [3] c = 3c C222 (21) C222 (21) C222 (21)

1; 1; 1; 1;

2; 3; 4; 2;

(3; 4) + ( 21 , 12 , 0) (2; 4) + ( 21 , 12 , 0) (2; 3) + ( 21 , 12 , 0) 3; 4

b, c, a c, a, b

0, 1/4, 0 1/4, 1/4, 0

3; 2 + (0, 0, 1) (2; 3) + (0, 0, 1) 2; 3 2; 3 + (0, 0, 1) 2; 3 2; 3 + (0, 0, 1) 3; 2 + (0, 0, 1) (2; 3) + (0, 0, 1)

a, b, 2c a, b, 2c a, b, 2c a, b, 2c a, b, 2c a, b, 2c a, b, 2c a, b, 2c

3/4, 0, 0 3/4, 0, 1/2

2; 3 (2; 3) + (2, 0, 0) (2; 3) + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 3 3; 2 + (0, 2, 0) 3; 2 + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2; 3 2; 3 + (0, 0, 2) 2; 3 + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

pa, b, c

u, 0, 0

a, pb, c

0, u, 0

a, b, pc

0, 0, u

• Series of maximal isomorphic subgroups [p] a = pa C222 (21) (2; 3) + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] b = pb C222 (21) 3; 2 + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] c = pc C222 (21) 2; 3 + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups

Copyright © 2011 International Union of Crystallography

162

0, 0, 1/2 0, 0, 1/2 0, 0, 1/2

C222

No. 21

CONTINUED

I Minimal translationengleiche supergroups [2] Cmmm (65); [2] Cccm (66); [2] Cmme (67); [2] Ccce (68); [2] P422 (89); [2] P421 2 (90); [2] P42 22 (93); [2] P42 21 2 (94); ¯ ¯ (116); [2] P4b2 ¯ (117); [2] P4n2 ¯ (118); [3] P622 (177); [3] P62 22 (180); [3] P64 22 (181) [2] P4m2 (115); [2] P4c2 II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] F222 (22) • Decreased unit cell

[2] a = 12 a, b = 12 b P222 (16)

F 222

No. 22

(Continued from the following page)

• Series of maximal isomorphic subgroups

[p] a = pa F222 (22)

[p] b = pb F222 (22) [p] c = pc F222 (22)

(2; 3) + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

pa, b, c

u, 0, 0

3; 2 + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, pb, c

0, u, 0

2; 3 + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, b, pc

0, 0, u

I Minimal translationengleiche supergroups ¯ ¯ (120); [3] F23 (196) [2] Fmmm (69); [2] Fddd (70); [2] I422 (97); [2] I41 22 (98); [2] I 4m2 (119); [2] I 4c2 II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations

none

• Decreased unit cell

[2] a = 12 a, b = 12 b, c = 12 c P222 (16)

163

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 22, pp. 163–164.

F 222

No. 22

D72

F 222

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t(0, 12 , 12 ); t( 21 , 0, 12 ); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

16

k

(0, 0, 0)+ (0, 12 , 12 )+ ( 21 , 0, 12 )+ ( 21 , 12 , 0)+

1

(1) x, y, z

(2) x, ¯ y, ¯z

I Maximal translationengleiche subgroups [2] F112 (5, A112) (1; 2)+ [2] F121 (5, C121) (1; 3)+ [2] F211 (5, C121) (1; 4)+

(3) x, ¯ y, z¯

(4) x, y, ¯ z¯

1/2(a − b), b, c a, b, 1/2(−a + c) c, a, 1/2(b − c)

II Maximal klassengleiche subgroups

• Loss of centring translations [2] A222 (21, C222) 1; 2; 3; 4; (1; 2; 3; 4) + (0, 12 , 12 ) [2] A222 (21, C222) 1; 4; (1; 4) + (0, 12 , 12 ); (2; 3) + ( 21 , 0, 12 ); (2; 3) + ( 21 , 12 , 0) [2] B222 (21, C222) 1; 2; 3; 4; (1; 2; 3; 4) + ( 21 , 0, 12 ) [2] B222 (21, C222) 1; 3; (1; 3) + ( 21 , 0, 12 ); (2; 4) + ( 21 , 12 , 0); (2; 4) + (0, 12 , 12 ) [2] C222 (21) 1; 2; 3; 4; (1; 2; 3; 4) + ( 21 , 12 , 0) [2] C222 (21) 1; 2; (1; 2) + ( 21 , 12 , 0); (3; 4) + (0, 12 , 12 ); (3; 4) + ( 21 , 0, 12 ) 1; 2; (1; 2) + (0, 12 , 12 ); (3; 4) + ( 21 , 0, 12 ); [2] A21 22 (20, C2221 ) (3; 4) + ( 21 , 12 , 0) 1; 3; (1; 3) + (0, 12 , 12 ); (2; 4) + ( 21 , 0, 12 ); [2] A21 22 (20, C2221 ) (2; 4) + ( 21 , 12 , 0) 1; 2; (1; 2) + ( 21 , 0, 12 ); (3; 4) + ( 21 , 12 , 0); [2] B221 2 (20, C2221 ) (3; 4) + (0, 12 , 12 ) 1; 4; (1; 4) + ( 21 , 0, 12 ); (2; 3) + ( 21 , 12 , 0); [2] B221 2 (20, C2221 ) (2; 3) + (0, 12 , 12 ) 1; 3; (1; 3) + ( 21 , 12 , 0); (2; 4) + (0, 12 , 12 ); [2] C2221 (20) (2; 4) + ( 21 , 0, 12 ) 1; 4; (1; 4) + ( 21 , 12 , 0); (2; 3) + (0, 12 , 12 ); [2] C2221 (20) (2; 3) + ( 21 , 0, 12 ) • Enlarged unit cell [3] a = 3a F222 (22) F222 (22) F222 (22) [3] b = 3b F222 (22) F222 (22) F222 (22) [3] c = 3c F222 (22) F222 (22) F222 (22)

b, c, a b, c, a

1/4, 1/4, 1/4

c, a, b c, a, b

1/4, 1/4, 1/4

1/4, 1/4, 1/4 b, c, a

1/4, 0, 1/4

b, c, a

0, 1/4, 0

c, a, b

0, 0, 1/4

c, a, b

1/4, 1/4, 0 0, 1/4, 1/4 1/4, 0, 0

2; 3 (2; 3) + (2, 0, 0) (2; 3) + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 3 3; 2 + (0, 2, 0) 3; 2 + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2; 3 2; 3 + (0, 0, 2) 2; 3 + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

(Continued on the preceding page)

Copyright © 2011 International Union of Crystallography

164

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 23, p. 165.

D82

I 222

No. 23

I 222

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t( 21 , 12 , 12 ); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

k

(0, 0, 0)+ ( 21 , 12 , 12 )+

1

(1) x, y, z

(2) x, ¯ y, ¯z

(3) x, ¯ y, z¯

I Maximal translationengleiche subgroups [2] I112 (5, A112) (1; 2)+ [2] I121 (5, C121) (1; 3)+ [2] I211 (5, C121) (1; 4)+

(4) x, y, ¯ z¯

b, −a − b, c −a − c, b, a −b − c, a, c

II Maximal klassengleiche subgroups

• Loss of centring translations [2] [2] [2] [2]

P21 21 2 (18) P21 221 (18, P21 21 2) P221 21 (18, P21 21 2) P222 (16)

• Enlarged unit cell [3] a = 3a I222 (23) I222 (23) I222 (23) [3] b = 3b I222 (23) I222 (23) I222 (23) [3] c = 3c I222 (23) I222 (23) I222 (23)

1; 1; 1; 1;

2; 3; 4; 2;

(3; 4) + ( 21 , 12 , 12 ) (2; 4) + ( 21 , 12 , 12 ) (2; 3) + ( 21 , 12 , 12 ) 3; 4

c, a, b b, c, a

0, 0, 1/4 0, 1/4, 0 1/4, 0, 0

2; 3 (2; 3) + (2, 0, 0) (2; 3) + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 3 3; 2 + (0, 2, 0) 3; 2 + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2; 3 2; 3 + (0, 0, 2) 2; 3 + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

pa, b, c

u, 0, 0

a, pb, c

0, u, 0

a, b, pc

0, 0, u

• Series of maximal isomorphic subgroups [p] a = pa I222 (23) (2; 3) + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] b = pb I222 (23) 3; 2 + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] c = pc I222 (23) 2; 3 + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups I Minimal translationengleiche supergroups ¯ [2] Immm (71); [2] Ibam (72); [2] I422 (97); [2] I 42m (121); [3] I23 (197) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations

none

• Decreased unit cell

[2] a = 12 a A222 (21, C222); [2] b = 12 b B222 (21, C222); [2] c = 12 c C222 (21)

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International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 24, p. 166.

I 21 2 1 2 1

No. 24

D92

I 21 21 21

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t( 21 , 12 , 12 ); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

d

(0, 0, 0)+ ( 21 , 12 , 12 )+

1

(1) x, y, z

¯ z + 12 (2) x¯ + 12 , y,

(3) x, ¯ y + 12 , z¯ + 12

I Maximal translationengleiche subgroups [2] I1121 (5, A112) (1; 2)+ (1; 3)+ [2] I121 1 (5, C121) (1; 4)+ [2] I21 11 (5, C121)

(4) x + 12 , y¯ + 12 , z¯

b, −a − b, c −a − c, b, a −b − c, a, c

0, 1/4, 0 1/4, 0, 0 0, 0, 1/4

c, a, b b, c, a

1/4, 0, 1/4 0, 1/4, 1/4 1/4, 1/4, 0

II Maximal klassengleiche subgroups

• Loss of centring translations [2] [2] [2] [2]

P21 21 21 (19) P2221 (17) P221 2 (17, P2221 ) P21 22 (17, P2221 )

1; 1; 1; 1;

2; 2; 3; 4;

3; 4 (3; 4) + ( 21 , 12 , 12 ) (2; 4) + ( 21 , 12 , 12 ) (2; 3) + ( 21 , 12 , 12 )

• Enlarged unit cell

[3] a = 3a I21 21 21 I21 21 21 I21 21 21 [3] b = 3b I21 21 21 I21 21 21 I21 21 21 [3] c = 3c I21 21 21 I21 21 21 I21 21 21

(24) (24) (24)

3; 2 + (1, 0, 0) 2 + (3, 0, 0); 3 + (2, 0, 0) 2 + (5, 0, 0); 3 + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

(24) (24) (24)

2; 3 + (0, 1, 0) 2 + (0, 2, 0); 3 + (0, 1, 0) 2 + (0, 4, 0); 3 + (0, 1, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

(24) (24) (24)

(2; 3) + (0, 0, 1) 2 + (0, 0, 1); 3 + (0, 0, 3) 2 + (0, 0, 1); 3 + (0, 0, 5)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

2 + ( 2p − 12 + 2u, 0, 0); 3 + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

pa, b, c

u, 0, 0

2 + (0, 2u, 0); 3 + (0, 2p − 12 , 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, pb, c

0, u, 0

2 + (0, 0, 2p − 12 ); 3 + (0, 0, 2p − 12 + 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, b, pc

0, 0, u

• Series of maximal isomorphic subgroups

[p] a = pa I21 21 21 (24) [p] b = pb I21 21 21 (24) [p] c = pc I21 21 21 (24)

I Minimal translationengleiche supergroups ¯ (122); [3] I21 3 (199) [2] Ibca (73); [2] Imma (74); [2] I41 22 (98); [2] I 42d II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations

none

• Decreased unit cell [2] a = 12 a A222 (21, C222); [2] b = 12 b B222 (21, C222); [2] c = 12 c C222 (21)

Copyright © 2011 International Union of Crystallography

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International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 25, pp. 167–168.

C2v1

Pmm2

Pmm2

No. 25

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

4

i

1

(1) x, y, z

(2) x, ¯ y, ¯z

I Maximal translationengleiche subgroups [2] P1m1 (6) 1; 3 [2] Pm11 (6, P1m1) 1; 4 [2] P112 (3) 1; 2

(3) x, y, ¯z

(4) x, ¯ y, z

c, a, b

II Maximal klassengleiche subgroups

• Enlarged unit cell

[2] a = 2a Pma2 (28) Pma2 (28) Pmm2 (25) Pmm2 (25) [2] b = 2b Pbm2 (28, Pma2) Pbm2 (28, Pma2) Pmm2 (25) Pmm2 (25) [2] c = 2c Pcc2 (27) Pmc21 (26) Pcm21 (26, Pmc21 ) Pmm2 (25) [2] b = 2b, c = 2c Aem2 (39) Aem2 (39) Amm2 (38) Amm2 (38) [2] a = 2a, c = 2c Bme2 (39, Aem2) Bme2 (39, Aem2) Bmm2 (38, Amm2) Bmm2 (38, Amm2) [2] a = 2a, b = 2b Cmm2 (35) Cmm2 (35) Cmm2 (35) Cmm2 (35) [2] a = 2a, b = 2b, c = 2c Fmm2 (42) Fmm2 (42) Fmm2 (42) Fmm2 (42) [3] a = 3a Pmm2 (25) Pmm2 (25) Pmm2 (25) [3] b = 3b Pmm2 (25) Pmm2 (25) Pmm2 (25) [3] c = 3c Pmm2 (25)

2; 3 + (1, 0, 0) (2; 3) + (1, 0, 0) 2; 3 3; 2 + (1, 0, 0)

2a, b, c 2a, b, c 2a, b, c 2a, b, c

2; 3 + (0, 1, 0) 3; 2 + (0, 1, 0) 2; 3 (2; 3) + (0, 1, 0)

−2b, a, c −2b, a, c a, 2b, c a, 2b, c

2; 3 + (0, 0, 1) (2; 3) + (0, 0, 1) 3; 2 + (0, 0, 1) 2; 3

a, b, 2c a, b, 2c −b, a, 2c a, b, 2c

2; 3 + (0, 1, 0) 3; 2 + (0, 1, 0) 2; 3 (2; 3) + (0, 1, 0)

a, 2b, 2c a, 2b, 2c a, 2b, 2c a, 2b, 2c

2; 3 + (1, 0, 0) (2; 3) + (1, 0, 0) 2; 3 3; 2 + (1, 0, 0)

−b, 2a, 2c −b, 2a, 2c −b, 2a, 2c −b, 2a, 2c

2; 3 3; 2 + (1, 0, 0) (2; 3) + (0, 1, 0) 2 + (1, 1, 0); 3 + (0, 1, 0)

2a, 2b, c 2a, 2b, c 2a, 2b, c 2a, 2b, c

1/2, 0, 0 0, 1/2, 0 1/2, 1/2, 0

2; 3 3; 2 + (1, 0, 0) (2; 3) + (0, 1, 0) 2 + (1, 1, 0); 3 + (0, 1, 0)

2a, 2b, 2c 2a, 2b, 2c 2a, 2b, 2c 2a, 2b, 2c

1/2, 0, 0 0, 1/2, 0 1/2, 1/2, 0

2; 3 3; 2 + (2, 0, 0) 3; 2 + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 3 (2; 3) + (0, 2, 0) (2; 3) + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2; 3

a, b, 3c

Copyright © 2011 International Union of Crystallography

167

1/2, 0, 0 1/2, 0, 0

0, 1/2, 0 0, 1/2, 0

0, 1/2, 0 0, 1/2, 0

1/2, 0, 0 1/2, 0, 0

Pmm2

No. 25

• Series of maximal isomorphic subgroups [p] a = pa Pmm2 (25) 3; 2 + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] b = pb Pmm2 (25) (2; 3) + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] c = pc Pmm2 (25) 2; 3 p prime no conjugate subgroups

CONTINUED

pa, b, c

u, 0, 0

a, pb, c

0, u, 0

a, b, pc

I Minimal translationengleiche supergroups ¯ [2] Pmmm (47); [2] Pmma (51); [2] Pmmn (59); [2] P4mm (99); [2] P42 mc (105); [2] P4m2 (115) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] Cmm2 (35); [2] Amm2 (38); [2] Bmm2 (38, Amm2); [2] Imm2 (44)

• Decreased unit cell

none

168

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 26, p. 169.

C2v2

P m c 21

P m c 21

No. 26

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

4

c

1

(1) x, y, z

(2) x, ¯ y, ¯ z + 12

I Maximal translationengleiche subgroups [2] P1c1 (7) 1; 3 [2] Pm11 (6, P1m1) 1; 4 1; 2 [2] P1121 (4)

(3) x, y, ¯ z + 12

(4) x, ¯ y, z

c, a, b

II Maximal klassengleiche subgroups

• Enlarged unit cell

[2] a = 2a Pmn21 (31) Pmn21 (31) Pmc21 (26) Pmc21 (26) [2] b = 2b Pbc21 (29, Pca21 ) Pbc21 (29, Pca21 ) Pmc21 (26) Pmc21 (26) [2] a = 2a, b = 2b Cmc21 (36) Cmc21 (36) Cmc21 (36) Cmc21 (36) [3] a = 3a Pmc21 (26) Pmc21 (26) Pmc21 (26) [3] b = 3b Pmc21 (26) Pmc21 (26) Pmc21 (26) [3] c = 3c Pmc21 (26)

2; 3 + (1, 0, 0) (2; 3) + (1, 0, 0) 2; 3 3; 2 + (1, 0, 0)

2a, b, c 2a, b, c 2a, b, c 2a, b, c

2; 3 + (0, 1, 0) 3; 2 + (0, 1, 0) 2; 3 (2; 3) + (0, 1, 0)

−2b, a, c −2b, a, c a, 2b, c a, 2b, c

2; 3 3; 2 + (1, 0, 0) (2; 3) + (0, 1, 0) 2 + (1, 1, 0); 3 + (0, 1, 0)

2a, 2b, c 2a, 2b, c 2a, 2b, c 2a, 2b, c

1/2, 0, 0 0, 1/2, 0 1/2, 1/2, 0

2; 3 3; 2 + (2, 0, 0) 3; 2 + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 3 (2; 3) + (0, 2, 0) (2; 3) + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

(2; 3) + (0, 0, 1)

a, b, 3c

1/2, 0, 0 1/2, 0, 0 0, 1/2, 0 0, 1/2, 0

• Series of maximal isomorphic subgroups

[p] a = pa Pmc21 (26) [p] b = pb Pmc21 (26) [p] c = pc Pmc21 (26)

3; 2 + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

pa, b, c

u, 0, 0

(2; 3) + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, pb, c

0, u, 0

(2; 3) + (0, 0, 2p − 12 ) prime p > 2 no conjugate subgroups

a, b, pc

I Minimal translationengleiche supergroups [2] Pmma (51); [2] Pbam (55); [2] Pbcm (57); [2] Pnma (62) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] Cmc21 (36); [2] Amm2 (38); [2] Bme2 (39, Aem2); [2] Ima2 (46) • Decreased unit cell

[2] c = 12 c Pmm2 (25)

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169

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 27, p. 170.

Pcc2

No. 27

C2v3

Pcc2

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

4

e

1

(1) x, y, z

(2) x, ¯ y, ¯z

I Maximal translationengleiche subgroups [2] P1c1 (7) 1; 3 [2] Pc11 (7, P1c1) 1; 4 [2] P112 (3) 1; 2

(3) x, y, ¯ z + 12

(4) x, ¯ y, z + 12

−b, a, c

II Maximal klassengleiche subgroups

• Enlarged unit cell [2] a = 2a Pcn2 (30, Pnc2) Pcn2 (30, Pnc2) Pcc2 (27) Pcc2 (27) [2] b = 2b Pnc2 (30) Pnc2 (30) Pcc2 (27) Pcc2 (27) [2] a = 2a, b = 2b Ccc2 (37) Ccc2 (37) Ccc2 (37) Ccc2 (37) [3] a = 3a Pcc2 (27) Pcc2 (27) Pcc2 (27) [3] b = 3b Pcc2 (27) Pcc2 (27) Pcc2 (27) [3] c = 3c Pcc2 (27)

2; 3 + (1, 0, 0) (2; 3) + (1, 0, 0) 2; 3 3; 2 + (1, 0, 0)

−b, 2a, c −b, 2a, c 2a, b, c 2a, b, c

2; 3 + (0, 1, 0) 3; 2 + (0, 1, 0) 2; 3 (2; 3) + (0, 1, 0)

a, 2b, c a, 2b, c a, 2b, c a, 2b, c

2; 3 3; 2 + (1, 0, 0) (2; 3) + (0, 1, 0) 2 + (1, 1, 0); 3 + (0, 1, 0)

2a, 2b, c 2a, 2b, c 2a, 2b, c 2a, 2b, c

1/2, 0, 0 0, 1/2, 0 1/2, 1/2, 0

2; 3 3; 2 + (2, 0, 0) 3; 2 + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 3 (2; 3) + (0, 2, 0) (2; 3) + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2; 3 + (0, 0, 1)

a, b, 3c

1/2, 0, 0 1/2, 0, 0 0, 1/2, 0 0, 1/2, 0

• Series of maximal isomorphic subgroups

[p] a = pa Pcc2 (27)

[p] b = pb Pcc2 (27) [p] c = pc Pcc2 (27)

3; 2 + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

pa, b, c

u, 0, 0

(2; 3) + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, pb, c

0, u, 0

2; 3 + (0, 0, 2p − 12 ) prime p > 2 no conjugate subgroups

a, b, pc

I Minimal translationengleiche supergroups ¯ (116) [2] Pccm (49); [2] Pcca (54); [2] Pccn (56); [2] P42 cm (101); [2] P4cc (103); [2] P4c2 II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] Ccc2 (37); [2] Aem2 (39); [2] Bme2 (39, Aem2); [2] Iba2 (45) • Decreased unit cell

[2] c = 12 c Pmm2 (25)

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International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 28, p. 171.

C2v4

Pma2

Pma2

No. 28

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

4

d

1

(1) x, y, z

(2) x, ¯ y, ¯z

I Maximal translationengleiche subgroups [2] P1a1 (7, P1c1) 1; 3 [2] Pm11 (6, P1m1) 1; 4 [2] P112 (3) 1; 2

(3) x + 12 , y, ¯z

(4) x¯ + 12 , y, z −a − c, b, a c, a, b

1/4, 0, 0

II Maximal klassengleiche subgroups

• Enlarged unit cell

[2] b = 2b Pba2 (32) Pba2 (32) Pma2 (28) Pma2 (28) [2] c = 2c Pmn21 (31) Pcn2 (30, Pnc2) Pca21 (29) Pma2 (28) [2] b = 2b, c = 2c Aea2 (41) Aea2 (41) Ama2 (40) Ama2 (40) [3] a = 3a Pma2 (28) Pma2 (28) Pma2 (28) [3] b = 3b Pma2 (28) Pma2 (28) Pma2 (28) [3] c = 3c Pma2 (28)

2; 3 + (0, 1, 0) 3; 2 + (0, 1, 0) 2; 3 (2; 3) + (0, 1, 0)

a, 2b, c a, 2b, c a, 2b, c a, 2b, c

(2; 3) + (0, 0, 1) 2; 3 + (0, 0, 1) 3; 2 + (0, 0, 1) 2; 3

a, b, 2c −b, a, 2c a, b, 2c a, b, 2c

2; 3 + (0, 1, 0) 3; 2 + (0, 1, 0) 2; 3 (2; 3) + (0, 1, 0)

a, 2b, 2c a, 2b, 2c a, 2b, 2c a, 2b, 2c

2; 3 + (1, 0, 0) 2 + (2, 0, 0); 3 + (1, 0, 0) 2 + (4, 0, 0); 3 + (1, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 3 (2; 3) + (0, 2, 0) (2; 3) + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2; 3

a, b, 3c

0, 1/2, 0 0, 1/2, 0 1/4, 0, 0

0, 1/2, 0 0, 1/2, 0

• Series of maximal isomorphic subgroups

[p] a = pa Pma2 (28)

[p] b = pb Pma2 (28) [p] c = pc Pma2 (28)

2 + (2u, 0, 0); 3 + ( 2p − 12 , 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

pa, b, c

u, 0, 0

(2; 3) + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, pb, c

0, u, 0

2; 3 p prime no conjugate subgroups

a, b, pc

I Minimal translationengleiche supergroups [2] Pccm (49); [2] Pmma (51); [2] Pmna (53); [2] Pbcm (57) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] Cmm2 (35); [2] Bme2 (39, Aem2); [2] Ama2 (40); [2] Ima2 (46) • Decreased unit cell

[2] a = 12 a Pmm2 (25)

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171

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 29, p. 172.

P c a 21

No. 29

C2v5

P c a 21

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

4

a

1

(1) x, y, z

(2) x, ¯ y, ¯ z + 12

(3) x + 12 , y, ¯z

I Maximal translationengleiche subgroups [2] P1a1 (7, P1c1) 1; 3 [2] Pc11 (7, P1c1) 1; 4 [2] P1121 (4) 1; 2

(4) x¯ + 12 , y, z + 12

−a − c, b, a −b, a, c

1/4, 0, 0

II Maximal klassengleiche subgroups

• Enlarged unit cell [2] b = 2b Pna21 (33) Pna21 (33) Pca21 (29) Pca21 (29) [3] a = 3a Pca21 (29) Pca21 (29) Pca21 (29) [3] b = 3b Pca21 (29) Pca21 (29) Pca21 (29) [3] c = 3c Pca21 (29)

2; 3 + (0, 1, 0) 3; 2 + (0, 1, 0) 2; 3 (2; 3) + (0, 1, 0)

a, 2b, c a, 2b, c a, 2b, c a, 2b, c

2; 3 + (1, 0, 0) 2 + (2, 0, 0); 3 + (1, 0, 0) 2 + (4, 0, 0); 3 + (1, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 3 (2; 3) + (0, 2, 0) (2; 3) + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

3; 2 + (0, 0, 1)

a, b, 3c

• Series of maximal isomorphic subgroups [p] a = pa 2 + (2u, 0, 0); 3 + ( 2p − 12 , 0, 0) Pca21 (29) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] b = pb (2; 3) + (0, 2u, 0) Pca21 (29) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] c = pc 3; 2 + (0, 0, 2p − 12 ) Pca21 (29) prime p > 2 no conjugate subgroups

II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] Ccm21 (36, Cmc21 ); [2] Bme2 (39, Aem2); [2] Aea2 (41); [2] Iba2 (45) • Decreased unit cell

[2] a = 12 a Pcm21 (26, Pmc21 ); [2] c = 12 c Pma2 (28)

Copyright © 2011 International Union of Crystallography

172

0, 1/2, 0

pa, b, c

u, 0, 0

a, pb, c

0, u, 0

a, b, pc

I Minimal translationengleiche supergroups [2] Pcca (54); [2] Pbcm (57); [2] Pbcn (60); [2] Pbca (61)

0, 1/2, 0

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 30, p. 173.

C2v6

Pnc2

Pnc2

No. 30

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

4

c

1

(1) x, y, z

(2) x, ¯ y, ¯z

(3) x, y¯ + 12 , z + 12

I Maximal translationengleiche subgroups [2] P1c1 (7) 1; 3 [2] Pn11 (7, P1c1) 1; 4 [2] P112 (3) 1; 2

(4) x, ¯ y + 12 , z + 12

b, a, −b − c

0, 1/4, 0

II Maximal klassengleiche subgroups

• Enlarged unit cell [2] a = 2a Pnn2 (34) Pnn2 (34) Pnc2 (30) Pnc2 (30) [3] a = 3a Pnc2 (30) Pnc2 (30) Pnc2 (30) [3] b = 3b Pnc2 (30) Pnc2 (30) Pnc2 (30) [3] c = 3c Pnc2 (30)

2; 3 + (1, 0, 0) (2; 3) + (1, 0, 0) 2; 3 3; 2 + (1, 0, 0)

2a, b, c 2a, b, c 2a, b, c 2a, b, c

2; 3 3; 2 + (2, 0, 0) 3; 2 + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 3 + (0, 1, 0) 2 + (0, 2, 0); 3 + (0, 3, 0) 2 + (0, 4, 0); 3 + (0, 5, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2; 3 + (0, 0, 1)

a, b, 3c

• Series of maximal isomorphic subgroups [p] a = pa Pnc2 (30) 3; 2 + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] b = pb Pnc2 (30) 2 + (0, 2u, 0); 3 + (0, 2p − 12 + 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] c = pc Pnc2 (30) 2; 3 + (0, 0, 2p − 12 ) prime p > 2 no conjugate subgroups I Minimal translationengleiche supergroups [2] Pban (50); [2] Pnna (52); [2] Pmna (53); [2] Pbcn (60) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] Ccc2 (37); [2] Amm2 (38); [2] Bbe2 (41, Aea2); [2] Ima2 (46) • Decreased unit cell

[2] b = 12 b Pcc2 (27); [2] c = 12 c Pbm2 (28, Pma2)

Copyright © 2011 International Union of Crystallography

173

1/2, 0, 0 1/2, 0, 0

pa, b, c

u, 0, 0

a, pb, c

0, u, 0

a, b, pc

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 31, p. 174.

P m n 21

No. 31

C2v7

P m n 21

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

4

b

1

(1) x, y, z

(2) x¯ + 12 , y, ¯ z + 12

I Maximal translationengleiche subgroups [2] P1n1 (7, P1c1) 1; 3 [2] Pm11 (6, P1m1) 1; 4 [2] P1121 (4) 1; 2

(3) x + 12 , y, ¯ z + 12

(4) x, ¯ y, z

c, b, −a − c c, a, b 1/4, 0, 0

II Maximal klassengleiche subgroups

• Enlarged unit cell [2] b = 2b Pbn21 (33, Pna21 ) Pbn21 (33, Pna21 ) Pmn21 (31) Pmn21 (31) [3] a = 3a Pmn21 (31) Pmn21 (31) Pmn21 (31) [3] b = 3b Pmn21 (31) Pmn21 (31) Pmn21 (31) [3] c = 3c Pmn21 (31)

2; 3 + (0, 1, 0) 3; 2 + (0, 1, 0) 2; 3 (2; 3) + (0, 1, 0)

−2b, a, c −2b, a, c a, 2b, c a, 2b, c

(2; 3) + (1, 0, 0) 2 + (3, 0, 0); 3 + (1, 0, 0) 2 + (5, 0, 0); 3 + (1, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 3 (2; 3) + (0, 2, 0) (2; 3) + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

(2; 3) + (0, 0, 1)

a, b, 3c

• Series of maximal isomorphic subgroups [p] a = pa Pmn21 (31) 2 + ( 2p − 12 + 2u, 0, 0); 3 + ( 2p − 12 , 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] b = pb Pmn21 (31) (2; 3) + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] c = pc Pmn21 (31) (2; 3) + (0, 0, 2p − 12 ) prime p > 2 no conjugate subgroups I Minimal translationengleiche supergroups [2] Pmna (53); [2] Pnnm (58); [2] Pmmn (59); [2] Pnma (62) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] Cmc21 (36); [2] Bmm2 (38, Amm2); [2] Ama2 (40); [2] Imm2 (44) • Decreased unit cell

[2] a = 12 a Pmc21 (26); [2] c = 12 c Pma2 (28)

Copyright © 2011 International Union of Crystallography

174

1/4, 0, 0 1/4, 1/2, 0 0, 1/2, 0

pa, b, c

u, 0, 0

a, pb, c

0, u, 0

a, b, pc

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 32, p. 175.

C2v8

Pba2

Pba2

No. 32

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

4

c

1

(1) x, y, z

(2) x, ¯ y, ¯z

(3) x + 12 , y¯ + 12 , z

I Maximal translationengleiche subgroups [2] P1a1 (7, P1c1) 1; 3 [2] Pb11 (7, P1c1) 1; 4 [2] P112 (3) 1; 2

(4) x¯ + 12 , y + 12 , z

−a − c, b, a c, a, b

0, 1/4, 0 1/4, 0, 0

II Maximal klassengleiche subgroups

• Enlarged unit cell [2] c = 2c Pnn2 (34) Pna21 (33) Pbn21 (33, Pna21 ) Pba2 (32) [3] a = 3a Pba2 (32) Pba2 (32) Pba2 (32) [3] b = 3b Pba2 (32) Pba2 (32) Pba2 (32) [3] c = 3c Pba2 (32)

2; 3 + (0, 0, 1) 3; 2 + (0, 0, 1) (2; 3) + (0, 0, 1) 2; 3

a, b, 2c a, b, 2c −b, a, 2c a, b, 2c

2; 3 + (1, 0, 0) 2 + (2, 0, 0); 3 + (1, 0, 0) 2 + (4, 0, 0); 3 + (1, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 3 + (0, 1, 0) 2 + (0, 2, 0); 3 + (0, 3, 0) 2 + (0, 4, 0); 3 + (0, 5, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2; 3

a, b, 3c

• Series of maximal isomorphic subgroups [p] a = pa Pba2 (32) 2 + (2u, 0, 0); 3 + ( 2p − 12 , 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] b = pb Pba2 (32) 2 + (0, 2u, 0); 3 + (0, 2p − 12 + 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] c = pc Pba2 (32) 2; 3 p prime no conjugate subgroups

pa, b, c

u, 0, 0

a, pb, c

0, u, 0

a, b, pc

I Minimal translationengleiche supergroups ¯ (117) [2] Pban (50); [2] Pcca (54); [2] Pbam (55); [2] P4bm (100); [2] P42 bc (106); [2] P4b2 II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] Cmm2 (35); [2] Aea2 (41); [2] Bbe2 (41, Aea2); [2] Iba2 (45) • Decreased unit cell

[2] a = 12 a Pbm2 (28, Pma2); [2] b = 12 b Pma2 (28)

Copyright © 2011 International Union of Crystallography

175

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 33, p. 176.

P n a 21

No. 33

C2v9

P n a 21

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

4

a

1

(1) x, y, z

(2) x, ¯ y, ¯ z + 12

(3) x + 12 , y¯ + 12 , z

I Maximal translationengleiche subgroups [2] P1a1 (7, P1c1) 1; 3 [2] Pn11 (7, P1c1) 1; 4 [2] P1121 (4) 1; 2

(4) x¯ + 12 , y + 12 , z + 12

−a − c, b, a b, a, −b − c

0, 1/4, 0 1/4, 0, 0

2; 3 + (1, 0, 0) 2 + (2, 0, 0); 3 + (1, 0, 0) 2 + (4, 0, 0); 3 + (1, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 3 + (0, 1, 0) 2 + (0, 2, 0); 3 + (0, 3, 0) 2 + (0, 4, 0); 3 + (0, 5, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

3; 2 + (0, 0, 1)

a, b, 3c

II Maximal klassengleiche subgroups

• Enlarged unit cell [3] a = 3a Pna21 (33) Pna21 (33) Pna21 (33) [3] b = 3b Pna21 (33) Pna21 (33) Pna21 (33) [3] c = 3c Pna21 (33)

• Series of maximal isomorphic subgroups [p] a = pa 2 + (2u, 0, 0); 3 + ( 2p − 12 , 0, 0) Pna21 (33) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] b = pb 2 + (0, 2u, 0); 3 + (0, 2p − 12 + 2u, 0) Pna21 (33) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] c = pc Pna21 (33) 3; 2 + (0, 0, 2p − 12 ) prime p > 2 no conjugate subgroups I Minimal translationengleiche supergroups [2] Pnna (52); [2] Pccn (56); [2] Pbcn (60); [2] Pnma (62) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] Ccm21 (36, Cmc21 ); [2] Ama2 (40); [2] Bbe2 (41, Aea2); [2] Ima2 (46) • Decreased unit cell

[2] b = 12 b Pca21 (29); [2] a = 12 a Pnm21 (31, Pmn21 ); [2] c = 12 c Pba2 (32)

Copyright © 2011 International Union of Crystallography

176

pa, b, c

u, 0, 0

a, pb, c

0, u, 0

a, b, pc

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 34, p. 177.

C2v10

Pnn2

Pnn2

No. 34

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

4

c

1

(1) x, y, z

(2) x, ¯ y, ¯z

(3) x + 12 , y¯ + 12 , z + 12

I Maximal translationengleiche subgroups [2] P1n1 (7, P1c1) 1; 3 [2] Pn11 (7, P1c1) 1; 4 [2] P112 (3) 1; 2

(4) x¯ + 12 , y + 12 , z + 12

c, b, −a − c b, a, −b − c

0, 1/4, 0 1/4, 0, 0

2; 3 3; 2 + (1, 0, 0) (2; 3) + (0, 1, 0) 2 + (1, 1, 0); 3 + (0, 1, 0)

2a, 2b, 2c 2a, 2b, 2c 2a, 2b, 2c 2a, 2b, 2c

1/2, 0, 0 0, 1/2, 0 1/2, 1/2, 0

2; 3 + (1, 0, 0) 2 + (2, 0, 0); 3 + (1, 0, 0) 2 + (4, 0, 0); 3 + (1, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 3 + (0, 1, 0) 2 + (0, 2, 0); 3 + (0, 3, 0) 2 + (0, 4, 0); 3 + (0, 5, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2; 3 + (0, 0, 1)

a, b, 3c

II Maximal klassengleiche subgroups

• Enlarged unit cell [2] a = 2a, b = 2b, c = 2c Fdd2 (43) Fdd2 (43) Fdd2 (43) Fdd2 (43) [3] a = 3a Pnn2 (34) Pnn2 (34) Pnn2 (34) [3] b = 3b Pnn2 (34) Pnn2 (34) Pnn2 (34) [3] c = 3c Pnn2 (34)

• Series of maximal isomorphic subgroups [p] a = pa Pnn2 (34) 2 + (2u, 0, 0); 3 + ( 2p − 12 , 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] b = pb Pnn2 (34) 2 + (0, 2u, 0); 3 + (0, 2p − 12 + 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] c = pc Pnn2 (34) 2; 3 + (0, 0, 2p − 12 ) prime p > 2 no conjugate subgroups

pa, b, c

u, 0, 0

a, pb, c

0, u, 0

a, b, pc

I Minimal translationengleiche supergroups ¯ (118) [2] Pnnn (48); [2] Pnna (52); [2] Pnnm (58); [2] P42 nm (102); [2] P4nc (104); [2] P4n2 II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] Ccc2 (37); [2] Ama2 (40); [2] Bbm2 (40, Ama2); [2] Imm2 (44) • Decreased unit cell

[2] a = 12 a Pnc2 (30); [2] b = 12 b Pcn2 (30, Pnc2); [2] c = 12 c Pba2 (32)

Copyright © 2011 International Union of Crystallography

177

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 35, p. 178.

Cmm2

No. 35

C2v11

Cmm2

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t( 21 , 12 , 0); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

f

(0, 0, 0)+ ( 21 , 12 , 0)+

1

(1) x, y, z

(2) x, ¯ y, ¯z

I Maximal translationengleiche subgroups [2] C1m1 (8) (1; 3)+ [2] Cm11 (8, C1m1) (1; 4)+ [2] C112 (3, P112) (1; 2)+

(3) x, y, ¯z

(4) x, ¯ y, z

−b, a, c 1/2(a − b), 1/2(a + b), c

II Maximal klassengleiche subgroups

• Loss of centring translations [2] [2] [2] [2]

Pba2 (32) Pbm2 (28, Pma2) Pma2 (28) Pmm2 (25)

• Enlarged unit cell [2] c = 2c Ima2 (46) Ibm2 (46, Ima2) Iba2 (45) Imm2 (44) Ccc2 (37) Cmc21 (36) Ccm21 (36, Cmc21 ) Cmm2 (35) [3] a = 3a Cmm2 (35) Cmm2 (35) Cmm2 (35) [3] b = 3b Cmm2 (35) Cmm2 (35) Cmm2 (35) [3] c = 3c Cmm2 (35)

1; 1; 1; 1;

2; 3; 4; 2;

(3; 4) + ( 21 , 12 , 0) (2; 4) + ( 21 , 12 , 0) (2; 3) + ( 21 , 12 , 0) 3; 4

−b, a, c

1/4, 1/4, 0 1/4, 1/4, 0

(2; 3) + (0, 0, 1) 3; 2 + (0, 0, 1) 2; 3 + (0, 0, 1) 2; 3 2; 3 + (0, 0, 1) (2; 3) + (0, 0, 1) 3; 2 + (0, 0, 1) 2; 3

a, b, 2c −b, a, 2c a, b, 2c a, b, 2c a, b, 2c a, b, 2c −b, a, 2c a, b, 2c

1/4, 1/4, 0 1/4, 1/4, 0

2; 3 3; 2 + (2, 0, 0) 3; 2 + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 3 (2; 3) + (0, 2, 0) (2; 3) + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2; 3

a, b, 3c

• Series of maximal isomorphic subgroups [p] a = pa Cmm2 (35) 3; 2 + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] b = pb Cmm2 (35) (2; 3) + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] c = pc Cmm2 (35) 2; 3 p prime no conjugate subgroups

pa, b, c

u, 0, 0

a, pb, c

0, u, 0

a, b, pc

I Minimal translationengleiche supergroups ¯ ¯ 1 m (113); (111); [2] P42 [2] Cmmm (65); [2] Cmme (67); [2] P4mm (99); [2] P4bm (100); [2] P42 cm (101); [2] P42 nm (102); [2] P42m [3] P6mm (183) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] Fmm2 (42)

• Decreased unit cell [2] a = 12 a, b = 12 b Pmm2 (25) Copyright © 2011 International Union of Crystallography

178

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 36, p. 179.

C2v12

C m c 21

C m c 21

No. 36

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t( 21 , 12 , 0); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

b

(0, 0, 0)+ ( 21 , 12 , 0)+

1

(1) x, y, z

(2) x, ¯ y, ¯ z + 12

I Maximal translationengleiche subgroups [2] C1c1 (9) (1; 3)+ [2] Cm11 (8, C1m1) (1; 4)+ [2] C1121 (4, P1121 ) (1; 2)+

(3) x, y, ¯ z + 12

(4) x, ¯ y, z

−b, a, c 1/2(a − b), 1/2(a + b), c

II Maximal klassengleiche subgroups

• Loss of centring translations [2] [2] [2] [2]

Pbn21 (33, Pna21 ) Pmn21 (31) Pbc21 (29, Pca21 ) Pmc21 (26)

1; 1; 1; 1;

2; 4; 3; 2;

(3; 4) + ( 21 , 12 , 0) (2; 3) + ( 21 , 12 , 0) (2; 4) + ( 21 , 12 , 0) 3; 4

−b, a, c −b, a, c

0, 1/4, 0 1/4, 1/4, 0

2; 3 3; 2 + (2, 0, 0) 3; 2 + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 3 (2; 3) + (0, 2, 0) (2; 3) + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

(2; 3) + (0, 0, 1)

a, b, 3c

• Enlarged unit cell

[3] a = 3a Cmc21 (36) Cmc21 (36) Cmc21 (36) [3] b = 3b Cmc21 (36) Cmc21 (36) Cmc21 (36) [3] c = 3c Cmc21 (36)

• Series of maximal isomorphic subgroups [p] a = pa Cmc21 (36) 3; 2 + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] b = pb Cmc21 (36) (2; 3) + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] c = pc Cmc21 (36) (2; 3) + (0, 0, 2p − 12 ) prime p > 2 no conjugate subgroups

u, 0, 0

a, pb, c

0, u, 0

a, b, pc

I Minimal translationengleiche supergroups [2] Cmcm (63); [2] Cmce (64); [3] P63 cm (185); [3] P63 mc (186) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] Fmm2 (42)

• Decreased unit cell [2] a = 12 a, b = 12 b Pmc21 (26); [2] c = 12 c Cmm2 (35)

Copyright © 2011 International Union of Crystallography

pa, b, c

179

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 37, p. 180.

Ccc2

No. 37

C2v13

Ccc2

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t( 21 , 12 , 0); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

d

(0, 0, 0)+ ( 21 , 12 , 0)+

1

(1) x, y, z

(2) x, ¯ y, ¯z

I Maximal translationengleiche subgroups [2] C1c1 (9) (1; 3)+ [2] Cc11 (9, C1c1) (1; 4)+ [2] C112 (3, P112) (1; 2)+

(3) x, y, ¯ z + 12

(4) x, ¯ y, z + 12

−b, a, c 1/2(a − b), 1/2(a + b), c

II Maximal klassengleiche subgroups

• Loss of centring translations [2] [2] [2] [2]

Pnn2 Pnc2 Pcn2 Pcc2

(34) (30) (30, Pnc2) (27)

1; 1; 1; 1;

2; 3; 4; 2;

(3; 4) + ( 21 , 12 , 0) (2; 4) + ( 21 , 12 , 0) (2; 3) + ( 21 , 12 , 0) 3; 4

−b, a, c

1/4, 1/4, 0 1/4, 1/4, 0

2; 3 3; 2 + (2, 0, 0) 3; 2 + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 3 (2; 3) + (0, 2, 0) (2; 3) + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2; 3 + (0, 0, 1)

a, b, 3c

• Enlarged unit cell

[3] a = 3a Ccc2 (37) Ccc2 (37) Ccc2 (37) [3] b = 3b Ccc2 (37) Ccc2 (37) Ccc2 (37) [3] c = 3c Ccc2 (37)

• Series of maximal isomorphic subgroups [p] a = pa Ccc2 (37) 3; 2 + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] b = pb Ccc2 (37) (2; 3) + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] c = pc Ccc2 (37) 2; 3 + (0, 0, 2p − 12 ) prime p > 2 no conjugate subgroups

pa, b, c

u, 0, 0

a, pb, c

0, u, 0

a, b, pc

I Minimal translationengleiche supergroups ¯ (112); [2] P42 ¯ 1 c (114); [2] Cccm (66); [2] Ccce (68); [2] P4cc (103); [2] P4nc (104); [2] P42 mc (105); [2] P42 bc (106); [2] P42c [3] P6cc (184) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] Fmm2 (42)

• Decreased unit cell [2] a = 12 a, b = 12 b Pcc2 (27); [2] c = 12 c Cmm2 (35)

Copyright © 2011 International Union of Crystallography

180

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 38, p. 181.

C2v14

Amm2

Amm2

No. 38

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t(0, 12 , 12 ); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

f

(0, 0, 0)+ (0, 12 , 12 )+

1

(1) x, y, z

(2) x, ¯ y, ¯z

I Maximal translationengleiche subgroups [2] A1m1 (8, C1m1) (1; 3)+ [2] Am11 (6, P1m1) (1; 4)+ [2] A112 (5) (1; 2)+

(3) x, y, ¯z

(4) x, ¯ y, z

c, b, −a − c 1/2(b + c), a, 1/2(b − c)

II Maximal klassengleiche subgroups

• Loss of centring translations [2] [2] [2] [2]

Pnm21 (31, Pmn21 ) Pnc2 (30) Pmc21 (26) Pmm2 (25)

• Enlarged unit cell [2] a = 2a Ima2 (46) Ima2 (46) Imm2 (44) Imm2 (44) Ama2 (40) Ama2 (40) Amm2 (38) Amm2 (38) [3] a = 3a Amm2 (38) Amm2 (38) Amm2 (38) [3] b = 3b Amm2 (38) Amm2 (38) Amm2 (38) [3] c = 3c Amm2 (38)

1; 1; 1; 1;

3; 2; 4; 2;

(2; 4) + (0, 12 , 12 ) (3; 4) + (0, 12 , 12 ) (2; 3) + (0, 12 , 12 ) 3; 4

−b, a, c 0, 1/4, 0

2; 3 + (1, 0, 0) (2; 3) + (1, 0, 0) 2; 3 3; 2 + (1, 0, 0) 2; 3 + (1, 0, 0) (2; 3) + (1, 0, 0) 2; 3 3; 2 + (1, 0, 0)

2a, b, c 2a, b, c 2a, b, c 2a, b, c 2a, b, c 2a, b, c 2a, b, c 2a, b, c

2; 3 3; 2 + (2, 0, 0) 3; 2 + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 3 (2; 3) + (0, 2, 0) (2; 3) + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2; 3

a, b, 3c

1/2, 0, 0 1/2, 0, 0 1/2, 0, 0 1/2, 0, 0

• Series of maximal isomorphic subgroups

[p] a = pa Amm2 (38)

[p] b = pb Amm2 (38) [p] c = pc Amm2 (38)

3; 2 + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

pa, b, c

u, 0, 0

(2; 3) + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, pb, c

0, u, 0

2; 3 prime p > 2 no conjugate subgroups

a, b, pc

I Minimal translationengleiche supergroups ¯ ¯ [2] Cmcm (63); [2] Cmmm (65); [3] P6m2 (187); [3] P62m (189) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] Fmm2 (42)

• Decreased unit cell

[2] b = 12 b, c = 12 c Pmm2 (25) Copyright © 2011 International Union of Crystallography

181

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 39, p. 182.

Aem2

No. 39

C2v15

Aem2

Former space-group symbol Abm2 Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t(0, 12 , 12 ); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

d

(0, 0, 0)+ (0, 12 , 12 )+

1

(1) x, y, z

(2) x, ¯ y, ¯z

I Maximal translationengleiche subgroups [2] A1m1 (8, C1m1) (1; 3)+ [2] Ae11 (7, P1c1) (1; 4)+ [2] A112 (5) (1; 2)+

(3) x, y¯ + 12 , z

(4) x, ¯ y + 12 , z c, b, −a − c 1/2(−b + c), a, b

0, 1/4, 0

−b, a, c −b, a, c

0, 1/4, 0

−b, a, c

0, 1/4, 0

II Maximal klassengleiche subgroups

• Loss of centring translations [2] [2] [2] [2]

Pbc21 (29, Pca21 ) Pbm2 (28, Pma2) Pcc2 (27) Pcm21 (26, Pmc21 )

• Enlarged unit cell [2] a = 2a Ibm2 (46, Ima2) Ibm2 (46, Ima2) Iba2 (45) Iba2 (45) Aea2 (41) Aea2 (41) Aem2 (39) Aem2 (39) [3] a = 3a Aem2 (39) Aem2 (39) Aem2 (39) [3] b = 3b Aem2 (39) Aem2 (39) Aem2 (39) [3] c = 3c Aem2 (39)

1; 1; 1; 1;

4; 2; 2; 3;

(2; 3) + (0, 12 , 12 ) 3; 4 (3; 4) + (0, 12 , 12 ) (2; 4) + (0, 12 , 12 )

2; 3 3; 2 + (1, 0, 0) 2; 3 + (1, 0, 0) (2; 3) + (1, 0, 0) 2; 3 + (1, 0, 0) (2; 3) + (1, 0, 0) 2; 3 3; 2 + (1, 0, 0)

−b, 2a, c −b, 2a, c 2a, b, c 2a, b, c 2a, b, c 2a, b, c 2a, b, c 2a, b, c

2; 3 3; 2 + (2, 0, 0) 3; 2 + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 3 + (0, 1, 0) 2 + (0, 2, 0); 3 + (0, 3, 0) 2 + (0, 4, 0); 3 + (0, 5, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2; 3

a, b, 3c

1/2, 0, 0 1/2, 0, 0 1/2, 0, 0 1/2, 0, 0

• Series of maximal isomorphic subgroups

[p] a = pa Aem2 (39)

[p] b = pb Aem2 (39) [p] c = pc Aem2 (39)

3; 2 + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

pa, b, c

u, 0, 0

2 + (0, 2u, 0); 3 + (0, 2p − 12 + 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, pb, c

0, u, 0

2; 3 prime p > 2 no conjugate subgroups

a, b, pc

I Minimal translationengleiche supergroups [2] Cmce (64); [2] Cmme (67) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] Fmm2 (42)

• Decreased unit cell [2] b = 12 b, c = 12 c Pmm2 (25) Copyright © 2011 International Union of Crystallography

182

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 40, p. 183.

C2v16

Ama2

Ama2

No. 40

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t(0, 12 , 12 ); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

c

(0, 0, 0)+ (0, 12 , 12 )+

1

(1) x, y, z

(2) x, ¯ y, ¯z

I Maximal translationengleiche subgroups [2] A1a1 (9, C1c1) (1; 3)+ [2] Am11 (6, P1m1) (1; 4)+ [2] A112 (5) (1; 2)+

¯z (3) x + 12 , y,

(4) x¯ + 12 , y, z

c, b, −a 1/2(b + c), a, 1/2(b − c)

1/4, 0, 0

II Maximal klassengleiche subgroups

• Loss of centring translations [2] [2] [2] [2]

Pnn2 (34) Pna21 (33) Pmn21 (31) Pma2 (28)

1; 1; 1; 1;

2; 3; 4; 2;

(3; 4) + (0, 12 , 12 ) (2; 4) + (0, 12 , 12 ) (2; 3) + (0, 12 , 12 ) 3; 4

0, 1/4, 0 1/4, 1/4, 0

• Enlarged unit cell

[3] a = 3a Ama2 (40) Ama2 (40) Ama2 (40) [3] b = 3b Ama2 (40) Ama2 (40) Ama2 (40) [3] c = 3c Ama2 (40)

2; 3 + (1, 0, 0) 2 + (2, 0, 0); 3 + (1, 0, 0) 2 + (4, 0, 0); 3 + (1, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 3 (2; 3) + (0, 2, 0) (2; 3) + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2; 3

a, b, 3c

• Series of maximal isomorphic subgroups [p] a = pa Ama2 (40) 2 + (2u, 0, 0); 3 + ( 2p − 12 , 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] b = pb Ama2 (40) (2; 3) + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] c = pc Ama2 (40) 2; 3 prime p > 2 no conjugate subgroups

u, 0, 0

a, pb, c

0, u, 0

a, b, pc

I Minimal translationengleiche supergroups ¯ (188); [3] P62c ¯ (190) [2] Cmcm (63); [2] Cccm (66); [3] P6c2 II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] Fmm2 (42)

• Decreased unit cell

[2] b = 12 b, c = 12 c Pma2 (28); [2] a = 12 a Amm2 (38)

Copyright © 2011 International Union of Crystallography

pa, b, c

183

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 41, p. 184.

Aea2

No. 41

C2v17

Aea2

Former space-group symbol Aba2 Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t(0, 12 , 12 ); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

b

(0, 0, 0)+ (0, 12 , 12 )+

1

(1) x, y, z

(2) x, ¯ y, ¯z

(3) x + 12 , y¯ + 12 , z

I Maximal translationengleiche subgroups [2] A1a1 (9, C1c1) (1; 3)+ [2] Ae11 (7, P1c1) (1; 4)+ [2] A112 (5) (1; 2)+

(4) x¯ + 12 , y + 12 , z

c, b, −a 1/2(−b + c), a, b

0, 1/4, 0 1/4, 0, 0

−b, a, c

0, 1/4, 0

II Maximal klassengleiche subgroups

• Loss of centring translations [2] [2] [2] [2]

Pbn21 (33, Pna21 ) Pba2 (32) Pcn2 (30, Pnc2) Pca21 (29)

1; 1; 1; 1;

4; 2; 2; 3;

(2; 3) + (0, 12 , 12 ) 3; 4 (3; 4) + (0, 12 , 12 ) (2; 4) + (0, 12 , 12 )

−b, a, c 0, 1/4, 0

• Enlarged unit cell

[3] a = 3a Aea2 (41) Aea2 (41) Aea2 (41) [3] b = 3b Aea2 (41) Aea2 (41) Aea2 (41) [3] c = 3c Aea2 (41)

2; 3 + (1, 0, 0) 2 + (2, 0, 0); 3 + (1, 0, 0) 2 + (4, 0, 0); 3 + (1, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 3 + (0, 1, 0) 2 + (0, 2, 0); 3 + (0, 3, 0) 2 + (0, 4, 0); 3 + (0, 5, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2; 3

a, b, 3c

• Series of maximal isomorphic subgroups [p] a = pa Aea2 (41) 2 + (2u, 0, 0); 3 + ( 2p − 12 , 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] b = pb Aea2 (41) 2 + (0, 2u, 0); 3 + (0, 2p − 12 + 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] c = pc Aea2 (41) 2; 3 prime p > 2 no conjugate subgroups I Minimal translationengleiche supergroups [2] Cmce (64); [2] Ccce (68) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] Fmm2 (42)

• Decreased unit cell

[2] b = 12 b, c = 12 c Pma2 (28); [2] a = 12 a Aem2 (39)

Copyright © 2011 International Union of Crystallography

184

pa, b, c

u, 0, 0

a, pb, c

0, u, 0

a, b, pc

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 42, pp. 185–186.

C2v18

F mm2

F mm2

No. 42

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t(0, 12 , 12 ); t( 21 , 0, 12 ); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

16

e

(0, 0, 0)+ (0, 12 , 12 )+ ( 21 , 0, 12 )+ ( 21 , 12 , 0)+

1

(1) x, y, z

(2) x, ¯ y, ¯z

I Maximal translationengleiche subgroups [2] F1m1 (8, C1m1) (1; 3)+ [2] Fm11 (8, C1m1) (1; 4)+ [2] F112 (5, A112) (1; 2)+

(3) x, y, ¯z

(4) x, ¯ y, z

a, b, 1/2(−a + c) c, a, 1/2(b − c) 1/2(a − b), b, c

II Maximal klassengleiche subgroups

• Loss of centring translations [2] Aea2 (41) [2] Bbe2 (41, Aea2) [2] Ama2 (40) [2] Bbm2 (40, Ama2) [2] Bme2 (39, Aem2) [2] Aem2 (39) [2] Amm2 (38) [2] Bmm2 (38, Amm2) [2] Ccc2 (37) [2] Ccm21 (36, Cmc21 ) [2] Cmc21 (36) [2] Cmm2 (35)

• Enlarged unit cell [3] a = 3a Fmm2 (42) Fmm2 (42) Fmm2 (42) [3] b = 3b Fmm2 (42) Fmm2 (42) Fmm2 (42) [3] c = 3c Fmm2 (42)

1; 2; (1; 2) + (0, 12 , 12 ); (3; 4) + ( 21 , 0, 12 ); (3; 4) + ( 21 , 12 , 0) 1; 2; (1; 2) + ( 21 , 0, 12 ); (3; 4) + (0, 12 , 12 ); (3; 4) + ( 21 , 12 , 0) 1; 4; (1; 4) + (0, 12 , 12 ); (2; 3) + ( 21 , 0, 12 ); (2; 3) + ( 21 , 12 , 0) 1; 3; (1; 3) + ( 21 , 0, 12 ); (2; 4) + (0, 12 , 12 ); (2; 4) + ( 21 , 12 , 0) 1; 4; (1; 4) + ( 21 , 0, 12 ); (2; 3) + (0, 12 , 12 ); (2; 3) + ( 21 , 12 , 0) 1; 3; (1; 3) + (0, 12 , 12 ); (2; 4) + ( 21 , 0, 12 ); (2; 4) + ( 21 , 12 , 0) 1; 2; 3; 4; (1; 2; 3; 4) + (0, 12 , 12 ) 1; 2; 3; 4; (1; 2; 3; 4) + ( 21 , 0, 12 ) 1; 2; (1; 2) + ( 21 , 12 , 0); (3; 4) + (0, 12 , 12 ); (3; 4) + ( 21 , 0, 12 ) 1; 3; (1; 3) + ( 21 , 12 , 0); (2; 4) + (0, 12 , 12 ); (2; 4) + ( 21 , 0, 12 ) 1; 4; (1; 4) + ( 21 , 12 , 0); (2; 3) + (0, 12 , 12 ); (2; 3) + ( 21 , 0, 12 ) 1; 2; 3; 4; (1; 2; 3; 4) + ( 21 , 12 , 0)

−b, a, c 1/4, 1/4, 0 −b, a, c

1/4, 1/4, 0

−b, a, c

1/4, 1/4, 0 1/4, 1/4, 0

−b, a, c 1/4, 1/4, 0 −b, a, c

1/4, 0, 0 0, 1/4, 0

2; 3 3; 2 + (2, 0, 0) 3; 2 + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 3 (2; 3) + (0, 2, 0) (2; 3) + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2; 3

a, b, 3c

• Series of maximal isomorphic subgroups

[p] a = pa Fmm2 (42)

[p] b = pb Fmm2 (42) [p] c = pc Fmm2 (42)

3; 2 + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

pa, b, c

u, 0, 0

(2; 3) + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, pb, c

0, u, 0

2; 3 prime p > 2 no conjugate subgroups

a, b, pc

Copyright © 2011 International Union of Crystallography

185

F mm2

No. 42

CONTINUED

I Minimal translationengleiche supergroups ¯ [2] Fmmm (69); [2] I4mm (107); [2] I4cm (108); [2] I 42m (121) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations

none

• Decreased unit cell [2] a = 12 a, b = 12 b, c = 12 c Pmm2 (25)

186

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 43, p. 187.

C2v19

Fdd2

Fdd2

No. 43

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t(0, 12 , 12 ); t( 21 , 0, 12 ); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

16

b

(0, 0, 0)+ (0, 12 , 12 )+ ( 21 , 0, 12 )+ ( 21 , 12 , 0)+

1

(1) x, y, z

(2) x, ¯ y, ¯z

(3) x + 14 , y¯ + 14 , z + 14

I Maximal translationengleiche subgroups [2] F1d1 (9, C1c1) (1; 3)+ [2] Fd11 (9, C1c1) (1; 4)+ [2] F112 (5, A112) (1; 2)+

(4) x¯ + 14 , y + 14 , z + 14

−c, b, 1/2(a + c) −b, a, 1/2(b + c) 1/2(a − b), b, c

0, 1/8, 0 1/8, 0, 0

II Maximal klassengleiche subgroups

• Loss of centring translations • Enlarged unit cell [3] a = 3a ⎧ ⎨ Fdd2 (43) Fdd2 (43) ⎩ Fdd2 (43) [3] b = 3b ⎧ ⎨ Fdd2 (43) Fdd2 (43) ⎩ Fdd2 (43) [3] c = 3c Fdd2 (43)

none

(2; 3) + ( 21 , 12 , 0) 2 + ( 25 , 12 , 0); 3 + ( 21 , 12 , 0) 2 + ( 29 , 12 , 0); 3 + ( 21 , 12 , 0)

3a, b, c 3a, b, c 3a, b, c

1/4, 1/4, 0 5/4, 1/4, 0 9/4, 1/4, 0

2 + ( 21 , 12 , 0); 3 + (0, 1, 0) 2 + ( 21 , 52 , 0); 3 + (0, 3, 0) 2 + ( 21 , 92 , 0); 3 + (0, 5, 0)

a, 3b, c a, 3b, c a, 3b, c

1/4, 1/4, 0 1/4, 5/4, 0 1/4, 9/4, 0

2 + ( 21 , 12 , 0); 3 + (0, 12 , 12 )

a, b, 3c

1/4, 1/4, 0

2 + (2u, 0, 0); 3 + ( 4p − 14 , 0, 0) prime p > 4; 0 ≤ u < p p conjugate subgroups for p = 4n + 1 2 + ( 21 + 2u, 12 , 0); 3 + ( 4p − 14 , 12 , 0) prime p > 2; 0 ≤ u < p p conjugate subgroups for p = 4n − 1

pa, b, c

u, 0, 0

pa, b, c

1/4 + u, 1/4, 0

2 + (0, 2u, 0); 3 + (0, 4p − 14 + 2u, 0) prime p > 4; 0 ≤ u < p p conjugate subgroups for p = 4n + 1 2 + ( 21 , 12 + 2u, 0); 3 + (0, 4p + 14 + 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups for p = 4n − 1

a, pb, c

0, u, 0

a, pb, c

1/4, 1/4 + u, 0

2; 3 + (0, 0, 4p − 14 ) prime p > 4; p = 4n + 1 no conjugate subgroups 2 + ( 21 , 12 , 0); 3 + (0, 12 , 4p − 14 ) prime p > 2; p = 4n − 1 no conjugate subgroups

a, b, pc

• Series of maximal isomorphic subgroups

[p] a = pa Fdd2 (43)

Fdd2 (43) [p] b = pb Fdd2 (43) Fdd2 (43) [p] c = pc Fdd2 (43) Fdd2 (43)

a, b, pc

I Minimal translationengleiche supergroups ¯ (122) [2] Fddd (70); [2] I41 md (109); [2] I41 cd (110); [2] I 42d II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations

none

• Decreased unit cell [2] a = 12 a, b = 12 b, c = 12 c Pnn2 (34) Copyright © 2011 International Union of Crystallography

187

1/4, 1/4, 0

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 44, p. 188.

I mm2

No. 44

C2v20

I mm2

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t( 21 , 12 , 12 ); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

e

(0, 0, 0)+ ( 21 , 12 , 12 )+

1

(1) x, y, z

(2) x, ¯ y, ¯z

(3) x, y, ¯z

I Maximal translationengleiche subgroups [2] I1m1 (8, C1m1) (1; 3)+ [2] Im11 (8, C1m1) (1; 4)+ [2] I112 (5, A112) (1; 2)+

(4) x, ¯ y, z

−a − c, b, a −b − c, a, c b, −a − b, c

II Maximal klassengleiche subgroups

• Loss of centring translations [2] [2] [2] [2]

Pnn2 (34) Pnm21 (31, Pmn21 ) Pmn21 (31) Pmm2 (25)

• Enlarged unit cell [3] a = 3a Imm2 (44) Imm2 (44) Imm2 (44) [3] b = 3b Imm2 (44) Imm2 (44) Imm2 (44) [3] c = 3c Imm2 (44)

1; 1; 1; 1;

2; 3; 4; 2;

(3; 4) + ( 21 , 12 , 12 ) (2; 4) + ( 21 , 12 , 12 ) (2; 3) + ( 21 , 12 , 12 ) 3; 4

−b, a, c

1/4, 0, 0 0, 1/4, 0

2; 3 3; 2 + (2, 0, 0) 3; 2 + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 3 (2; 3) + (0, 2, 0) (2; 3) + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2; 3

a, b, 3c

• Series of maximal isomorphic subgroups

[p] a = pa Imm2 (44)

[p] b = pb Imm2 (44) [p] c = pc Imm2 (44)

3; 2 + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

pa, b, c

u, 0, 0

(2; 3) + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, pb, c

0, u, 0

2; 3 prime p > 2 no conjugate subgroups

a, b, pc

I Minimal translationengleiche supergroups ¯ [2] Immm (71); [2] Imma (74); [2] I4mm (107); [2] I41 md (109); [2] I 4m2 (119) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations

none

• Decreased unit cell

[2] c = 12 c Cmm2 (35); [2] a = 12 a Amm2 (38); [2] b = 12 b Bmm2 (38, Amm2)

Copyright © 2011 International Union of Crystallography

188

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 45, p. 189.

C2v21

I ba2

No. 45

I ba2

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t( 21 , 12 , 12 ); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

c

(0, 0, 0)+ ( 21 , 12 , 12 )+

1

(1) x, y, z

(2) x, ¯ y, ¯z

(3) x + 12 , y¯ + 12 , z

I Maximal translationengleiche subgroups [2] I1a1 (9, C1c1) (1; 3)+ [2] Ib11 (9, C1c1) (1; 4)+ [2] I112 (5, A112) (1; 2)+

(4) x¯ + 12 , y + 12 , z

a − c, b, c −b − c, a, c b, −a − b, c

II Maximal klassengleiche subgroups

• Loss of centring translations [2] [2] [2] [2]

Pba2 (32) Pca21 (29) Pbc21 (29, Pca21 ) Pcc2 (27)

1; 1; 1; 1;

2; 3; 4; 2;

3; 4 (2; 4) + ( 21 , 12 , 12 ) (2; 3) + ( 21 , 12 , 12 ) (3; 4) + ( 21 , 12 , 12 )

−b, a, c

1/4, 1/4, 0 1/4, 1/4, 0

2; 3 + (1, 0, 0) 2 + (2, 0, 0); 3 + (1, 0, 0) 2 + (4, 0, 0); 3 + (1, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 3 + (0, 1, 0) 2 + (0, 2, 0); 3 + (0, 3, 0) 2 + (0, 4, 0); 3 + (0, 5, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2; 3

a, b, 3c

• Enlarged unit cell

[3] a = 3a Iba2 (45) Iba2 (45) Iba2 (45) [3] b = 3b Iba2 (45) Iba2 (45) Iba2 (45) [3] c = 3c Iba2 (45)

• Series of maximal isomorphic subgroups [p] a = pa Iba2 (45) 2 + (2u, 0, 0); 3 + ( 2p − 12 , 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] b = pb Iba2 (45) 2 + (0, 2u, 0); 3 + (0, 2p − 12 + 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] c = pc Iba2 (45) 2; 3 prime p > 2 no conjugate subgroups

pa, b, c

u, 0, 0

a, pb, c

0, u, 0

a, b, pc

I Minimal translationengleiche supergroups ¯ (120) [2] Ibam (72); [2] Ibca (73); [2] I4cm (108); [2] I41 cd (110); [2] I 4c2 II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations

none

• Decreased unit cell [2] c = 12 c Cmm2 (35); [2] a = 12 a Aem2 (39); [2] b = 12 b Bme2 (39, Aem2)

Copyright © 2011 International Union of Crystallography

189

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 46, p. 190.

I ma2

No. 46

C2v22

I ma2

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t( 21 , 12 , 12 ); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

c

(0, 0, 0)+ ( 21 , 12 , 12 )+

1

(1) x, y, z

(2) x, ¯ y, ¯z

¯z (3) x + 12 , y,

I Maximal translationengleiche subgroups [2] I1a1 (9, C1c1) (1; 3)+ [2] Im11 (8, C1m1) (1; 4)+ [2] I112 (5, A112) (1; 2)+

(4) x¯ + 12 , y, z

−a − c, b, a −b − c, a, c b, −a − b, c

1/4, 0, 0

II Maximal klassengleiche subgroups

• Loss of centring translations [2] [2] [2] [2]

Pna21 (33) Pnc2 (30) Pma2 (28) Pmc21 (26)

1; 1; 1; 1;

3; 2; 2; 4;

(2; 4) + ( 21 , 12 , 12 ) (3; 4) + ( 21 , 12 , 12 ) 3; 4 (2; 3) + ( 21 , 12 , 12 )

1/4, 1/4, 0

1/4, 1/4, 0

• Enlarged unit cell

[3] a = 3a Ima2 (46) Ima2 (46) Ima2 (46) [3] b = 3b Ima2 (46) Ima2 (46) Ima2 (46) [3] c = 3c Ima2 (46)

2; 3 + (1, 0, 0) 2 + (2, 0, 0); 3 + (1, 0, 0) 2 + (4, 0, 0); 3 + (1, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 3 (2; 3) + (0, 2, 0) (2; 3) + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2; 3

a, b, 3c

• Series of maximal isomorphic subgroups [p] a = pa Ima2 (46) 2 + (2u, 0, 0); 3 + ( 2p − 12 , 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] b = pb Ima2 (46) (2; 3) + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] c = pc Ima2 (46) 2; 3 prime p > 2 no conjugate subgroups

pa, b, c

u, 0, 0

a, pb, c

0, u, 0

a, b, pc

I Minimal translationengleiche supergroups [2] Ibam (72); [2] Imma (74) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations

none

• Decreased unit cell

[2] c = 12 c Cmm2 (35); [2] a = 12 a Amm2 (38); [2] b = 12 b Bme2 (39, Aem2)

Copyright © 2011 International Union of Crystallography

190

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 47, pp. 191–192.

D12h

P 2/m 2/m 2/m

No. 47

Pmmm

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

d

1

(1) x, y, z (5) x, ¯ y, ¯ z¯

(2) x, ¯ y, ¯z (6) x, y, z¯

I Maximal translationengleiche subgroups [2] Pmm2 (25) 1; 2; 7; 8 [2] Pm2m (25, Pmm2) 1; 3; 6; 8 [2] P2mm (25, Pmm2) 1; 4; 6; 7 [2] P222 (16) 1; 2; 3; 4 [2] P112/m (10) 1; 2; 5; 6 [2] P12/m1 (10) 1; 3; 5; 7 [2] P2/m11 (10, P12/m1) 1; 4; 5; 8

(3) x, ¯ y, z¯ (7) x, y, ¯z

(4) x, y, ¯ z¯ (8) x, ¯ y, z

c, a, b b, c, a

c, a, b

II Maximal klassengleiche subgroups

• Enlarged unit cell

[2] a = 2a Pmma (51) Pmma (51) Pmam (51, Pmma) Pmam (51, Pmma) Pmaa (49, Pccm) Pmaa (49, Pccm) Pmmm (47) Pmmm (47) [2] b = 2b Pbmm (51, Pmma) Pbmm (51, Pmma) Pmmb (51, Pmma) Pmmb (51, Pmma) Pbmb (49, Pccm) Pbmb (49, Pccm) Pmmm (47) Pmmm (47) [2] c = 2c Pcmm (51, Pmma) Pcmm (51, Pmma) Pmcm (51, Pmma) Pmcm (51, Pmma) Pccm (49) Pccm (49) Pmmm (47) Pmmm (47) [2] b = 2b, c = 2c Aemm (67, Cmme) Aemm (67, Cmme) Aemm (67, Cmme) Aemm (67, Cmme) Ammm (65, Cmmm) Ammm (65, Cmmm) Ammm (65, Cmmm) Ammm (65, Cmmm) [2] a = 2a, c = 2c Bmem (67, Cmme) Bmem (67, Cmme) Bmem (67, Cmme) Bmem (67, Cmme) Bmmm (65, Cmmm) Bmmm (65, Cmmm) Bmmm (65, Cmmm) Bmmm (65, Cmmm)

3; 5; 2 + (1, 0, 0) 2; (3; 5) + (1, 0, 0) 2; 5; 3 + (1, 0, 0) 3; (2; 5) + (1, 0, 0) 5; (2; 3) + (1, 0, 0) 2; 3; 5 + (1, 0, 0) 2; 3; 5 (2; 3; 5) + (1, 0, 0)

2a, b, c 2a, b, c 2a, −c, b 2a, −c, b b, c, 2a b, c, 2a 2a, b, c 2a, b, c

2; 5; 3 + (0, 1, 0) (2; 3; 5) + (0, 1, 0) 5; (2; 3) + (0, 1, 0) 2; (3; 5) + (0, 1, 0) 3; 5; 2 + (0, 1, 0) 2; 3; 5 + (0, 1, 0) 2; 3; 5 3; (2; 5) + (0, 1, 0)

2b, c, a 2b, c, a −2b, a, c −2b, a, c c, a, 2b c, a, 2b a, 2b, c a, 2b, c

3; 5; 2 + (0, 0, 1) (2; 3; 5) + (0, 0, 1) 5; (2; 3) + (0, 0, 1) 3; (2; 5) + (0, 0, 1) 2; 5; 3 + (0, 0, 1) 2; 3; 5 + (0, 0, 1) 2; 3; 5 2; (3; 5) + (0, 0, 1)

2c, b, −a 2c, b, −a 2c, a, b 2c, a, b a, b, 2c a, b, 2c a, b, 2c a, b, 2c

3; 5; 2 + (0, 0, 1) (2; 3; 5) + (0, 0, 1) 2; 5; 3 + (0, 0, 1) 2; 3; 5 + (0, 0, 1) 2; 3; 5 2; (3; 5) + (0, 0, 1) 5; (2; 3) + (0, 0, 1) 3; (2; 5) + (0, 0, 1)

2b, 2c, a 2b, 2c, a 2b, 2c, a 2b, 2c, a 2b, 2c, a 2b, 2c, a 2b, 2c, a 2b, 2c, a

2; 5; 3 + (1, 0, 0) 3; (2; 5) + (1, 0, 0) 5; (2; 3) + (1, 0, 0) 2; 3; 5 + (1, 0, 0) 2; 3; 5 (2; 3; 5) + (1, 0, 0) 3; 5; 2 + (1, 0, 0) 2; (3; 5) + (1, 0, 0)

2c, 2a, b 2c, 2a, b 2c, 2a, b 2c, 2a, b 2c, 2a, b 2c, 2a, b 2c, 2a, b 2c, 2a, b

Copyright © 2011 International Union of Crystallography

191

1/2, 0, 0 1/2, 0, 0 1/2, 0, 0 1/2, 0, 0 0, 1/2, 0 0, 1/2, 0 0, 1/2, 0 0, 1/2, 0 0, 0, 1/2 0, 0, 1/2 0, 0, 1/2 0, 0, 1/2 0, 0, 1/2 0, 1/2, 1/2 0, 1/2, 0 0, 0, 1/2 0, 1/2, 1/2 0, 1/2, 0 1/2, 0, 0 1/2, 0, 1/2 0, 0, 1/2 1/2, 0, 0 1/2, 0, 1/2 0, 0, 1/2

Pmmm [2] a = 2a, b = 2b Cmme (67) Cmme (67) Cmme (67) Cmme (67) Cmmm (65) Cmmm (65) Cmmm (65) Cmmm (65) [2] a = 2a, b = 2b, c = 2c Fmmm (69) Fmmm (69) Fmmm (69) Fmmm (69) Fmmm (69) Fmmm (69) Fmmm (69) Fmmm (69) [3] a = 3a Pmmm (47) Pmmm (47) Pmmm (47) [3] b = 3b Pmmm (47) Pmmm (47) Pmmm (47) [3] c = 3c Pmmm (47) Pmmm (47) Pmmm (47)

No. 47

CONTINUED

3; 5; 2 + (1, 0, 0) 2; (3; 5) + (1, 0, 0) 5; (2; 3) + (1, 0, 0) 2; 3; 5 + (1, 0, 0) 2; 3; 5 (2; 3; 5) + (1, 0, 0) 2; 5; 3 + (1, 0, 0) 3; (2; 5) + (1, 0, 0)

2a, 2b, c 2a, 2b, c 2a, 2b, c 2a, 2b, c 2a, 2b, c 2a, 2b, c 2a, 2b, c 2a, 2b, c

2; 3; 5 (2; 3; 5) + (1, 0, 0) 3; 5; 2 + (1, 0, 0) 2; (3; 5) + (1, 0, 0) 2; 5; 3 + (1, 0, 0) 3; (2; 5) + (1, 0, 0) 5; (2; 3) + (1, 0, 0) 2; 3; 5 + (1, 0, 0)

2a, 2b, 2c 2a, 2b, 2c 2a, 2b, 2c 2a, 2b, 2c 2a, 2b, 2c 2a, 2b, 2c 2a, 2b, 2c 2a, 2b, 2c

1/2, 0, 0 1/2, 0, 1/2 0, 0, 1/2 1/2, 1/2, 0 0, 1/2, 0 0, 1/2, 1/2 1/2, 1/2, 1/2

2; 3; 5 (2; 3; 5) + (2, 0, 0) (2; 3; 5) + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 3; 5 3; (2; 5) + (0, 2, 0) 3; (2; 5) + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2; 3; 5 2; (3; 5) + (0, 0, 2) 2; (3; 5) + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

pa, b, c

u, 0, 0

a, pb, c

0, u, 0

a, b, pc

0, 0, u

• Series of maximal isomorphic subgroups [p] a = pa Pmmm (47) (2; 3; 5) + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] b = pb Pmmm (47) 3; (2; 5) + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] c = pc Pmmm (47) 2; (3; 5) + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups I Minimal translationengleiche supergroups [2] P4/mmm (123); [2] P42 /mmc (131); [3] Pm3¯ (200) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] Ammm (65, Cmmm); [2] Bmmm (65, Cmmm); [2] Cmmm (65); [2] Immm (71) • Decreased unit cell

none

192

1/2, 1/2, 0 0, 1/2, 0 1/2, 0, 0 1/2, 0, 0 1/2, 1/2, 0 0, 1/2, 0

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 48, pp. 193–195.

D22h

P 2/n 2/n 2/n

ORIGIN CHOICE

No. 48

Pnnn

1, Origin at 2 2 2, at 14 , 14 , 14 from 1¯

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

m

1

(1) x, y, z (5) x¯ + 12 , y¯ + 12 , z¯ + 12

(2) x, ¯ y, ¯z (6) x + 12 , y + 12 , z¯ + 12

I Maximal translationengleiche subgroups [2] Pnn2 (34) 1; 2; 7; 8 [2] Pn2n (34, Pnn2) 1; 3; 6; 8 [2] P2nn (34, Pnn2) 1; 4; 6; 7 [2] P222 (16) 1; 2; 3; 4 [2] P112/n (13, P112/a) 1; 2; 5; 6 [2] P12/n1 (13, P12/c1) 1; 3; 5; 7 [2] P2/n11 (13, P12/c1) 1; 4; 5; 8

(3) x, ¯ y, z¯ (7) x + 12 , y¯ + 12 , z + 12

(4) x, y, ¯ z¯ (8) x¯ + 12 , y + 12 , z + 12

c, a, b b, c, a −a − b, a, c c, b, −a − c −b, a, b + c

1/4, 1/4, 1/4 1/4, 1/4, 1/4 1/4, 1/4, 1/4

2; 3; 5 2; (3; 5) + (0, 0, 1) (2; 3; 5) + (0, 0, 1) 3; (2; 5) + (0, 0, 1) 2; 5; 3 + (0, 0, 1) 3; 5; 2 + (0, 0, 1) 5; (2; 3) + (0, 0, 1) 2; 3; 5 + (0, 0, 1)

2a, 2b, 2c 2a, 2b, 2c 2a, 2b, 2c 2a, 2b, 2c 2a, 2b, 2c 2a, 2b, 2c 2a, 2b, 2c 2a, 2b, 2c

0, 0, 1/2 1/2, 0, 0 0, 1/2, 0 1/2, 1/2, 0 1/2, 0, 1/2 0, 1/2, 1/2 1/2, 1/2, 1/2

2; 3; 5 + (1, 0, 0) (2; 3) + (2, 0, 0); 5 + (3, 0, 0) (2; 3) + (4, 0, 0); 5 + (5, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 3; 5 + (0, 1, 0) 3; 2 + (0, 2, 0); 5 + (0, 3, 0) 3; 2 + (0, 4, 0); 5 + (0, 5, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2; 3; 5 + (0, 0, 1) 2; 3 + (0, 0, 2); 5 + (0, 0, 3) 2; 3 + (0, 0, 4); 5 + (0, 0, 5)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

pa, b, c

u, 0, 0

a, pb, c

0, u, 0

a, b, pc

0, 0, u

II Maximal klassengleiche subgroups

• Enlarged unit cell

[2] a = 2a, b = 2b, c = 2c Fddd (70) Fddd (70) Fddd (70) Fddd (70) Fddd (70) Fddd (70) Fddd (70) Fddd (70) [3] a = 3a Pnnn (48) Pnnn (48) Pnnn (48) [3] b = 3b Pnnn (48) Pnnn (48) Pnnn (48) [3] c = 3c Pnnn (48) Pnnn (48) Pnnn (48)

• Series of maximal isomorphic subgroups [p] a = pa Pnnn (48) (2; 3) + (2u, 0, 0); 5 + ( 2p − 12 + 2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] b = pb Pnnn (48) 3; 2 + (0, 2u, 0); 5 + (0, 2p − 12 + 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] c = pc Pnnn (48) 2; 3 + (0, 0, 2u); 5 + (0, 0, 2p − 12 + 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups

Copyright © 2011 International Union of Crystallography

193

Pnnn

ORIGIN CHOICE

1

No. 48

CONTINUED

I Minimal translationengleiche supergroups [2] P4/nnc (126); [2] P42 /nnm (134); [3] Pn3¯ (201) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] Immm (71); [2] Amaa (66, Cccm); [2] Bbmb (66, Cccm); [2] Cccm (66) • Decreased unit cell [2] a = 12 a Pncb (50, Pban); [2] b = 12 b Pcna (50, Pban); [2] c = 12 c Pban (50)

Pnnn

ORIGIN CHOICE

2

No. 48

I Minimal translationengleiche supergroups [2] P4/nnc (126); [2] P42 /nnm (134); [3] Pn3¯ (201) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] Immm (71); [2] Amaa (66, Cccm); [2] Bbmb (66, Cccm); [2] Cccm (66) • Decreased unit cell [2] a = 12 a Pncb (50, Pban); [2] b = 12 b Pcna (50, Pban); [2] c = 12 c Pban (50)

194

(Continued from the facing page)

D22h

P 2/n 2/n 2/n

ORIGIN CHOICE

No. 48

Pnnn

2, Origin at 1¯ at n n n, at − 14 , − 14 , − 14 from 2 2 2

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

m

1

(1) x, y, z (5) x, ¯ y, ¯ z¯

(2) x¯ + 12 , y¯ + 12 , z (6) x + 12 , y + 12 , z¯

(3) x¯ + 12 , y, z¯ + 12 (7) x + 12 , y, ¯ z + 12

I Maximal translationengleiche subgroups [2] Pnn2 (34) 1; 2; 7; 8 [2] Pn2n (34, Pnn2) 1; 3; 6; 8 [2] P2nn (34, Pnn2) 1; 4; 6; 7 [2] P222 (16) 1; 2; 3; 4 [2] P112/n (13, P112/a) 1; 2; 5; 6 [2] P12/n1 (13, P12/c1) 1; 3; 5; 7 [2] P2/n11 (13, P12/c1) 1; 4; 5; 8

(4) x, y¯ + 12 , z¯ + 12 (8) x, ¯ y + 12 , z + 12

c, a, b b, c, a −a − b, a, c c, b, −a − c −b, a, b + c

1/4, 1/4, 1/4 1/4, 1/4, 1/4 1/4, 1/4, 1/4 1/4, 1/4, 1/4

II Maximal klassengleiche subgroups

• Enlarged unit cell [2] a = 2a, b = 2b, c = 2c Fddd (70) Fddd (70) Fddd (70) Fddd (70) Fddd (70) Fddd (70) Fddd (70) Fddd (70) [3] a = 3a Pnnn (48) Pnnn (48) Pnnn (48) [3] b = 3b Pnnn (48) Pnnn (48) Pnnn (48) [3] c = 3c Pnnn (48) Pnnn (48) Pnnn (48)

2; 3; 5 2; (3; 5) + (0, 0, 1) (2; 3; 5) + (0, 0, 1) 3; (2; 5) + (0, 0, 1) 2; 5; 3 + (0, 0, 1) 3; 5; 2 + (0, 0, 1) 5; (2; 3) + (0, 0, 1) 2; 3; 5 + (0, 0, 1)

2a, 2b, 2c 2a, 2b, 2c 2a, 2b, 2c 2a, 2b, 2c 2a, 2b, 2c 2a, 2b, 2c 2a, 2b, 2c 2a, 2b, 2c

0, 0, 1/2 1/2, 0, 0 0, 1/2, 0 1/2, 1/2, 0 1/2, 0, 1/2 0, 1/2, 1/2 1/2, 1/2, 1/2

5; (2; 3) + (1, 0, 0) (2; 3) + (3, 0, 0); 5 + (2, 0, 0) (2; 3) + (5, 0, 0); 5 + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

3; 5; 2 + (0, 1, 0) 3; 2 + (0, 3, 0); 5 + (0, 2, 0) 3; 2 + (0, 5, 0); 5 + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2; 5; 3 + (0, 0, 1) 2; 3 + (0, 0, 3); 5 + (0, 0, 2) 2; 3 + (0, 0, 5); 5 + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

(2; 3) + ( 2p − 12 + 2u, 0, 0); 5 + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

pa, b, c

u, 0, 0

3; 2 + (0, 2p − 12 + 2u, 0); 5 + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, pb, c

0, u, 0

2; 3 + (0, 0, 2p − 12 + 2u); 5 + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, b, pc

0, 0, u

• Series of maximal isomorphic subgroups

[p] a = pa Pnnn (48)

[p] b = pb Pnnn (48) [p] c = pc Pnnn (48)

(Continued on the facing page)

195

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 49, pp. 196–197.

Pccm

No. 49

D32h

P 2/c 2/c 2/m

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

r

1

(1) x, y, z (5) x, ¯ y, ¯ z¯

(2) x, ¯ y, ¯z (6) x, y, z¯

I Maximal translationengleiche subgroups [2] Pc2m (28, Pma2) 1; 3; 6; 8 [2] P2cm (28, Pma2) 1; 4; 6; 7 [2] Pcc2 (27) 1; 2; 7; 8 [2] P222 (16) 1; 2; 3; 4 [2] P12/c1 (13) 1; 3; 5; 7 [2] P2/c11 (13, P12/c1) 1; 4; 5; 8 [2] P112/m (10) 1; 2; 5; 6

(3) x, ¯ y, z¯ + 12 (7) x, y, ¯ z + 12

(4) x, y, ¯ z¯ + 12 (8) x, ¯ y, z + 12

c, a, b c, b, −a

0, 0, 1/4 0, 0, 1/4 0, 0, 1/4

−b, a, c

II Maximal klassengleiche subgroups

• Enlarged unit cell [2] a = 2a Pcca (54) Pcca (54) Pcnm (53, Pmna) Pcnm (53, Pmna) Pcna (50, Pban) Pcna (50, Pban) Pccm (49) Pccm (49) [2] b = 2b Pccb (54, Pcca) Pccb (54, Pcca) Pncm (53, Pmna) Pncm (53, Pmna) Pncb (50, Pban) Pncb (50, Pban) Pccm (49) Pccm (49) [2] a = 2a, b = 2b Ccce (68) Ccce (68) Ccce (68) Ccce (68) Cccm (66) Cccm (66) Cccm (66) Cccm (66) [3] a = 3a Pccm (49) Pccm (49) Pccm (49) [3] b = 3b Pccm (49) Pccm (49) Pccm (49) [3] c = 3c Pccm (49) Pccm (49) Pccm (49)

3; 5; 2 + (1, 0, 0) 2; (3; 5) + (1, 0, 0) 2; 5; 3 + (1, 0, 0) 3; (2; 5) + (1, 0, 0) 5; (2; 3) + (1, 0, 0) 2; 3; 5 + (1, 0, 0) 2; 3; 5 (2; 3; 5) + (1, 0, 0)

2a, b, c 2a, b, c c, b, −2a c, b, −2a c, 2a, b c, 2a, b 2a, b, c 2a, b, c

5; (2; 3) + (0, 1, 0) 2; (3; 5) + (0, 1, 0) 2; 5; 3 + (0, 1, 0) (2; 3; 5) + (0, 1, 0) 3; 5; 2 + (0, 1, 0) 2; 3; 5 + (0, 1, 0) 2; 3; 5 3; (2; 5) + (0, 1, 0)

−2b, a, c −2b, a, c c, a, 2b c, a, 2b 2b, c, a 2b, c, a a, 2b, c a, 2b, c

3; 5; 2 + (1, 0, 0) 5; (2; 3) + (1, 0, 0) 2; 3; 5 + (1, 0, 0) 2; (3; 5) + (1, 0, 0) 2; 3; 5 (2; 3; 5) + (1, 0, 0) 2; 5; 3 + (1, 0, 0) 3; (2; 5) + (1, 0, 0)

2a, 2b, c 2a, 2b, c 2a, 2b, c 2a, 2b, c 2a, 2b, c 2a, 2b, c 2a, 2b, c 2a, 2b, c

2; 3; 5 (2; 3; 5) + (2, 0, 0) (2; 3; 5) + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 3; 5 3; (2; 5) + (0, 2, 0) 3; (2; 5) + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2; 5; 3 + (0, 0, 1) 2; 3 + (0, 0, 3); 5 + (0, 0, 2) 2; 3 + (0, 0, 5); 5 + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

Copyright © 2011 International Union of Crystallography

196

1/2, 0, 0 1/2, 0, 0 1/2, 0, 0 1/2, 0, 0

0, 1/2, 0 0, 1/2, 0 0, 1/2, 0 0, 1/2, 0

1/2, 1/2, 0 0, 1/2, 0 1/2, 0, 0 1/2, 0, 0 1/2, 1/2, 0 0, 1/2, 0

CONTINUED

Pccm

No. 49

• Series of maximal isomorphic subgroups [p] a = pa Pccm (49) (2; 3; 5) + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] b = pb Pccm (49) 3; (2; 5) + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] c = pc Pccm (49) 2; 3 + (0, 0, 2p − 12 + 2u); 5 + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups

pa, b, c

u, 0, 0

a, pb, c

0, u, 0

a, b, pc

0, 0, u

I Minimal translationengleiche supergroups [2] P4/mcc (124); [2] P42 /mcm (132) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] Cccm (66); [2] Aemm (67, Cmme); [2] Bmem (67, Cmme); [2] Ibam (72)

• Decreased unit cell

[2] c = 12 c Pmmm (47)

(Continued from the following page)

No. 50

I Minimal translationengleiche supergroups [2] P4/nbm (125); [2] P42 /nbc (133) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] Cmmm (65); [2] Aeaa (68, Ccce); [2] Bbeb (68, Ccce); [2] Ibam (72)

• Decreased unit cell [2] a = 12 a Pbmb (49, Pccm); [2] b = 12 b Pmaa (49, Pccm)

197

ORIGIN CHOICE

1

Pban

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 50, pp. 197–200.

Pban ORIGIN CHOICE

No. 50

D42h

P 2/b 2/a 2/n

1, Origin at 2 2 2/n, at 14 , 14 , 0 from 1¯

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

m

1

(1) x, y, z (5) x¯ + 12 , y¯ + 12 , z¯

(2) x, ¯ y, ¯z (6) x + 12 , y + 12 , z¯

I Maximal translationengleiche subgroups [2] Pba2 (32) 1; 2; 7; 8 [2] Pb2n (30, Pnc2) 1; 3; 6; 8 [2] P2an (30, Pnc2) 1; 4; 6; 7 [2] P222 (16) 1; 2; 3; 4 [2] P112/n (13, P112/a) 1; 2; 5; 6 [2] P12/a1 (13, P12/c1) 1; 3; 5; 7 [2] P2/b11 (13, P12/c1) 1; 4; 5; 8

(3) x, ¯ y, z¯ (7) x + 12 , y¯ + 12 , z

(4) x, y, ¯ z¯ (8) x¯ + 12 , y + 12 , z

c, a, b c, b, −a −a − b, a, c −a − c, b, a c, a, b

1/4, 1/4, 0 1/4, 1/4, 0 1/4, 1/4, 0

3; 5; 2 + (0, 0, 1) (2; 3; 5) + (0, 0, 1) 5; (2; 3) + (0, 0, 1) 3; (2; 5) + (0, 0, 1) 2; 3; 5 2; (3; 5) + (0, 0, 1) 2; 5; 3 + (0, 0, 1) 2; 3; 5 + (0, 0, 1)

a, −2c, b a, −2c, b b, 2c, a b, 2c, a a, b, 2c a, b, 2c a, b, 2c a, b, 2c

1/4, 1/4, 0 1/4, 1/4, 1/2 1/4, 1/4, 0 1/4, 1/4, 1/2

2; 3; 5 + (1, 0, 0) (2; 3) + (2, 0, 0); 5 + (3, 0, 0) (2; 3) + (4, 0, 0); 5 + (5, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 3; 5 + (0, 1, 0) 3; 2 + (0, 2, 0); 5 + (0, 3, 0) 3; 2 + (0, 4, 0); 5 + (0, 5, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2; 3; 5 2; (3; 5) + (0, 0, 2) 2; (3; 5) + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

pa, b, c

u, 0, 0

a, pb, c

0, u, 0

a, b, pc

0, 0, u

II Maximal klassengleiche subgroups

• Enlarged unit cell

[2] c = 2c Pnan (52, Pnna) Pnan (52, Pnna) Pbnn (52, Pnna) Pbnn (52, Pnna) Pban (50) Pban (50) Pnnn (48) Pnnn (48) [3] a = 3a Pban (50) Pban (50) Pban (50) [3] b = 3b Pban (50) Pban (50) Pban (50) [3] c = 3c Pban (50) Pban (50) Pban (50)

• Series of maximal isomorphic subgroups [p] a = pa Pban (50) (2; 3) + (2u, 0, 0); 5 + ( 2p − 12 + 2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] b = pb Pban (50) 3; 2 + (0, 2u, 0); 5 + (0, 2p − 12 + 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] c = pc Pban (50) 2; (3; 5) + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups (Continued on the preceding page)

Copyright © 2011 International Union of Crystallography

198

0, 0, 1/2 0, 0, 1/2

D42h

P 2/b 2/a 2/n

ORIGIN CHOICE

No. 50

Pban

2, Origin at 1¯ at b a n, at − 14 , − 14 , 0 from 2 2 2

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

m

1

(1) x, y, z (5) x, ¯ y, ¯ z¯

(2) x¯ + 12 , y¯ + 12 , z (6) x + 12 , y + 12 , z¯

I Maximal translationengleiche subgroups [2] Pba2 (32) 1; 2; 7; 8 [2] Pb2n (30, Pnc2) 1; 3; 6; 8 [2] P2an (30, Pnc2) 1; 4; 6; 7 [2] P222 (16) 1; 2; 3; 4 [2] P112/n (13, P112/a) 1; 2; 5; 6 [2] P12/a1 (13, P12/c1) 1; 3; 5; 7 [2] P2/b11 (13, P12/c1) 1; 4; 5; 8

(3) x¯ + 12 , y, z¯ (7) x + 12 , y, ¯z

(4) x, y¯ + 12 , z¯ (8) x, ¯ y + 12 , z

c, a, b c, b, −a −a − b, a, c −a − c, b, a c, a, b

1/4, 1/4, 0 1/4, 1/4, 0 1/4, 1/4, 0 1/4, 1/4, 0

II Maximal klassengleiche subgroups

• Enlarged unit cell [2] c = 2c Pnan (52, Pnna) Pnan (52, Pnna) Pbnn (52, Pnna) Pbnn (52, Pnna) Pban (50) Pban (50) Pnnn (48) Pnnn (48) [3] a = 3a Pban (50) Pban (50) Pban (50) [3] b = 3b Pban (50) Pban (50) Pban (50) [3] c = 3c Pban (50) Pban (50) Pban (50)

3; 5; 2 + (0, 0, 1) (2; 3; 5) + (0, 0, 1) 5; (2; 3) + (0, 0, 1) 3; (2; 5) + (0, 0, 1) 2; 3; 5 2; (3; 5) + (0, 0, 1) 2; 5; 3 + (0, 0, 1) 2; 3; 5 + (0, 0, 1)

a, −2c, b a, −2c, b b, 2c, a b, 2c, a a, b, 2c a, b, 2c a, b, 2c a, b, 2c

5; (2; 3) + (1, 0, 0) (2; 3) + (3, 0, 0); 5 + (2, 0, 0) (2; 3) + (5, 0, 0); 5 + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

3; 5; 2 + (0, 1, 0) 3; 2 + (0, 3, 0); 5 + (0, 2, 0) 3; 2 + (0, 5, 0); 5 + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2; 3; 5 2; (3; 5) + (0, 0, 2) 2; (3; 5) + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

(2; 3) + ( 2p − 12 + 2u, 0, 0); 5 + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

pa, b, c

u, 0, 0

3; 2 + (0, 2p − 12 + 2u, 0); 5 + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, pb, c

0, u, 0

2; (3; 5) + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, b, pc

0, 0, u

0, 0, 1/2 0, 0, 1/2 0, 0, 1/2 0, 0, 1/2

• Series of maximal isomorphic subgroups

[p] a = pa Pban (50)

[p] b = pb Pban (50) [p] c = pc Pban (50)

199

Pban

ORIGIN CHOICE

2

No. 50

CONTINUED

I Minimal translationengleiche supergroups [2] P4/nbm (125); [2] P42 /nbc (133) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] Cmmm (65); [2] Aeaa (68, Ccce); [2] Bbeb (68, Ccce); [2] Ibam (72) • Decreased unit cell

[2] a = 12 a Pbmb (49, Pccm); [2] b = 12 b Pmaa (49, Pccm)

Pmma

No. 51

(Continued from the facing page)

• Series of maximal isomorphic subgroups [p] a = pa Pmma (51) 2 + ( 2p − 12 + 2u, 0, 0); (3; 5) + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] b = pb Pmma (51) 3; (2; 5) + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] c = pc Pmma (51) 2; (3; 5) + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups

pa, b, c

u, 0, 0

a, pb, c

0, u, 0

a, b, pc

0, 0, u

I Minimal translationengleiche supergroups

none

II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] Amma (63, Cmcm); [2] Bmmm (65, Cmmm); [2] Cmme (67); [2] Imma (74) • Decreased unit cell

[2] a = 12 a Pmmm (47)

200

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 51, pp. 200–201.

D52h

P 21/m 2/m 2/a

No. 51

Pmma

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

l

1

(1) x, y, z (5) x, ¯ y, ¯ z¯

(2) x¯ + 12 , y, ¯z 1 (6) x + 2 , y, z¯

I Maximal translationengleiche subgroups [2] Pm2a (28, Pma2) 1; 3; 6; 8 1; 4; 6; 7 [2] P21 ma (26, Pmc21 ) [2] Pmm2 (25) 1; 2; 7; 8 1; 2; 3; 4 [2] P21 22 (17, P2221 ) [2] P112/a (13) 1; 2; 5; 6 1; 4; 5; 8 [2] P21 /m11 (11, P121 /m1) [2] P12/m1 (10) 1; 3; 5; 7

(3) x, ¯ y, z¯ (7) x, y, ¯z

(4) x + 12 , y, ¯ z¯ 1 (8) x¯ + 2 , y, z

a, −c, b b, c, a 1/4, 0, 0 b, c, a c, a, b

II Maximal klassengleiche subgroups

• Enlarged unit cell [2] b = 2b Pmmn (59) Pmmn (59) Pbma (57, Pbcm) Pbma (57, Pbcm) Pbmn (53, Pmna) Pbmn (53, Pmna) Pmma (51) Pmma (51) [2] c = 2c Pmca (57, Pbcm) Pmca (57, Pbcm) Pcma (55, Pbam) Pcma (55, Pbam) Pcca (54) Pcca (54) Pmma (51) Pmma (51) [2] b = 2b, c = 2c Aema (64, Cmce) Aema (64, Cmce) Aema (64, Cmce) Aema (64, Cmce) Amma (63, Cmcm) Amma (63, Cmcm) Amma (63, Cmcm) Amma (63, Cmcm) [3] a = 3a Pmma (51) Pmma (51) Pmma (51) [3] b = 3b Pmma (51) Pmma (51) Pmma (51) [3] c = 3c Pmma (51) Pmma (51) Pmma (51)

5; (2; 3) + (0, 1, 0) 2; (3; 5) + (0, 1, 0) 2; 5; 3 + (0, 1, 0) (2; 3; 5) + (0, 1, 0) 3; 5; 2 + (0, 1, 0) 2; 3; 5 + (0, 1, 0) 2; 3; 5 3; (2; 5) + (0, 1, 0)

a, 2b, c a, 2b, c c, a, 2b c, a, 2b 2b, c, a 2b, c, a a, 2b, c a, 2b, c

5; (2; 3) + (0, 0, 1) 3; (2; 5) + (0, 0, 1) 3; 5; 2 + (0, 0, 1) (2; 3; 5) + (0, 0, 1) 2; 5; 3 + (0, 0, 1) 2; 3; 5 + (0, 0, 1) 2; 3; 5 2; (3; 5) + (0, 0, 1)

b, 2c, a b, 2c, a 2c, a, b 2c, a, b a, b, 2c a, b, 2c a, b, 2c a, b, 2c

3; 5; 2 + (0, 0, 1) (2; 3; 5) + (0, 0, 1) 2; 5; 3 + (0, 0, 1) 2; 3; 5 + (0, 0, 1) 2; 3; 5 2; (3; 5) + (0, 0, 1) 5; (2; 3) + (0, 0, 1) 3; (2; 5) + (0, 0, 1)

2b, 2c, a 2b, 2c, a 2b, 2c, a 2b, 2c, a 2b, 2c, a 2b, 2c, a 2b, 2c, a 2b, 2c, a

3; 5; 2 + (1, 0, 0) 2 + (3, 0, 0); (3; 5) + (2, 0, 0) 2 + (5, 0, 0); (3; 5) + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 3; 5 3; (2; 5) + (0, 2, 0) 3; (2; 5) + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2; 3; 5 2; (3; 5) + (0, 0, 2) 2; (3; 5) + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

(Continued on the facing page)

Copyright © 2011 International Union of Crystallography

201

0, 1/2, 0 0, 1/2, 0 0, 1/2, 0 0, 1/2, 0

0, 0, 1/2 0, 0, 1/2 0, 0, 1/2 0, 0, 1/2

0, 0, 1/2 0, 1/2, 1/2 0, 1/2, 0 0, 0, 1/2 0, 1/2, 1/2 0, 1/2, 0

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 52, p. 202.

Pnna

D62h

P 2/n 21/n 2/a

No. 52

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

e

1

(1) x, y, z (5) x, ¯ y, ¯ z¯

(2) x¯ + 12 , y, ¯z 1 (6) x + 2 , y, z¯

(3) x¯ + 12 , y + 12 , z¯ + 12 (7) x + 12 , y¯ + 12 , z + 12

I Maximal translationengleiche subgroups [2] Pnn2 (34) 1; 2; 7; 8 [2] Pn21 a (33, Pna21 ) 1; 3; 6; 8 [2] P2na (30, Pnc2) 1; 4; 6; 7 1; 2; 3; 4 [2] P221 2 (17, P2221 ) 1; 3; 5; 7 [2] P121 /n1 (14, P121 /c1) [2] P112/a (13) 1; 2; 5; 6 [2] P2/n11 (13, P12/c1) 1; 4; 5; 8

(4) x, y¯ + 12 , z¯ + 12 (8) x, ¯ y + 12 , z + 12

a, −c, b b, c, a c, a, b c, b, −a − c

1/4, 0, 0 1/4, 0, 1/4 0, 1/4, 1/4 1/4, 0, 1/4

−b, a, b + c

II Maximal klassengleiche subgroups

• Enlarged unit cell [3] a = 3a Pnna (52) Pnna (52) Pnna (52) [3] b = 3b Pnna (52) Pnna (52) Pnna (52) [3] c = 3c Pnna (52) Pnna (52) Pnna (52)

5; (2; 3) + (1, 0, 0) (2; 3) + (3, 0, 0); 5 + (2, 0, 0) (2; 3) + (5, 0, 0); 5 + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 5; 3 + (0, 1, 0) (2; 5) + (0, 2, 0); 3 + (0, 1, 0) (2; 5) + (0, 4, 0); 3 + (0, 1, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2; 5; 3 + (0, 0, 1) 2; 3 + (0, 0, 3); 5 + (0, 0, 2) 2; 3 + (0, 0, 5); 5 + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

(2; 3) + ( 2p − 12 + 2u, 0, 0); 5 + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

pa, b, c

u, 0, 0

(2; 5) + (0, 2u, 0); 3 + (0, 2p − 12 , 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, pb, c

0, u, 0

2; 3 + (0, 0, 2p − 12 + 2u); 5 + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, b, pc

0, 0, u

• Series of maximal isomorphic subgroups

[p] a = pa Pnna (52)

[p] b = pb Pnna (52) [p] c = pc Pnna (52)

I Minimal translationengleiche supergroups

none

II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] Bbmm (63, Cmcm); [2] Amaa (66, Cccm); [2] Ccce (68); [2] Imma (74)

• Decreased unit cell

[2] a = 12 a Pncm (53, Pmna); [2] b = 12 b Pcna (50, Pban); [2] c = 12 c Pbaa (54, Pcca)

Copyright © 2011 International Union of Crystallography

202

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 53, p. 203.

D72h

P 2/m 2/n 21/a

Pmna

No. 53

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

i

1

(1) x, y, z (5) x, ¯ y, ¯ z¯

(2) x¯ + 12 , y, ¯ z + 12 1 (6) x + 2 , y, z¯ + 12

I Maximal translationengleiche subgroups 1; 2; 7; 8 [2] Pmn21 (31) [2] P2na (30, Pnc2) 1; 4; 6; 7 [2] Pm2a (28, Pma2) 1; 3; 6; 8 1; 2; 3; 4 [2] P2221 (17) 1; 2; 5; 6 [2] P1121 /a (14) [2] P12/n1 (13, P12/c1) 1; 3; 5; 7 [2] P2/m11 (10, P12/m1) 1; 4; 5; 8

(3) x¯ + 12 , y, z¯ + 12 (7) x + 12 , y, ¯ z + 12

(4) x, y, ¯ z¯ (8) x, ¯ y, z

b, c, a a, −c, b

1/4, 0, 1/4 1/4, 0, 0

c, b, −a − c c, a, b

II Maximal klassengleiche subgroups

• Enlarged unit cell [2] b = 2b Pbna (60, Pbcn) Pbna (60, Pbcn) Pmnn (58, Pnnm) Pmnn (58, Pnnm) Pmna (53) Pmna (53) Pbnn (52, Pnna) Pbnn (52, Pnna) [3] a = 3a Pmna (53) Pmna (53) Pmna (53) [3] b = 3b Pmna (53) Pmna (53) Pmna (53) [3] c = 3c Pmna (53) Pmna (53) Pmna (53)

2; 5; 3 + (0, 1, 0) (2; 3; 5) + (0, 1, 0) 5; (2; 3) + (0, 1, 0) 2; (3; 5) + (0, 1, 0) 2; 3; 5 3; (2; 5) + (0, 1, 0) 3; 5; 2 + (0, 1, 0) 2; 3; 5 + (0, 1, 0)

c, a, 2b c, a, 2b 2b, c, a 2b, c, a a, 2b, c a, 2b, c 2b, c, a 2b, c, a

5; (2; 3) + (1, 0, 0) (2; 3) + (3, 0, 0); 5 + (2, 0, 0) (2; 3) + (5, 0, 0); 5 + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 3; 5 3; (2; 5) + (0, 2, 0) 3; (2; 5) + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

5; (2; 3) + (0, 0, 1) 2 + (0, 0, 1); 3 + (0, 0, 3); 5 + (0, 0, 2) 2 + (0, 0, 1); 3 + (0, 0, 5); 5 + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

0, 1/2, 0 0, 1/2, 0 0, 1/2, 0 0, 1/2, 0

• Series of maximal isomorphic subgroups [p] a = pa Pmna (53) (2; 3) + ( 2p − 12 + 2u, 0, 0); 5 + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] b = pb Pmna (53) 3; (2; 5) + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] c = pc Pmna (53) 2 + (0, 0, 2p − 12 ); 3 + (0, 0, 2p − 12 + 2u); 5 + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups

pa, b, c

u, 0, 0

a, pb, c

0, u, 0

a, b, pc

0, 0, u

I Minimal translationengleiche supergroups

none

II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] Cmce (64); [2] Bmmm (65, Cmmm); [2] Amaa (66, Cccm); [2] Imma (74)

• Decreased unit cell [2] c = 12 c Pmaa (49, Pccm); [2] a = 12 a Pmcm (51, Pmma) Copyright © 2011 International Union of Crystallography

203

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 54, p. 204.

Pcca

D82h

P 21/c 2/c 2/a

No. 54

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

f

1

(1) x, y, z (5) x, ¯ y, ¯ z¯

(2) x¯ + 12 , y, ¯z 1 (6) x + 2 , y, z¯

(3) x, ¯ y, z¯ + 12 (7) x, y, ¯ z + 12

I Maximal translationengleiche subgroups [2] Pc2a (32, Pba2) 1; 3; 6; 8 [2] P21 ca (29, Pca21 ) 1; 4; 6; 7 [2] Pcc2 (27) 1; 2; 7; 8 1; 2; 3; 4 [2] P21 22 (17, P2221 ) [2] P21 /c11 (14, P121 /c1) 1; 4; 5; 8 [2] P112/a (13) 1; 2; 5; 6 [2] P12/c1 (13) 1; 3; 5; 7

(4) x + 12 , y, ¯ z¯ + 12 (8) x¯ + 12 , y, z + 12 a, −c, b c, b, −a b, c, a −b, a, c

0, 0, 1/4 0, 0, 1/4 1/4, 0, 0 0, 0, 1/4

II Maximal klassengleiche subgroups

• Enlarged unit cell [2] b = 2b Pnca (60, Pbcn) Pnca (60, Pbcn) Pccn (56) Pccn (56) Pcca (54) Pcca (54) Pncn (52, Pnna) Pncn (52, Pnna) [3] a = 3a Pcca (54) Pcca (54) Pcca (54) [3] b = 3b Pcca (54) Pcca (54) Pcca (54) [3] c = 3c Pcca (54) Pcca (54) Pcca (54)

2; 5; 3 + (0, 1, 0) (2; 3; 5) + (0, 1, 0) 5; (2; 3) + (0, 1, 0) 2; (3; 5) + (0, 1, 0) 2; 3; 5 3; (2; 5) + (0, 1, 0) 3; 5; 2 + (0, 1, 0) 2; 3; 5 + (0, 1, 0)

2b, c, a 2b, c, a a, 2b, c a, 2b, c a, 2b, c a, 2b, c c, a, 2b c, a, 2b

3; 5; 2 + (1, 0, 0) 2 + (3, 0, 0); (3; 5) + (2, 0, 0) 2 + (5, 0, 0); (3; 5) + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 3; 5 3; (2; 5) + (0, 2, 0) 3; (2; 5) + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2; 5; 3 + (0, 0, 1) 2; 3 + (0, 0, 3); 5 + (0, 0, 2) 2; 3 + (0, 0, 5); 5 + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

2 + ( 2p − 12 + 2u, 0, 0); (3; 5) + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

pa, b, c

u, 0, 0

3; (2; 5) + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, pb, c

0, u, 0

2; 3 + (0, 0, 2p − 12 + 2u); 5 + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, b, pc

0, 0, u

0, 1/2, 0 0, 1/2, 0 0, 1/2, 0 0, 1/2, 0

• Series of maximal isomorphic subgroups

[p] a = pa Pcca (54)

[p] b = pb Pcca (54) [p] c = pc Pcca (54)

I Minimal translationengleiche supergroups

none

II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] Aema (64, Cmce); [2] Bmem (67, Cmme); [2] Ccce (68); [2] Ibca (73)

• Decreased unit cell [2] a = 12 a Pccm (49); [2] c = 12 c Pmma (51) Copyright © 2011 International Union of Crystallography

204

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 55, p. 205.

D92h

P 21/b 21/a 2/m

No. 55

Pbam

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

i

1

(1) x, y, z (5) x, ¯ y, ¯ z¯

(2) x, ¯ y, ¯z (6) x, y, z¯

(3) x¯ + 12 , y + 12 , z¯ (7) x + 12 , y¯ + 12 , z

I Maximal translationengleiche subgroups [2] Pba2 (32) 1; 2; 7; 8 1; 3; 6; 8 [2] Pb21 m (26, Pmc21 ) 1; 4; 6; 7 [2] P21 am (26, Pmc21 ) 1; 2; 3; 4 [2] P21 21 2 (18) 1; 3; 5; 7 [2] P121 /a1 (14, P121 /c1) 1; 4; 5; 8 [2] P21 /b11 (14, P121 /c1) [2] P112/m (10) 1; 2; 5; 6

(4) x + 12 , y¯ + 12 , z¯ (8) x¯ + 12 , y + 12 , z

c, a, b c, b, −a

1/4, 0, 0 0, 1/4, 0

−a − c, b, a c, a, b

II Maximal klassengleiche subgroups

• Enlarged unit cell [2] c = 2c Pnam (62, Pnma) Pnam (62, Pnma) Pbnm (62, Pnma) Pbnm (62, Pnma) Pnnm (58) Pnnm (58) Pbam (55) Pbam (55) [3] a = 3a Pbam (55) Pbam (55) Pbam (55) [3] b = 3b Pbam (55) Pbam (55) Pbam (55) [3] c = 3c Pbam (55) Pbam (55) Pbam (55)

3; 5; 2 + (0, 0, 1) (2; 3; 5) + (0, 0, 1) 5; (2; 3) + (0, 0, 1) 3; (2; 5) + (0, 0, 1) 2; 5; 3 + (0, 0, 1) 2; 3; 5 + (0, 0, 1) 2; 3; 5 2; (3; 5) + (0, 0, 1)

a, −2c, b a, −2c, b b, 2c, a b, 2c, a a, b, 2c a, b, 2c a, b, 2c a, b, 2c

2; 5; 3 + (1, 0, 0) (2; 5) + (2, 0, 0); 3 + (3, 0, 0) (2; 5) + (4, 0, 0); 3 + (5, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 5; 3 + (0, 1, 0) (2; 5) + (0, 2, 0); 3 + (0, 1, 0) (2; 5) + (0, 4, 0); 3 + (0, 1, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2; 3; 5 2; (3; 5) + (0, 0, 2) 2; (3; 5) + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

pa, b, c

u, 0, 0

a, pb, c

0, u, 0

a, b, pc

0, 0, u

• Series of maximal isomorphic subgroups [p] a = pa Pbam (55) (2; 5) + (2u, 0, 0); 3 + ( 2p − 12 + 2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] b = pb Pbam (55) (2; 5) + (0, 2u, 0); 3 + (0, 2p − 12 , 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] c = pc Pbam (55) 2; (3; 5) + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups I Minimal translationengleiche supergroups [2] P4/mbm (127); [2] P42 /mbc (135) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] Aeam (64, Cmce); [2] Bbem (64, Cmce); [2] Cmmm (65); [2] Ibam (72)

• Decreased unit cell [2] a = 12 a Pbmm (51, Pmma); [2] b = 12 b Pmam (51, Pmma) Copyright © 2011 International Union of Crystallography

205

0, 0, 1/2 0, 0, 1/2 0, 0, 1/2 0, 0, 1/2

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 56, p. 206.

Pccn

D102h

P 21/c 21/c 2/n

No. 56

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

e

1

(1) x, y, z (5) x, ¯ y, ¯ z¯

(2) x¯ + 12 , y¯ + 12 , z (6) x + 12 , y + 12 , z¯

(3) x, ¯ y + 12 , z¯ + 12 (7) x, y¯ + 12 , z + 12

I Maximal translationengleiche subgroups [2] Pc21 n (33, Pna21 ) 1; 3; 6; 8 [2] P21 cn (33, Pna21 ) 1; 4; 6; 7 [2] Pcc2 (27) 1; 2; 7; 8 1; 2; 3; 4 [2] P21 21 2 (18) 1; 3; 5; 7 [2] P121 /c1 (14) 1; 4; 5; 8 [2] P21 /c11 (14, P121 /c1) [2] P112/n (13, P112/a) 1; 2; 5; 6

(4) x + 12 , y, ¯ z¯ + 12 (8) x¯ + 12 , y, z + 12

c, a, b c, b, −a

0, 0, 1/4 0, 0, 1/4 1/4, 1/4, 0 1/4, 1/4, 1/4

−b, a, c −a − b, a, c

II Maximal klassengleiche subgroups

• Enlarged unit cell [3] a = 3a Pccn (56) Pccn (56) Pccn (56) [3] b = 3b Pccn (56) Pccn (56) Pccn (56) [3] c = 3c Pccn (56) Pccn (56) Pccn (56)

3; 5; 2 + (1, 0, 0) 2 + (3, 0, 0); (3; 5) + (2, 0, 0) 2 + (5, 0, 0); (3; 5) + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

5; (2; 3) + (0, 1, 0) 2 + (0, 3, 0); 3 + (0, 1, 0); 5 + (0, 2, 0) 2 + (0, 5, 0); 3 + (0, 1, 0); 5 + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2; 5; 3 + (0, 0, 1) 2; 3 + (0, 0, 3); 5 + (0, 0, 2) 2; 3 + (0, 0, 5); 5 + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

2 + ( 2p − 12 + 2u, 0, 0); (3; 5) + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

pa, b, c

u, 0, 0

2 + (0, 2p − 12 + 2u, 0); 3 + (0, 2p − 12 , 0); 5 + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, pb, c

0, u, 0

2; 3 + (0, 0, 2p − 12 + 2u); 5 + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, b, pc

0, 0, u

• Series of maximal isomorphic subgroups

[p] a = pa Pccn (56)

[p] b = pb Pccn (56)

[p] c = pc Pccn (56)

I Minimal translationengleiche supergroups [2] P4/ncc (130); [2] P42 /ncm (138) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] Aema (64, Cmce); [2] Bmeb (64, Cmce); [2] Cccm (66); [2] Ibam (72) • Decreased unit cell

[2] a = 12 a Pccb (54, Pcca); [2] b = 12 b Pcca (54); [2] c = 12 c Pmmn (59)

Copyright © 2011 International Union of Crystallography

206

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 57, p. 207.

D112h

P 2/b 21/c 21/m

No. 57

Pbcm

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

e

1

(1) x, y, z (5) x, ¯ y, ¯ z¯

(2) x, ¯ y, ¯ z + 12 (6) x, y, z¯ + 12

(3) x, ¯ y + 12 , z¯ + 12 (7) x, y¯ + 12 , z + 12

I Maximal translationengleiche subgroups 1; 2; 7; 8 [2] Pbc21 (29, Pca21 ) [2] P2cm (28, Pma2) 1; 4; 6; 7 1; 3; 6; 8 [2] Pb21 m (26, Pmc21 ) 1; 2; 3; 4 [2] P221 21 (18, P21 21 2) 1; 3; 5; 7 [2] P121 /c1 (14) [2] P2/b11 (13, P12/c1) 1; 4; 5; 8 1; 2; 5; 6 [2] P1121 /m (11, P1121 /m)

(4) x, y¯ + 12 , z¯ (8) x, ¯ y + 12 , z

−b, a, c c, b, −a c, a, b b, c, a

0, 1/4, 0 0, 0, 1/4 0, 1/4, 0

c, a, b

II Maximal klassengleiche subgroups

• Enlarged unit cell [2] a = 2a Pbnm (62, Pnma) Pbnm (62, Pnma) Pbca (61) Pbca (61) Pbna (60, Pbcn) Pbna (60, Pbcn) Pbcm (57) Pbcm (57) [3] a = 3a Pbcm (57) Pbcm (57) Pbcm (57) [3] b = 3b Pbcm (57) Pbcm (57) Pbcm (57) [3] c = 3c Pbcm (57) Pbcm (57) Pbcm (57)

2; 5; 3 + (1, 0, 0) 3; (2; 5) + (1, 0, 0) 3; 5; 2 + (1, 0, 0) 2; (3; 5) + (1, 0, 0) 5; (2; 3) + (1, 0, 0) 2; 3; 5 + (1, 0, 0) 2; 3; 5 (2; 3; 5) + (1, 0, 0)

b, c, 2a b, c, 2a 2a, b, c 2a, b, c c, 2a, b c, 2a, b 2a, b, c 2a, b, c

2; 3; 5 (2; 3; 5) + (2, 0, 0) (2; 3; 5) + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 5; 3 + (0, 1, 0) (2; 5) + (0, 2, 0); 3 + (0, 1, 0) (2; 5) + (0, 4, 0); 3 + (0, 1, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

5; (2; 3) + (0, 0, 1) 2 + (0, 0, 1); 3 + (0, 0, 3); 5 + (0, 0, 2) 2 + (0, 0, 1); 3 + (0, 0, 5); 5 + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

1/2, 0, 0 1/2, 0, 0 1/2, 0, 0 1/2, 0, 0

• Series of maximal isomorphic subgroups [p] a = pa Pbcm (57) (2; 3; 5) + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] b = pb Pbcm (57) (2; 5) + (0, 2u, 0); 3 + (0, 2p − 12 , 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] c = pc Pbcm (57) 2 + (0, 0, 2p − 12 ); 3 + (0, 0, 2p − 12 + 2u); 5 + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups

pa, b, c

u, 0, 0

a, pb, c

0, u, 0

a, b, pc

0, 0, u

I Minimal translationengleiche supergroups

none

II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] Cmcm (63); [2] Bbem (64, Cmce); [2] Aemm (67, Cmme); [2] Ibam (72)

• Decreased unit cell [2] b = 12 b Pmcm (51, Pmma); [2] c = 12 c Pbmm (51, Pmma) Copyright © 2011 International Union of Crystallography

207

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 58, p. 208.

Pnnm

D122h

P 21/n 21/n 2/m

No. 58

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

h

1

(1) x, y, z (5) x, ¯ y, ¯ z¯

(2) x, ¯ y, ¯z (6) x, y, z¯

(3) x¯ + 12 , y + 12 , z¯ + 12 (7) x + 12 , y¯ + 12 , z + 12

I Maximal translationengleiche subgroups [2] Pnn2 (34) 1; 2; 7; 8 [2] Pn21 m (31, Pmn21 ) 1; 3; 6; 8 1; 4; 6; 7 [2] P21 nm (31, Pmn21 ) [2] P21 21 2 (18) 1; 2; 3; 4 1; 3; 5; 7 [2] P121 /n1 (14, P121 /c1) 1; 4; 5; 8 [2] P21 /n11 (14, P121 /c1) [2] P112/m (10) 1; 2; 5; 6

(4) x + 12 , y¯ + 12 , z¯ + 12 (8) x¯ + 12 , y + 12 , z + 12

c, a, b −c, b, a c, b, −a − c −b, a, b + c

1/4, 0, 0 0, 1/4, 0 0, 0, 1/4

II Maximal klassengleiche subgroups

• Enlarged unit cell [3] a = 3a Pnnm (58) Pnnm (58) Pnnm (58) [3] b = 3b Pnnm (58) Pnnm (58) Pnnm (58) [3] c = 3c Pnnm (58) Pnnm (58) Pnnm (58)

2; 5; 3 + (1, 0, 0) (2; 5) + (2, 0, 0); 3 + (3, 0, 0) (2; 5) + (4, 0, 0); 3 + (5, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 5; 3 + (0, 1, 0) (2; 5) + (0, 2, 0); 3 + (0, 1, 0) (2; 5) + (0, 4, 0); 3 + (0, 1, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2; 5; 3 + (0, 0, 1) 2; 3 + (0, 0, 3); 5 + (0, 0, 2) 2; 3 + (0, 0, 5); 5 + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

(2; 5) + (2u, 0, 0); 3 + ( 2p − 12 + 2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

pa, b, c

u, 0, 0

(2; 5) + (0, 2u, 0); 3 + (0, 2p − 12 , 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, pb, c

0, u, 0

2; 3 + (0, 0, 2p − 12 + 2u); 5 + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, b, pc

0, 0, u

• Series of maximal isomorphic subgroups

[p] a = pa Pnnm (58)

[p] b = pb Pnnm (58) [p] c = pc Pnnm (58)

I Minimal translationengleiche supergroups [2] P4/mnc (128); [2] P42 /mnm (136) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] Amam (63, Cmcm); [2] Bbmm (63, Cmcm); [2] Cccm (66); [2] Immm (71) • Decreased unit cell [2] a = 12 a Pncm (53, Pmna); [2] b = 12 b Pcnm (53, Pmna); [2] c = 12 c Pbam (55)

Copyright © 2011 International Union of Crystallography

208

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 59, pp. 209–211.

D132h

P 21/m 21/m 2/n

ORIGIN CHOICE

Pmmn

No. 59

1, Origin at m m 2/n, at 14 , 14 , 0 from 1¯

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

g

1

(1) x, y, z (5) x¯ + 12 , y¯ + 12 , z¯

(2) x, ¯ y, ¯z (6) x + 12 , y + 12 , z¯

I Maximal translationengleiche subgroups [2] Pm21 n (31, Pmn21 ) 1; 3; 6; 8 1; 4; 6; 7 [2] P21 mn (31, Pmn21 ) [2] Pmm2 (25) 1; 2; 7; 8 1; 2; 3; 4 [2] P21 21 2 (18) [2] P112/n (13, P112/a) 1; 2; 5; 6 1; 3; 5; 7 [2] P121 /m1 (11) 1; 4; 5; 8 [2] P21 /m11 (11, P121 /m1)

(3) x¯ + 12 , y + 12 , z¯ (7) x, y, ¯z

(4) x + 12 , y¯ + 12 , z¯ (8) x, ¯ y, z

a, −c, b b, c, a −a − b, a, c c, a, b

1/4, 1/4, 0 1/4, 1/4, 0 1/4, 1/4, 0

II Maximal klassengleiche subgroups

• Enlarged unit cell [2] c = 2c Pcmn (62, Pnma) Pcmn (62, Pnma) Pmcn (62, Pnma) Pmcn (62, Pnma) Pmmn (59) Pmmn (59) Pccn (56) Pccn (56) [3] a = 3a Pmmn (59) Pmmn (59) Pmmn (59) [3] b = 3b Pmmn (59) Pmmn (59) Pmmn (59) [3] c = 3c Pmmn (59) Pmmn (59) Pmmn (59)

3; 5; 2 + (0, 0, 1) (2; 3; 5) + (0, 0, 1) 5; (2; 3) + (0, 0, 1) 3; (2; 5) + (0, 0, 1) 2; 3; 5 2; (3; 5) + (0, 0, 1) 2; 5; 3 + (0, 0, 1) 2; 3; 5 + (0, 0, 1)

2c, b, −a 2c, b, −a 2c, a, b 2c, a, b a, b, 2c a, b, 2c a, b, 2c a, b, 2c

0, 0, 1/2 1/4, 1/4, 0 1/4, 1/4, 1/2

2; (3; 5) + (1, 0, 0) 2 + (2, 0, 0); (3; 5) + (3, 0, 0) 2 + (4, 0, 0); (3; 5) + (5, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; (3; 5) + (0, 1, 0) 2 + (0, 2, 0); 3 + (0, 1, 0); 5 + (0, 3, 0) 2 + (0, 4, 0); 3 + (0, 1, 0); 5 + (0, 5, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2; 3; 5 2; (3; 5) + (0, 0, 2) 2; (3; 5) + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

2 + (2u, 0, 0); (3; 5) + ( 2p − 12 + 2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

pa, b, c

u, 0, 0

2 + (0, 2u, 0); 3 + (0, 2p − 12 , 0); 5 + (0, 2p − 12 + 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, pb, c

0, u, 0

2; (3; 5) + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, b, pc

0, 0, u

1/4, 1/4, 0 1/4, 1/4, 1/2 1/4, 1/4, 0 1/4, 1/4, 1/2

• Series of maximal isomorphic subgroups

[p] a = pa Pmmn (59)

[p] b = pb Pmmn (59)

[p] c = pc Pmmn (59)

Copyright © 2011 International Union of Crystallography

209

Pmmn

ORIGIN CHOICE

1

No. 59

CONTINUED

I Minimal translationengleiche supergroups [2] P4/nmm (129); [2] P42 /nmc (137) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] Amma (63, Cmcm); [2] Bmmb (63, Cmcm); [2] Cmmm (65); [2] Immm (71) • Decreased unit cell

[2] a = 12 a Pmmb (51, Pmma); [2] b = 12 b Pmma (51)

Pmmn

ORIGIN CHOICE

2

No. 59

I Minimal translationengleiche supergroups [2] P4/nmm (129); [2] P42 /nmc (137) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] Amma (63, Cmcm); [2] Bmmb (63, Cmcm); [2] Cmmm (65); [2] Immm (71) • Decreased unit cell [2] a = 12 a Pmmb (51, Pmma); [2] b = 12 b Pmma (51)

210

(Continued from the facing page)

D132h

P 21/m 21/m 2/n

ORIGIN CHOICE

Pmmn

No. 59

2, Origin at 1¯ at 21 21 n, at − 14 , − 14 , 0 from m m 2

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

g

1

(1) x, y, z (5) x, ¯ y, ¯ z¯

(2) x¯ + 12 , y¯ + 12 , z (6) x + 12 , y + 12 , z¯

I Maximal translationengleiche subgroups [2] Pm21 n (31, Pmn21 ) 1; 3; 6; 8 1; 4; 6; 7 [2] P21 mn (31, Pmn21 ) [2] Pmm2 (25) 1; 2; 7; 8 1; 2; 3; 4 [2] P21 21 2 (18) [2] P112/n (13, P112/a) 1; 2; 5; 6 1; 3; 5; 7 [2] P121 /m1 (11) 1; 4; 5; 8 [2] P21 /m11 (11, P121 /m1)

(3) x, ¯ y + 12 , z¯ (7) x, y¯ + 12 , z

(4) x + 12 , y, ¯ z¯ 1 (8) x¯ + 2 , y, z

a, −c, b b, c, a −a − b, a, c

1/4, 1/4, 0 1/4, 1/4, 0 1/4, 1/4, 0 1/4, 1/4, 0

c, a, b

II Maximal klassengleiche subgroups

• Enlarged unit cell [2] c = 2c Pcmn (62, Pnma) Pcmn (62, Pnma) Pmcn (62, Pnma) Pmcn (62, Pnma) Pmmn (59) Pmmn (59) Pccn (56) Pccn (56) [3] a = 3a Pmmn (59) Pmmn (59) Pmmn (59) [3] b = 3b Pmmn (59) Pmmn (59) Pmmn (59) [3] c = 3c Pmmn (59) Pmmn (59) Pmmn (59)

3; 5; 2 + (0, 0, 1) (2; 3; 5) + (0, 0, 1) 5; (2; 3) + (0, 0, 1) 3; (2; 5) + (0, 0, 1) 2; 3; 5 2; (3; 5) + (0, 0, 1) 2; 5; 3 + (0, 0, 1) 2; 3; 5 + (0, 0, 1)

2c, b, −a 2c, b, −a 2c, a, b 2c, a, b a, b, 2c a, b, 2c a, b, 2c a, b, 2c

3; 5; 2 + (1, 0, 0) 2 + (3, 0, 0); (3; 5) + (2, 0, 0) 2 + (5, 0, 0); (3; 5) + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

5; (2; 3) + (0, 1, 0) 2 + (0, 3, 0); 3 + (0, 1, 0); 5 + (0, 2, 0) 2 + (0, 5, 0); 3 + (0, 1, 0); 5 + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2; 3; 5 2; (3; 5) + (0, 0, 2) 2; (3; 5) + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

2 + ( 2p − 12 + 2u, 0, 0); (3; 5) + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

pa, b, c

u, 0, 0

2 + (0, 2p − 12 + 2u, 0); 3 + (0, 2p − 12 , 0); 5 + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, pb, c

0, u, 0

2; (3; 5) + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, b, pc

0, 0, u

0, 0, 1/2 0, 0, 1/2 0, 0, 1/2 0, 0, 1/2

• Series of maximal isomorphic subgroups

[p] a = pa Pmmn (59)

[p] b = pb Pmmn (59)

[p] c = pc Pmmn (59)

(Continued on the facing page)

211

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 60, p. 212.

Pbcn

D142h

P 21/b 2/c 21/n

No. 60

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

d

1

(1) x, y, z (5) x, ¯ y, ¯ z¯

(2) x¯ + 12 , y¯ + 12 , z + 12 (6) x + 12 , y + 12 , z¯ + 12

I Maximal translationengleiche subgroups 1; 4; 6; 7 [2] P21 cn (33, Pna21 ) [2] Pb2n (30, Pnc2) 1; 3; 6; 8 1; 2; 7; 8 [2] Pbc21 (29, Pca21 ) [2] P21 221 (18, P21 21 2) 1; 2; 3; 4 1; 2; 5; 6 [2] P1121 /n (14, P1121 /a) 1; 4; 5; 8 [2] P21 /b11 (14, P121 /c1) [2] P12/c1 (13) 1; 3; 5; 7

(3) x, ¯ y, z¯ + 12 (7) x, y, ¯ z + 12

(4) x + 12 , y¯ + 12 , z¯ (8) x¯ + 12 , y + 12 , z

c, b, −a c, a, b −b, a, c c, a, b −a − b, a, c c, a, b

0, 1/4, 0 0, 0, 1/4 1/4, 1/4, 0 0, 1/4, 1/4

II Maximal klassengleiche subgroups

• Enlarged unit cell [3] a = 3a Pbcn (60) Pbcn (60) Pbcn (60) [3] b = 3b Pbcn (60) Pbcn (60) Pbcn (60) [3] c = 3c Pbcn (60) Pbcn (60) Pbcn (60)

3; 5; 2 + (1, 0, 0) 2 + (3, 0, 0); (3; 5) + (2, 0, 0) 2 + (5, 0, 0); (3; 5) + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

3; 5; 2 + (0, 1, 0) 3; 2 + (0, 3, 0); 5 + (0, 2, 0) 3; 2 + (0, 5, 0); 5 + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

5; (2; 3) + (0, 0, 1) 2 + (0, 0, 1); 3 + (0, 0, 3); 5 + (0, 0, 2) 2 + (0, 0, 1); 3 + (0, 0, 5); 5 + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

2 + ( 2p − 12 + 2u, 0, 0); (3; 5) + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

pa, b, c

u, 0, 0

3; 2 + (0, 2p − 12 + 2u, 0); 5 + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, pb, c

0, u, 0

2 + (0, 0, 2p − 12 ); 3 + (0, 0, 2p − 12 + 2u); 5 + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, b, pc

0, 0, u

• Series of maximal isomorphic subgroups

[p] a = pa Pbcn (60)

[p] b = pb Pbcn (60) [p] c = pc Pbcn (60)

I Minimal translationengleiche supergroups

none

II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] Cmcm (63); [2] Aema (64, Cmce); [2] Bbeb (68, Ccce); [2] Ibam (72) • Decreased unit cell [2] c = 12 c Pbmn (53, Pmna); [2] a = 12 a Pbcb (54, Pcca); [2] b = 12 b Pmca (57, Pbcm)

Copyright © 2011 International Union of Crystallography

212

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 61, p. 213.

D152h

P 21/b 21/c 21/a

No. 61

Pbca

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

c

1

(1) x, y, z (5) x, ¯ y, ¯ z¯

(2) x¯ + 12 , y, ¯ z + 12 (6) x + 12 , y, z¯ + 12

(3) x, ¯ y + 12 , z¯ + 12 (7) x, y¯ + 12 , z + 12

I Maximal translationengleiche subgroups 1; 2; 7; 8 [2] Pbc21 (29, Pca21 ) 1; 3; 6; 8 [2] Pb21 a (29, Pca21 ) 1; 4; 6; 7 [2] P21 ca (29, Pca21 ) 1; 2; 3; 4 [2] P21 21 21 (19) 1; 2; 5; 6 [2] P1121 /a (14) 1; 3; 5; 7 [2] P121 /c1 (14) 1; 4; 5; 8 [2] P21 /b11 (14, P121 /c1)

(4) x + 12 , y¯ + 12 , z¯ (8) x¯ + 12 , y + 12 , z

−b, a, c a, −c, b c, b, −a

1/4, 0, 0 0, 0, 1/4 0, 1/4, 0

c, a, b

II Maximal klassengleiche subgroups

• Enlarged unit cell [3] a = 3a Pbca (61) Pbca (61) Pbca (61) [3] b = 3b Pbca (61) Pbca (61) Pbca (61) [3] c = 3c Pbca (61) Pbca (61) Pbca (61)

3; 5; 2 + (1, 0, 0) 2 + (3, 0, 0); (3; 5) + (2, 0, 0) 2 + (5, 0, 0); (3; 5) + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 5; 3 + (0, 1, 0) (2; 5) + (0, 2, 0); 3 + (0, 1, 0) (2; 5) + (0, 4, 0); 3 + (0, 1, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

5; (2; 3) + (0, 0, 1) 2 + (0, 0, 1); 3 + (0, 0, 3); 5 + (0, 0, 2) 2 + (0, 0, 1); 3 + (0, 0, 5); 5 + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

2 + ( 2p − 12 + 2u, 0, 0); (3; 5) + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

pa, b, c

u, 0, 0

(2; 5) + (0, 2u, 0); 3 + (0, 2p − 12 , 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, pb, c

0, u, 0

2 + (0, 0, 2p − 12 ); 3 + (0, 0, 2p − 12 + 2u); 5 + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, b, pc

0, 0, u

• Series of maximal isomorphic subgroups

[p] a = pa Pbca (61)

[p] b = pb Pbca (61) [p] c = pc Pbca (61)

I Minimal translationengleiche supergroups [3] Pa3¯ (205) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] Aema (64, Cmce); [2] Bbem (64, Cmce); [2] Cmce (64); [2] Ibca (73) • Decreased unit cell [2] a = 12 a Pbcm (57); [2] b = 12 b Pmca (57, Pbcm); [2] c = 12 c Pbma (57, Pbcm)

Copyright © 2011 International Union of Crystallography

213

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 62, p. 214.

Pnma

D162h

P 21/n 21/m 21/a

No. 62

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

d

1

(1) x, y, z (5) x, ¯ y, ¯ z¯

(2) x¯ + 12 , y, ¯ z + 12 (6) x + 12 , y, z¯ + 12

(3) x, ¯ y + 12 , z¯ (7) x, y¯ + 12 , z

I Maximal translationengleiche subgroups [2] Pn21 a (33, Pna21 ) 1; 3; 6; 8 1; 2; 7; 8 [2] Pnm21 (31, Pmn21 ) 1; 4; 6; 7 [2] P21 ma (26, Pmc21 ) 1; 2; 3; 4 [2] P21 21 21 (19) 1; 2; 5; 6 [2] P1121 /a (14) 1; 4; 5; 8 [2] P21 /n11 (14, P121 /c1) 1; 3; 5; 7 [2] P121 /m1 (11)

(4) x + 12 , y¯ + 12 , z¯ + 12 (8) x¯ + 12 , y + 12 , z + 12

a, −c, b −b, a, c b, c, a

1/4, 1/4, 0 0, 1/4, 1/4 0, 0, 1/4

−b, a, b + c

II Maximal klassengleiche subgroups

• Enlarged unit cell [3] a = 3a Pnma (62) Pnma (62) Pnma (62) [3] b = 3b Pnma (62) Pnma (62) Pnma (62) [3] c = 3c Pnma (62) Pnma (62) Pnma (62)

3; 5; 2 + (1, 0, 0) 2 + (3, 0, 0); (3; 5) + (2, 0, 0) 2 + (5, 0, 0); (3; 5) + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 5; 3 + (0, 1, 0) (2; 5) + (0, 2, 0); 3 + (0, 1, 0) (2; 5) + (0, 4, 0); 3 + (0, 1, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

3; 5; 2 + (0, 0, 1) 2 + (0, 0, 1); (3; 5) + (0, 0, 2) 2 + (0, 0, 1); (3; 5) + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

2 + ( 2p − 12 + 2u, 0, 0); (3; 5) + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

pa, b, c

u, 0, 0

(2; 5) + (0, 2u, 0); 3 + (0, 2p − 12 , 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, pb, c

0, u, 0

2 + (0, 0, 2p − 12 ); (3; 5) + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, b, pc

0, 0, u

• Series of maximal isomorphic subgroups

[p] a = pa Pnma (62)

[p] b = pb Pnma (62) [p] c = pc Pnma (62)

I Minimal translationengleiche supergroups

none

II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] Amma (63, Cmcm); [2] Bbmm (63, Cmcm); [2] Ccme (64, Cmce); [2] Imma (74)

• Decreased unit cell

[2] b = 12 b Pcma (55, Pbam); [2] c = 12 c Pbma (57, Pbcm); [2] a = 12 a Pnmm (59, Pmmn)

Copyright © 2011 International Union of Crystallography

214

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 63, pp. 215–216.

D172h

C 2/m 2/c 21/m

Cmcm

No. 63

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t( 21 , 12 , 0); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

16

h

(0, 0, 0)+ ( 21 , 12 , 0)+

1

(1) x, y, z (5) x, ¯ y, ¯ z¯

(2) x, ¯ y, ¯ z + 12 (6) x, y, z¯ + 12

I Maximal translationengleiche subgroups [2] C2cm (40, Ama2) (1; 4; 6; 7)+ [2] Cm2m (38, Amm2) (1; 3; 6; 8)+ [2] Cmc21 (36) (1; 2; 7; 8)+ (1; 2; 3; 4)+ [2] C2221 (20) [2] C12/c1 (15) (1; 3; 5; 7)+ [2] C2/m11 (12, C12/m1) (1; 4; 5; 8)+ (1; 2; 5; 6)+ [2] C1121 /m (11, P1121 /m)

(3) x, ¯ y, z¯ + 12 (7) x, y, ¯ z + 12

(4) x, y, ¯ z¯ (8) x, ¯ y, z

c, b, −a c, a, b

0, 0, 1/4

−b, a, c 1/2(a − b), 1/2(a + b), c

II Maximal klassengleiche subgroups

• Loss of centring translations [2] [2] [2] [2] [2] [2] [2] [2]

Pbnm (62, Pnma) Pmcn (62, Pnma) Pbcn (60) Pmnm (59, Pmmn) Pmnn (58, Pnnm) Pbcm (57) Pbnn (52, Pnna) Pmcm (51, Pmma)

1; 1; 1; 1; 1; 1; 1; 1;

2; 2; 3; 3; 4; 4; 2; 2;

5; 7; 5; 6; 5; 6; 3; 3;

6; 8; 7; 8; 8; 7; 4; 4;

(3; 4; 7; 8) + ( 21 , 12 , 0) (3; 4; 5; 6) + ( 21 , 12 , 0) (2; 4; 6; 8) + ( 21 , 12 , 0) (2; 4; 5; 7) + ( 21 , 12 , 0) (2; 3; 6; 7) + ( 21 , 12 , 0) (2; 3; 5; 8) + ( 21 , 12 , 0) (5; 6; 7; 8) + ( 21 , 12 , 0) 5; 6; 7; 8

b, c, a c, a, b c, a, b b, c, a b, c, a c, a, b

1/4, 1/4, 0 1/4, 1/4, 0 1/4, 1/4, 0 1/4, 1/4, 0

• Enlarged unit cell

[3] a = 3a Cmcm (63) Cmcm (63) Cmcm (63) [3] b = 3b Cmcm (63) Cmcm (63) Cmcm (63) [3] c = 3c Cmcm (63) Cmcm (63) Cmcm (63)

2; 3; 5 (2; 3; 5) + (2, 0, 0) (2; 3; 5) + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 3; 5 3; (2; 5) + (0, 2, 0) 3; (2; 5) + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

5; (2; 3) + (0, 0, 1) 2 + (0, 0, 1); 3 + (0, 0, 3); 5 + (0, 0, 2) 2 + (0, 0, 1); 3 + (0, 0, 5); 5 + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

pa, b, c

u, 0, 0

a, pb, c

0, u, 0

a, b, pc

0, 0, u

• Series of maximal isomorphic subgroups [p] a = pa Cmcm (63) (2; 3; 5) + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] b = pb Cmcm (63) 3; (2; 5) + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] c = pc Cmcm (63) 2 + (0, 0, 2p − 12 ); 3 + (0, 0, 2p − 12 + 2u); 5 + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups

Copyright © 2011 International Union of Crystallography

215

Cmcm

No. 63

CONTINUED

I Minimal translationengleiche supergroups [3] P63 /mcm (193); [3] P63 /mmc (194) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] Fmmm (69) • Decreased unit cell

[2] a = 12 a, b = 12 b Pmcm (51, Pmma); [2] c = 12 c Cmmm (65)

Cmce

No. 64

(Continued from the facing page)

I Minimal translationengleiche supergroups

none

II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] Fmmm (69)

• Decreased unit cell

[2] a = 12 a, b = 12 b Pmcm (51, Pmma); [2] c = 12 c Cmme (67)

216

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 64, pp. 216–217.

D182h

C 2/m 2/c 21/e

Cmce

No. 64

Former space-group symbol Cmca

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t( 21 , 12 , 0); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

16

g

(0, 0, 0)+ ( 21 , 12 , 0)+

1

(1) x, y, z (5) x, ¯ y, ¯ z¯

(2) x, ¯ y¯ + 12 , z + 12 (6) x, y + 12 , z¯ + 12

I Maximal translationengleiche subgroups [2] C2ce (41, Aea2) (1; 4; 6; 7)+ [2] Cm2e (39, Aem2) (1; 3; 6; 8)+ [2] Cmc21 (36) (1; 2; 7; 8)+ (1; 2; 3; 4)+ [2] C2221 (20) [2] C12/c1 (15) (1; 3; 5; 7)+ (1; 2; 5; 6)+ [2] C1121 /e (14, P1121 /a) [2] C2/m11 (12, C12/m1) (1; 4; 5; 8)+

(3) x, ¯ y + 12 , z¯ + 12 (7) x, y¯ + 12 , z + 12

(4) x, y, ¯ z¯ (8) x, ¯ y, z

c, b, −a c, a, b

a, 1/2(−a + b), c −b, a, c

1/4, 0, 1/4 0, 1/4, 0 1/4, 0, 0 1/4, 1/4, 0

II Maximal klassengleiche subgroups

• Loss of centring translations [2] [2] [2] [2] [2] [2] [2] [2]

Pmnb (62, Pnma) Pbca (61) Pbna (60, Pbcn) Pmca (57, Pbcm) Pbnb (56, Pccn) Pmcb (55, Pbam) Pbcb (54, Pcca) Pmna (53)

1; 1; 1; 1; 1; 1; 1; 1;

3; 3; 2; 2; 2; 2; 4; 4;

6; 5; 3; 7; 5; 3; 6; 5;

8; 7; 4; 8; 6; 4; 7; 8;

(2; 4; 5; 7) + ( 21 , 12 , 0) (2; 4; 6; 8) + ( 21 , 12 , 0) (5; 6; 7; 8) + ( 21 , 12 , 0) (3; 4; 5; 6) + ( 21 , 12 , 0) (3; 4; 7; 8) + ( 21 , 12 , 0) 5; 6; 7; 8 (2; 3; 5; 8) + ( 21 , 12 , 0) (2; 3; 6; 7) + ( 21 , 12 , 0)

−b, a, c

1/4, 1/4, 0

c, a, b b, c, a c, a, b b, c, a c, a, b

1/4, 1/4, 0 1/4, 1/4, 0

1/4, 1/4, 0

• Enlarged unit cell

[3] a = 3a Cmce (64) Cmce (64) Cmce (64) [3] b = 3b Cmce (64) Cmce (64) Cmce (64) [3] c = 3c Cmce (64) Cmce (64) Cmce (64)

2; 3; 5 (2; 3; 5) + (2, 0, 0) (2; 3; 5) + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

5; (2; 3) + (0, 1, 0) 2 + (0, 3, 0); 3 + (0, 1, 0); 5 + (0, 2, 0) 2 + (0, 5, 0); 3 + (0, 1, 0); 5 + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

5; (2; 3) + (0, 0, 1) 2 + (0, 0, 1); 3 + (0, 0, 3); 5 + (0, 0, 2) 2 + (0, 0, 1); 3 + (0, 0, 5); 5 + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

(2; 3; 5) + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

pa, b, c

u, 0, 0

2 + (0, 2p − 12 + 2u, 0); 3 + (0, 2p − 12 , 0); 5 + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, pb, c

0, u, 0

2 + (0, 0, 2p − 12 ); 3 + (0, 0, 2p − 12 + 2u); 5 + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, b, pc

0, 0, u

• Series of maximal isomorphic subgroups

[p] a = pa Cmce (64)

[p] b = pb Cmce (64)

[p] c = pc Cmce (64)

(Continued on the facing page)

Copyright © 2011 International Union of Crystallography

217

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 65, pp. 218–219.

Cmmm

No. 65

D192h

C 2/m 2/m 2/m

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t( 21 , 12 , 0); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

16

r

(0, 0, 0)+ ( 21 , 12 , 0)+

1

(1) x, y, z (5) x, ¯ y, ¯ z¯

(2) x, ¯ y, ¯z (6) x, y, z¯

I Maximal translationengleiche subgroups [2] Cm2m (38, Amm2) (1; 3; 6; 8)+ [2] C2mm (38, Amm2) (1; 4; 6; 7)+ [2] Cmm2 (35) (1; 2; 7; 8)+ [2] C222 (21) (1; 2; 3; 4)+ [2] C12/m1 (12) (1; 3; 5; 7)+ [2] C2/m11 (12, C12/m1) (1; 4; 5; 8)+ [2] C112/m (10, P112/m) (1; 2; 5; 6)+

(3) x, ¯ y, z¯ (7) x, y, ¯z

(4) x, y, ¯ z¯ (8) x, ¯ y, z

c, a, b c, b, −a

−b, a, c 1/2(a − b), 1/2(a + b), c

II Maximal klassengleiche subgroups

• Loss of centring translations [2] [2] [2] [2] [2] [2] [2] [2]

Pmmn (59) Pbam (55) Pbmn (53, Pmna) Pman (53, Pmna) Pmam (51, Pmma) Pbmm (51, Pmma) Pban (50) Pmmm (47)

• Enlarged unit cell [2] c = 2c Ibmm (74, Imma) Ibmm (74, Imma) Imam (74, Imma) Imam (74, Imma) Ibam (72) Ibam (72) Immm (71) Immm (71) Cccm (66) Cccm (66) Cmmm (65) Cmmm (65) Ccmm (63, Cmcm) Ccmm (63, Cmcm) Cmcm (63) Cmcm (63) [3] a = 3a Cmmm (65) Cmmm (65) Cmmm (65) [3] b = 3b Cmmm (65) Cmmm (65) Cmmm (65) [3] c = 3c Cmmm (65) Cmmm (65) Cmmm (65)

1; 1; 1; 1; 1; 1; 1; 1;

2; 2; 3; 4; 3; 4; 2; 2;

7; 5; 5; 5; 6; 6; 3; 3;

8; 6; 7; 8; 8; 7; 4; 4;

(3; 4; 5; 6) + ( 21 , 12 , 0) (3; 4; 7; 8) + ( 21 , 12 , 0) (2; 4; 6; 8) + ( 21 , 12 , 0) (2; 3; 6; 7) + ( 21 , 12 , 0) (2; 4; 5; 7) + ( 21 , 12 , 0) (2; 3; 5; 8) + ( 21 , 12 , 0) (5; 6; 7; 8) + ( 21 , 12 , 0) 5; 6; 7; 8

1/4, 1/4, 0 b, c, a a, −c, b a, −c, b b, c, a

1/4, 1/4, 0 1/4, 1/4, 0 1/4, 1/4, 0

3; 5; 2 + (0, 0, 1) (2; 3; 5) + (0, 0, 1) 5; (2; 3) + (0, 0, 1) 3; (2; 5) + (0, 0, 1) 2; 5; 3 + (0, 0, 1) 2; 3; 5 + (0, 0, 1) 2; 3; 5 2; (3; 5) + (0, 0, 1) 2; 5; 3 + (0, 0, 1) 2; 3; 5 + (0, 0, 1) 2; 3; 5 2; (3; 5) + (0, 0, 1) 3; 5; 2 + (0, 0, 1) (2; 3; 5) + (0, 0, 1) 5; (2; 3) + (0, 0, 1) 3; (2; 5) + (0, 0, 1)

b, 2c, a b, 2c, a a, −2c, b a, −2c, b a, b, 2c a, b, 2c a, b, 2c a, b, 2c a, b, 2c a, b, 2c a, b, 2c a, b, 2c −b, a, 2c −b, a, 2c a, b, 2c a, b, 2c

2; 3; 5 (2; 3; 5) + (2, 0, 0) (2; 3; 5) + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 3; 5 3; (2; 5) + (0, 2, 0) 3; (2; 5) + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2; 3; 5 2; (3; 5) + (0, 0, 2) 2; (3; 5) + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

Copyright © 2011 International Union of Crystallography

218

0, 0, 1/2 0, 0, 1/2 0, 0, 1/2 0, 0, 1/2 0, 0, 1/2 0, 0, 1/2 0, 0, 1/2 0, 0, 1/2

CONTINUED

Cmmm

No. 65

• Series of maximal isomorphic subgroups [p] a = pa Cmmm (65) (2; 3; 5) + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] b = pb Cmmm (65) 3; (2; 5) + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] c = pc Cmmm (65) 2; (3; 5) + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups

pa, b, c

u, 0, 0

a, pb, c

0, u, 0

a, b, pc

0, 0, u

I Minimal translationengleiche supergroups [2] P4/mmm (123); [2] P4/mbm (127); [2] P42 /mcm (132); [2] P42 /mnm (136); [3] P6/mmm (191) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] Fmmm (69)

• Decreased unit cell

[2] a = 12 a, b = 12 b Pmmm (47)

No. 66

(Continued from the following page)

I Minimal translationengleiche supergroups [2] P4/mcc (124); [2] P4/mnc (128); [2] P42 /mmc (131); [2] P42 /mbc (135); [3] P6/mcc (192) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] Fmmm (69)

• Decreased unit cell [2] a = 12 a, b = 12 b Pccm (49); [2] c = 12 c Cmmm (65)

219

Cccm

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 66, pp. 219–220.

Cccm

No. 66

D202h

C 2/c 2/c 2/m

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t( 21 , 12 , 0); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

16

m

(0, 0, 0)+ ( 21 , 12 , 0)+

1

(1) x, y, z (5) x, ¯ y, ¯ z¯

(2) x, ¯ y, ¯z (6) x, y, z¯

I Maximal translationengleiche subgroups [2] Cc2m (40, Ama2) (1; 3; 6; 8)+ [2] C2cm (40, Ama2) (1; 4; 6; 7)+ [2] Ccc2 (37) (1; 2; 7; 8)+ [2] C222 (21) (1; 2; 3; 4)+ [2] C12/c1 (15) (1; 3; 5; 7)+ [2] C2/c11 (15, C12/c1) (1; 4; 5; 8)+ [2] C112/m (10, P112/m) (1; 2; 5; 6)+

(3) x, ¯ y, z¯ + 12 (7) x, y, ¯ z + 12

(4) x, y, ¯ z¯ + 12 (8) x, ¯ y, z + 12

c, a, b c, b, −a

0, 0, 1/4 0, 0, 1/4 0, 0, 1/4

−b, a, c 1/2(a − b), 1/2(a + b), c

II Maximal klassengleiche subgroups

• Loss of centring translations [2] [2] [2] [2] [2] [2] [2] [2]

Pnnm (58) Pccn (56) Pcnm (53, Pmna) Pncm (53, Pmna) Pncn (52, Pnna) Pcnn (52, Pnna) Pccm (49) Pnnn (48)

1; 1; 1; 1; 1; 1; 1; 1;

2; 2; 3; 4; 3; 4; 2; 2;

5; 7; 6; 6; 5; 5; 3; 3;

6; 8; 8; 7; 7; 8; 4; 4;

(3; 4; 7; 8) + ( 21 , 12 , 0) (3; 4; 5; 6) + ( 21 , 12 , 0) (2; 4; 5; 7) + ( 21 , 12 , 0) (2; 3; 5; 8) + ( 21 , 12 , 0) (2; 4; 6; 8) + ( 21 , 12 , 0) (2; 3; 6; 7) + ( 21 , 12 , 0) 5; 6; 7; 8 (5; 6; 7; 8) + ( 21 , 12 , 0)

c, b, −a c, a, b c, a, b c, b, −a

1/4, 1/4, 0 1/4, 1/4, 0 1/4, 1/4, 0

1/4, 1/4, 0

• Enlarged unit cell

[3] a = 3a Cccm (66) Cccm (66) Cccm (66) [3] b = 3b Cccm (66) Cccm (66) Cccm (66) [3] c = 3c Cccm (66) Cccm (66) Cccm (66)

2; 3; 5 (2; 3; 5) + (2, 0, 0) (2; 3; 5) + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 3; 5 3; (2; 5) + (0, 2, 0) 3; (2; 5) + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2; 5; 3 + (0, 0, 1) 2; 3 + (0, 0, 3); 5 + (0, 0, 2) 2; 3 + (0, 0, 5); 5 + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

pa, b, c

u, 0, 0

a, pb, c

0, u, 0

a, b, pc

0, 0, u

• Series of maximal isomorphic subgroups [p] a = pa Cccm (66) (2; 3; 5) + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] b = pb Cccm (66) 3; (2; 5) + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] c = pc Cccm (66) 2; 3 + (0, 0, 2p − 12 + 2u); 5 + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups (Continued on the preceding page)

Copyright © 2011 International Union of Crystallography

220

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 67, pp. 221–222.

D212h

C 2/m 2/m 2/e

Cmme

No. 67

Former space-group symbol Cmma

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t( 21 , 12 , 0); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

16

o

(0, 0, 0)+ ( 21 , 12 , 0)+

1

(1) x, y, z (5) x, ¯ y, ¯ z¯

(2) x, ¯ y¯ + 12 , z (6) x, y + 12 , z¯

I Maximal translationengleiche subgroups [2] Cm2e (39, Aem2) (1; 3; 6; 8)+ [2] C2me (39, Aem2) (1; 4; 6; 7)+ [2] Cmm2 (35) (1; 2; 7; 8)+ [2] C222 (21) (1; 2; 3; 4)+ [2] C112/e (13, P112/a) (1; 2; 5; 6)+ [2] C12/m1 (12) (1; 3; 5; 7)+ [2] C2/m11 (12, C12/m1) (1; 4; 5; 8)+

(3) x, ¯ y + 12 , z¯ (7) x, y¯ + 12 , z

(4) x, y, ¯ z¯ (8) x, ¯ y, z

c, a, b c, b, −a a, 1/2(−a + b), c −b, a, c

1/4, 0, 0 0, 1/4, 0 1/4, 0, 0 1/4, 1/4, 0

II Maximal klassengleiche subgroups

• Loss of centring translations [2] [2] [2] [2] [2] [2] [2] [2]

Pbma (57, Pbcm) Pmab (57, Pbcm) Pbaa (54, Pcca) Pbab (54, Pcca) Pmmb (51, Pmma) Pmma (51) Pmaa (49, Pccm) Pbmb (49, Pccm)

1; 1; 1; 1; 1; 1; 1; 1;

3; 3; 2; 2; 2; 2; 4; 4;

5; 6; 3; 5; 3; 7; 5; 6;

7; 8; 4; 6; 4; 8; 8; 7;

(2; 4; 6; 8) + ( 21 , 12 , 0) (2; 4; 5; 7) + ( 21 , 12 , 0) (5; 6; 7; 8) + ( 21 , 12 , 0) (3; 4; 7; 8) + ( 21 , 12 , 0) 5; 6; 7; 8 (3; 4; 5; 6) + ( 21 , 12 , 0) (2; 3; 6; 7) + ( 21 , 12 , 0) (2; 3; 5; 8) + ( 21 , 12 , 0)

c, a, b c, b, −a b, c, a a, −c, b −b, a, c

1/4, 1/4, 0 1/4, 1/4, 0

1/4, 1/4, 0 b, c, a c, a, b

1/4, 1/4, 0

• Enlarged unit cell

[2] c = 2c Imma (74) Imma (74) Ibca (73) Ibca (73) Ibmb (72, Ibam) Ibmb (72, Ibam) Imaa (72, Ibam) Imaa (72, Ibam) Ccce (68) Ccce (68) Cmme (67) Cmme (67) Ccme (64, Cmce) Ccme (64, Cmce) Cmce (64) Cmce (64) [3] a = 3a Cmme (67) Cmme (67) Cmme (67) [3] b = 3b Cmme (67) Cmme (67) Cmme (67) [3] c = 3c Cmme (67) Cmme (67) Cmme (67)

2; 3; 5 2; (3; 5) + (0, 0, 1) 2; 5; 3 + (0, 0, 1) 2; 3; 5 + (0, 0, 1) 3; 5; 2 + (0, 0, 1) (2; 3; 5) + (0, 0, 1) 5; (2; 3) + (0, 0, 1) 3; (2; 5) + (0, 0, 1) 2; 5; 3 + (0, 0, 1) 2; 3; 5 + (0, 0, 1) 2; 3; 5 2; (3; 5) + (0, 0, 1) 3; 5; 2 + (0, 0, 1) (2; 3; 5) + (0, 0, 1) 5; (2; 3) + (0, 0, 1) 3; (2; 5) + (0, 0, 1)

a, b, 2c a, b, 2c a, b, 2c a, b, 2c 2c, a, b 2c, a, b b, 2c, a b, 2c, a a, b, 2c a, b, 2c a, b, 2c a, b, 2c −b, a, 2c −b, a, 2c a, b, 2c a, b, 2c

2; 3; 5 (2; 3; 5) + (2, 0, 0) (2; 3; 5) + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

5; (2; 3) + (0, 1, 0) 2 + (0, 3, 0); 3 + (0, 1, 0); 5 + (0, 2, 0) 2 + (0, 5, 0); 3 + (0, 1, 0); 5 + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2; 3; 5 2; (3; 5) + (0, 0, 2) 2; (3; 5) + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

Copyright © 2011 International Union of Crystallography

221

0, 0, 1/2 0, 0, 1/2 1/4, 1/4, 1/2 1/4, 1/4, 0 0, 0, 1/2 1/4, 1/4, 0 1/4, 1/4, 1/2 0, 0, 1/2 1/4, 1/4, 0 1/4, 1/4, 1/2 0, 0, 1/2

Cmme

No. 67

• Series of maximal isomorphic subgroups [p] a = pa Cmme (67) (2; 3; 5) + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] b = pb Cmme (67) 2 + (0, 2p − 12 + 2u, 0); 3 + (0, 2p − 12 , 0); 5 + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] c = pc Cmme (67) 2; (3; 5) + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups I Minimal translationengleiche supergroups [2] P4/nbm (125); [2] P4/nmm (129); [2] P42 /nnm (134); [2] P42 /ncm (138) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] Fmmm (69)

• Decreased unit cell [2] a = 12 a, b = 12 b Pmmm (47)

222

CONTINUED

pa, b, c

u, 0, 0

a, pb, c

0, u, 0

a, b, pc

0, 0, u

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 68, pp. 223–225.

D222h

C 2/c 2/c 2/e

Ccce

No. 68

Former space-group symbol Ccca

ORIGIN CHOICE

1, Origin at 2 2 2 at 2/n 2/n 2, at 0, 14 , 14 from 1¯

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t( 21 , 12 , 0); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

16

i

(0, 0, 0)+ ( 21 , 12 , 0)+

1

(1) x, y, z (5) x, ¯ y¯ + 12 , z¯ + 12

(2) x¯ + 12 , y¯ + 12 , z (6) x + 12 , y, z¯ + 12

I Maximal translationengleiche subgroups [2] Cc2e (41, Aea2) (1; 3; 6; 8)+ [2] C2ce (41, Aea2) (1; 4; 6; 7)+ [2] Ccc2 (37) (1; 2; 7; 8)+ [2] C222 (21) (1; 2; 3; 4)+ [2] C12/c1 (15) (1; 3; 5; 7)+ [2] C2/c11 (15, C12/c1) (1; 4; 5; 8)+ [2] C112/e (13, P112/a) (1; 2; 5; 6)+

(3) x, ¯ y, z¯ (7) x, y¯ + 12 , z + 12

(4) x + 12 , y¯ + 12 , z¯ (8) x¯ + 12 , y, z + 12

c, a, b c, b, −a 1/4, 1/4, 0 −b, a, c a, 1/2(−a + b), c

0, 1/4, 1/4 1/4, 0, 1/4 1/4, 0, 1/4

II Maximal klassengleiche subgroups

• Loss of centring translations [2] [2] [2] [2] [2] [2] [2] [2]

Pcnb Pnca Pcca Pccb Pnnb Pnna Pncb Pcna

(60, Pbcn) (60, Pbcn) (54) (54, Pcca) (52, Pnna) (52) (50, Pban) (50, Pban)

• Enlarged unit cell [3] a = 3a Ccce (68) Ccce (68) Ccce (68) [3] b = 3b Ccce (68) Ccce (68) Ccce (68) [3] c = 3c Ccce (68) Ccce (68) Ccce (68)

1; 1; 1; 1; 1; 1; 1; 1;

4; 4; 2; 2; 2; 2; 3; 3;

5; 6; 3; 7; 3; 5; 5; 6;

8; 7; 4; 8; 4; 6; 7; 8;

(2; 3; 6; 7) + ( 21 , 12 , 0) (2; 3; 5; 8) + ( 21 , 12 , 0) 5; 6; 7; 8 (3; 4; 5; 6) + ( 21 , 12 , 0) (5; 6; 7; 8) + ( 21 , 12 , 0) (3; 4; 7; 8) + ( 21 , 12 , 0) (2; 4; 6; 8) + ( 21 , 12 , 0) (2; 4; 5; 7) + ( 21 , 12 , 0)

a, −c, b b, c, a −b, a, c −b, a, c

0, 1/4, 1/4 1/4, 0, 1/4 0, 1/4, 1/4 1/4, 0, 1/4 1/4, 0, 1/4 0, 1/4, 1/4

b, c, a c, a, b

3; 5; 2 + (1, 0, 0) 2 + (3, 0, 0); (3; 5) + (2, 0, 0) 2 + (5, 0, 0); (3; 5) + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

3; (2; 5) + (0, 1, 0) 3; (2; 5) + (0, 3, 0) 3; (2; 5) + (0, 5, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2; 3; 5 + (0, 0, 1) 2; 3 + (0, 0, 2); 5 + (0, 0, 3) 2; 3 + (0, 0, 4); 5 + (0, 0, 5)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

2 + ( 2p − 12 + 2u, 0, 0); (3; 5) + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

pa, b, c

u, 0, 0

3; (2; 5) + (0, 2p − 12 + 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, pb, c

0, u, 0

2; 3 + (0, 0, 2u); 5 + (0, 0, 2p − 12 + 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, b, pc

0, 0, u

• Series of maximal isomorphic subgroups

[p] a = pa Ccce (68)

[p] b = pb Ccce (68) [p] c = pc Ccce (68)

Copyright © 2011 International Union of Crystallography

223

Ccce

ORIGIN CHOICE

1

No. 68

CONTINUED

I Minimal translationengleiche supergroups [2] P4/nnc (126); [2] P4/ncc (130); [2] P42 /nbc (133); [2] P42 /nmc (137) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] Fmmm (69) • Decreased unit cell

[2] a = 12 a, b = 12 b Pccm (49); [2] c = 12 c Cmme (67)

Ccce

ORIGIN CHOICE

2

No. 68

I Minimal translationengleiche supergroups [2] P4/nnc (126); [2] P4/ncc (130); [2] P42 /nbc (133); [2] P42 /nmc (137) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] Fmmm (69) • Decreased unit cell [2] a = 12 a, b = 12 b Pccm (49); [2] c = 12 c Cmme (67)

224

(Continued from the facing page)

D222h

C 2/c 2/c 2/e

No. 68

Ccce

Former space-group symbol Ccca

ORIGIN CHOICE

2, Origin at 1¯ at 2/n c a, at 0, − 14 , − 14 from 2 2 2

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t( 21 , 12 , 0); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

16

i

(0, 0, 0)+ ( 21 , 12 , 0)+

1

(1) x, y, z (5) x, ¯ y, ¯ z¯

¯z (2) x¯ + 12 , y, (6) x + 12 , y, z¯

(3) x, ¯ y, z¯ + 12 (7) x, y, ¯ z + 12

I Maximal translationengleiche subgroups [2] Cc2e (41, Aea2) (1; 3; 6; 8)+ [2] C2ce (41, Aea2) (1; 4; 6; 7)+ [2] Ccc2 (37) (1; 2; 7; 8)+ [2] C222 (21) (1; 2; 3; 4)+ [2] C12/c1 (15) (1; 3; 5; 7)+ [2] C2/c11 (15, C12/c1) (1; 4; 5; 8)+ [2] C112/e (13, P112/a) (1; 2; 5; 6)+

(4) x + 12 , y, ¯ z¯ + 12 (8) x¯ + 12 , y, z + 12

c, a, b c, b, −a

0, 1/4, 1/4 0, 1/4, 1/4 1/4, 0, 1/4 0, 1/4, 1/4

−b, a, c a, 1/2(−a + b), c

1/4, 1/4, 0 1/4, 1/4, 0

a, −c, b b, c, a

1/4, 1/4, 0

−b, a, c −b, a, c

1/4, 1/4, 0 1/4, 1/4, 0

b, c, a c, a, b

1/4, 1/4, 0

3; 5; 2 + (1, 0, 0) 2 + (3, 0, 0); (3; 5) + (2, 0, 0) 2 + (5, 0, 0); (3; 5) + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 3; 5 3; (2; 5) + (0, 2, 0) 3; (2; 5) + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2; 5; 3 + (0, 0, 1) 2; 3 + (0, 0, 3); 5 + (0, 0, 2) 2; 3 + (0, 0, 5); 5 + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

2 + ( 2p − 12 + 2u, 0, 0); (3; 5) + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

pa, b, c

u, 0, 0

3; (2; 5) + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, pb, c

0, u, 0

2; 3 + (0, 0, 2p − 12 + 2u); 5 + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, b, pc

0, 0, u

II Maximal klassengleiche subgroups

• Loss of centring translations [2] [2] [2] [2] [2] [2] [2] [2]

Pcnb Pnca Pcca Pccb Pnnb Pnna Pncb Pcna

(60, Pbcn) (60, Pbcn) (54) (54, Pcca) (52, Pnna) (52) (50, Pban) (50, Pban)

• Enlarged unit cell [3] a = 3a Ccce (68) Ccce (68) Ccce (68) [3] b = 3b Ccce (68) Ccce (68) Ccce (68) [3] c = 3c Ccce (68) Ccce (68) Ccce (68)

1; 1; 1; 1; 1; 1; 1; 1;

4; 4; 2; 2; 2; 2; 3; 3;

5; 6; 3; 7; 3; 5; 5; 6;

8; 7; 4; 8; 4; 6; 7; 8;

(2; 3; 6; 7) + ( 21 , 12 , 0) (2; 3; 5; 8) + ( 21 , 12 , 0) 5; 6; 7; 8 (3; 4; 5; 6) + ( 21 , 12 , 0) (5; 6; 7; 8) + ( 21 , 12 , 0) (3; 4; 7; 8) + ( 21 , 12 , 0) (2; 4; 6; 8) + ( 21 , 12 , 0) (2; 4; 5; 7) + ( 21 , 12 , 0)

• Series of maximal isomorphic subgroups

[p] a = pa Ccce (68)

[p] b = pb Ccce (68) [p] c = pc Ccce (68)

(Continued on the facing page)

225

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 69, pp. 226–227.

F mmm

No. 69

F 2/m 2/m 2/m

D232h

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t(0, 12 , 12 ); t( 21 , 0, 12 ); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

32

p

(0, 0, 0)+ (0, 12 , 12 )+ ( 21 , 0, 12 )+ ( 21 , 12 , 0)+

1

(1) x, y, z (5) x, ¯ y, ¯ z¯

(2) x, ¯ y, ¯z (6) x, y, z¯

(3) x, ¯ y, z¯ (7) x, y, ¯z

I Maximal translationengleiche subgroups [2] Fmm2 (42) (1; 2; 7; 8)+ [2] Fm2m (42, Fmm2) (1; 3; 6; 8)+ [2] F2mm (42, Fmm2) (1; 4; 6; 7)+ [2] F222 (22) (1; 2; 3; 4)+ [2] F112/m (12, A112/m) (1; 2; 5; 6)+ [2] F12/m1 (12, C12/m1) (1; 3; 5; 7)+ [2] F2/m11 (12, C12/m1) (1; 4; 5; 8)+

(4) x, y, ¯ z¯ (8) x, ¯ y, z

c, a, b b, c, a 1/2(a − b), b, c a, b, 1/2(−a + c) −b, a, 1/2(b + c)

II Maximal klassengleiche subgroups

• Loss of centring translations [2] Aeaa (68, Ccce) [2] Bbeb (68, Ccce) [2] Ccce (68) [2] Cmme (67) [2] Bmem (67, Cmme) [2] Aemm (67, Cmme) [2] Cccm (66) [2] Bbmb (66, Cccm) [2] Amaa (66, Cccm) [2] Ammm (65, Cmmm) [2] Bmmm (65, Cmmm) [2] Cmmm (65) [2] Aeam (64, Cmce) [2] Bbem (64, Cmce) [2] Aema (64, Cmce) [2] Ccme (64, Cmce) [2] Bmeb (64, Cmce) [2] Cmce (64)

1; 2; 3; 4; (1; 2; 3; 4) + (0, 12 , 12 ); (5; 6; 7; 8) + ( 21 , 0, 12 ); (5; 6; 7; 8) + ( 21 , 12 , 0) 1; 2; 3; 4; (1; 2; 3; 4) + ( 21 , 0, 12 ); (5; 6; 7; 8) + (0, 12 , 12 ); (5; 6; 7; 8) + ( 21 , 12 , 0) 1; 2; 3; 4; (1; 2; 3; 4) + ( 21 , 12 , 0); (5; 6; 7; 8) + (0, 12 , 12 ); (5; 6; 7; 8) + ( 21 , 0, 12 ) 1; 2; 7; 8; (1; 2; 7; 8) + ( 21 , 12 , 0); (3; 4; 5; 6) + (0, 12 , 12 ); (3; 4; 5; 6) + ( 21 , 0, 12 ) 1; 3; 6; 8; (1; 3; 6; 8) + ( 21 , 0, 12 ); (2; 4; 5; 7) + (0, 12 , 12 ); (2; 4; 5; 7) + ( 21 , 12 , 0) 1; 4; 6; 7; (1; 4; 6; 7) + (0, 12 , 12 ); (2; 3; 5; 8) + ( 21 , 0, 12 ); (2; 3; 5; 8) + ( 21 , 12 , 0) 1; 2; 5; 6; (1; 2; 5; 6) + ( 21 , 12 , 0); (3; 4; 7; 8) + (0, 12 , 12 ); (3; 4; 7; 8) + ( 21 , 0, 12 ) 1; 3; 5; 7; (1; 3; 5; 7) + ( 21 , 0, 12 ); (2; 4; 6; 8) + (0, 12 , 12 ); (2; 4; 6; 8) + ( 21 , 12 , 0) 1; 4; 5; 8; (1; 4; 5; 8) + (0, 12 , 12 ); (2; 3; 6; 7) + ( 21 , 0, 12 ); (2; 3; 6; 7) + ( 21 , 12 , 0) 1; 2; 3; 4; 5; 6; 7; 8; (1; 2; 3; 4; 5; 6; 7; 8) + (0, 12 , 12 ) 1; 2; 3; 4; 5; 6; 7; 8; (1; 2; 3; 4; 5; 6; 7; 8) + ( 21 , 0, 12 ) 1; 2; 3; 4; 5; 6; 7; 8; (1; 2; 3; 4; 5; 6; 7; 8) + ( 21 , 12 , 0) 1; 2; 5; 6; (1; 2; 5; 6) + (0, 12 , 12 ); (3; 4; 7; 8) + ( 21 , 0, 12 ); (3; 4; 7; 8) + ( 21 , 12 , 0) 1; 2; 5; 6; (1; 2; 5; 6) + ( 21 , 0, 12 ); (3; 4; 7; 8) + (0, 12 , 12 ); (3; 4; 7; 8) + ( 21 , 12 , 0) 1; 3; 5; 7; (1; 3; 5; 7) + (0, 12 , 12 ); (2; 4; 6; 8) + ( 21 , 0, 12 ); (2; 4; 6; 8) + ( 21 , 12 , 0) 1; 3; 5; 7; (1; 3; 5; 7) + ( 21 , 12 , 0); (2; 4; 6; 8) + (0, 12 , 12 ); (2; 4; 6; 8) + ( 21 , 0, 12 ) 1; 4; 5; 8; (1; 4; 5; 8) + ( 21 , 0, 12 ); (2; 3; 6; 7) + (0, 12 , 12 ); (2; 3; 6; 7) + ( 21 , 12 , 0) 1; 4; 5; 8; (1; 4; 5; 8) + ( 21 , 12 , 0); (2; 3; 6; 7) + (0, 12 , 12 ); (2; 3; 6; 7) + ( 21 , 0, 12 )

Copyright © 2011 International Union of Crystallography

226

c, b, −a

1/4, 1/4, 0

c, a, b

1/4, 1/4, 0 0, 1/4, 1/4 0, 1/4, 1/4

c, a, b

1/4, 1/4, 0

b, c, a

1/4, 0, 1/4 1/4, 1/4, 0

c, a, b

1/4, 0, 1/4

b, c, a

0, 1/4, 1/4

b, c, a c, a, b

c, b, −a c, a, b b, c, a −b, a, c a, −c, b

[2] Amam (63, Cmcm) [2] Amma (63, Cmcm) [2] Bmmb (63, Cmcm) [2] Bbmm (63, Cmcm) [2] Cmcm (63) [2] Ccmm (63, Cmcm)

• Enlarged unit cell [3] a = 3a Fmmm (69) Fmmm (69) Fmmm (69) [3] b = 3b Fmmm (69) Fmmm (69) Fmmm (69) [3] c = 3c Fmmm (69) Fmmm (69) Fmmm (69)

F mmm

No. 69

CONTINUED

1; 3; 6; 8; (1; 3; 6; 8) + (0, 12 , 12 ); (2; 4; 5; 7) + ( 21 , 0, 12 ); (2; 4; 5; 7) + ( 21 , 12 , 0) 1; 2; 7; 8; (1; 2; 7; 8) + (0, 12 , 12 ); (3; 4; 5; 6) + ( 21 , 0, 12 ); (3; 4; 5; 6) + ( 21 , 12 , 0) 1; 2; 7; 8; (1; 2; 7; 8) + ( 21 , 0, 12 ); (3; 4; 5; 6) + (0, 12 , 12 ); (3; 4; 5; 6) + ( 21 , 12 , 0) 1; 4; 6; 7; (1; 4; 6; 7) + ( 21 , 0, 12 ); (2; 3; 5; 8) + (0, 12 , 12 ); (2; 3; 5; 8) + ( 21 , 12 , 0) 1; 3; 6; 8; (1; 3; 6; 8) + ( 21 , 12 , 0); (2; 4; 5; 7) + (0, 12 , 12 ); (2; 4; 5; 7) + ( 21 , 0, 12 ) 1; 4; 6; 7; (1; 4; 6; 7) + ( 21 , 12 , 0); (2; 3; 5; 8) + (0, 12 , 12 ); (2; 3; 5; 8) + ( 21 , 0, 12 )

c, b, −a

1/4, 1/4, 0

b, c, a

1/4, 0, 1/4

a, −c, b

0, 1/4, 1/4

c, a, b

1/4, 1/4, 0 0, 1/4, 1/4

−b, a, c

1/4, 0, 1/4

2; 3; 5 (2; 3; 5) + (2, 0, 0) (2; 3; 5) + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 3; 5 3; (2; 5) + (0, 2, 0) 3; (2; 5) + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2; 3; 5 2; (3; 5) + (0, 0, 2) 2; (3; 5) + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

pa, b, c

u, 0, 0

a, pb, c

0, u, 0

a, b, pc

0, 0, u

• Series of maximal isomorphic subgroups [p] a = pa Fmmm (69) (2; 3; 5) + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] b = pb Fmmm (69) 3; (2; 5) + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] c = pc Fmmm (69) 2; (3; 5) + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups I Minimal translationengleiche supergroups [2] I4/mmm (139); [2] I4/mcm (140); [3] Fm3¯ (202) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations

none

• Decreased unit cell

[2] a = 12 a, b = 12 b, c = 12 c Pmmm (47)

227

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 70, pp. 228–230.

Fddd ORIGIN CHOICE

No. 70

D242h

F 2/d 2/d 2/d

1, Origin at 2 2 2, at − 18 , − 18 , − 18 from 1¯

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t(0, 12 , 12 ); t( 21 , 0, 12 ); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

32

h

1

(0, 0, 0)+ (0, 12 , 12 )+ ( 21 , 0, 12 )+ ( 21 , 12 , 0)+ (1) x, y, z (5) x¯ + 14 , y¯ + 14 , z¯ + 14

(2) x, ¯ y, ¯z (6) x + 14 , y + 14 , z¯ + 14

I Maximal translationengleiche subgroups [2] Fdd2 (43) (1; 2; 7; 8)+ [2] Fd2d (43, Fdd2) (1; 3; 6; 8)+ [2] F2dd (43, Fdd2) (1; 4; 6; 7)+ [2] F222 (22) (1; 2; 3; 4)+ [2] F112/d (15, A112/a) (1; 2; 5; 6)+ [2] F12/d1 (15, C12/c1) (1; 3; 5; 7)+ [2] F2/d11 (15, C12/c1) (1; 4; 5; 8)+

(3) x, ¯ y, z¯ (7) x + 14 , y¯ + 14 , z + 14

(4) x, y, ¯ z¯ (8) x¯ + 14 , y + 14 , z + 14

c, a, b b, c, a 1/2(a − b), b, c −c, b, 1/2(a + c) −b, a, 1/2(b + c)

1/8, 3/8, 3/8 1/8, 1/8, 1/8 1/8, 1/8, 1/8

II Maximal klassengleiche subgroups

• Loss of centring translations • Enlarged unit cell [3] a = 3a ⎧ ⎨ Fddd (70) Fddd (70) ⎩ Fddd (70) [3] b = 3b ⎧ ⎨ Fddd (70) Fddd (70) ⎩ Fddd (70) [3] c = 3c ⎧ ⎨ Fddd (70) Fddd (70) ⎩ Fddd (70)

none

2 + ( 21 , 12 , 0); 3 + ( 21 , 0, 12 ); 5 + (1, 12 , 12 ) 2 + ( 25 , 12 , 0); 3 + ( 25 , 0, 12 ); 5 + (3, 12 , 12 ) 2 + ( 29 , 12 , 0); 3 + ( 29 , 0, 12 ); 5 + (5, 12 , 12 )

3a, b, c 3a, b, c 3a, b, c

1/4, 1/4, 1/4 5/4, 1/4, 1/4 9/4, 1/4, 1/4

2 + ( 21 , 12 , 0); 3 + ( 21 , 0, 12 ); 5 + ( 21 , 1, 12 ) 2 + ( 21 , 52 , 0); 3 + ( 21 , 0, 12 ); 5 + ( 21 , 3, 12 ) 2 + ( 21 , 92 , 0); 3 + ( 21 , 0, 12 ); 5 + ( 21 , 5, 12 )

a, 3b, c a, 3b, c a, 3b, c

1/4, 1/4, 1/4 1/4, 5/4, 1/4 1/4, 9/4, 1/4

2 + ( 21 , 12 , 0); 3 + ( 21 , 0, 12 ); 5 + ( 21 , 12 , 1) 2 + ( 21 , 12 , 0); 3 + ( 21 , 0, 52 ); 5 + ( 21 , 12 , 3) 2 + ( 21 , 12 , 0); 3 + ( 21 , 0, 92 ); 5 + ( 21 , 12 , 5)

a, b, 3c a, b, 3c a, b, 3c

1/4, 1/4, 1/4 1/4, 1/4, 5/4 1/4, 1/4, 9/4

(2; 3) + (2u, 0, 0); 5 + ( 4p − 14 + 2u, 0, 0) prime p > 4; 0 ≤ u < p p conjugate subgroups for p = 4n + 1 2 + ( 21 + 2u, 12 , 0); 3 + ( 21 + 2u, 0, 12 ); 5 + ( 4p + 14 + 2u, 12 , 12 ) prime p > 2; 0 ≤ u < p p conjugate subgroups for p = 4n − 1

pa, b, c

u, 0, 0

pa, b, c

1/4 + u, 1/4, 1/4

3; 2 + (0, 2u, 0); 5 + (0, 4p − 14 + 2u, 0) prime p > 4; 0 ≤ u < p p conjugate subgroups for p = 4n + 1 2 + ( 21 , 12 + 2u, 0); 3 + ( 21 , 0, 12 ); 5 + ( 21 , 4p + 14 + 2u, 12 ) prime p > 2; 0 ≤ u < p p conjugate subgroups for p = 4n − 1

a, pb, c

0, u, 0

a, pb, c

1/4, 1/4 + u, 1/4

2; 3 + (0, 0, 2u); 5 + (0, 0, 4p − 14 + 2u) prime p > 4; 0 ≤ u < p p conjugate subgroups for p = 4n + 1 2 + ( 21 , 12 , 0); 3 + ( 21 , 0, 12 + 2u); 5 + ( 21 , 12 , 4p + 14 + 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups for p = 4n − 1

a, b, pc

0, 0, u

a, b, pc

1/4, 1/4, 1/4 + u

• Series of maximal isomorphic subgroups

[p] a = pa Fddd (70)

Fddd (70)

[p] b = pb Fddd (70) Fddd (70)

[p] c = pc Fddd (70) Fddd (70)

Copyright © 2011 International Union of Crystallography

228

CONTINUED

No. 70

ORIGIN CHOICE

1

Fddd

2

Fddd

I Minimal translationengleiche supergroups [2] I41 /amd (141); [2] I41 /acd (142); [3] Fd 3¯ (203) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations

none

• Decreased unit cell [2] a = 12 a, b = 12 b, c = 12 c Pnnn (48)

(Continued from the following page)

No. 70

ORIGIN CHOICE

I Minimal translationengleiche supergroups [2] I41 /amd (141); [2] I41 /acd (142); [3] Fd 3¯ (203) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations

none

• Decreased unit cell

[2] a = 12 a, b = 12 b, c = 12 c Pnnn (48)

229

Fddd ORIGIN CHOICE

No. 70

D242h

F 2/d 2/d 2/d

2, Origin at 1¯ at d d d, at 18 , 18 , 18 from 2 2 2

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t(0, 12 , 12 ); t( 21 , 0, 12 ); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

32

h

(0, 0, 0)+ (0, 12 , 12 )+ ( 21 , 0, 12 )+ ( 21 , 12 , 0)+

1

(1) x, y, z (5) x, ¯ y, ¯ z¯

(2) x¯ + 34 , y¯ + 34 , z (6) x + 14 , y + 14 , z¯

(3) x¯ + 34 , y, z¯ + 34 (7) x + 14 , y, ¯ z + 14

I Maximal translationengleiche subgroups [2] Fdd2 (43) (1; 2; 7; 8)+ [2] Fd2d (43, Fdd2) (1; 3; 6; 8)+ [2] F2dd (43, Fdd2) (1; 4; 6; 7)+ [2] F222 (22) (1; 2; 3; 4)+ [2] F112/d (15, A112/a) (1; 2; 5; 6)+ [2] F12/d1 (15, C12/c1) (1; 3; 5; 7)+ [2] F2/d11 (15, C12/c1) (1; 4; 5; 8)+

(4) x, y¯ + 34 , z¯ + 34 (8) x, ¯ y + 14 , z + 14

c, a, b b, c, a 1/2(a − b), b, c −c, b, 1/2(a + c) −b, a, 1/2(b + c)

3/8, 3/8, 0 3/8, 0, 3/8 0, 3/8, 3/8 1/8, 1/8, 1/8 0, 1/4, 1/4

II Maximal klassengleiche subgroups

• Loss of centring translations

none

• Enlarged unit cell

[3] a = 3a ⎧ ⎨ Fddd (70) Fddd (70) ⎩ Fddd (70) [3] b = 3b ⎧ ⎨ Fddd (70) Fddd (70) ⎩ Fddd (70) [3] c = 3c ⎧ ⎨ Fddd (70) Fddd (70) ⎩ Fddd (70)

2 + ( 23 , 12 , 0); 3 + ( 23 , 0, 12 ); 5 + (0, 12 , 12 ) 2 + ( 27 , 12 , 0); 3 + ( 27 , 0, 12 ); 5 + (2, 12 , 12 ) 1 11 1 1 1 2 + ( 11 2 , 2 , 0); 3 + ( 2 , 0, 2 ); 5 + (4, 2 , 2 )

3a, b, c 3a, b, c 3a, b, c

0, 1/4, 1/4 1, 1/4, 1/4 2, 1/4, 1/4

2 + ( 21 , 32 , 0); (3; 5) + ( 21 , 0, 12 ) 2 + ( 21 , 72 , 0); 3 + ( 21 , 0, 12 ); 5 + ( 21 , 2, 12 ) 1 1 1 1 2 + ( 21 , 11 2 , 0); 3 + ( 2 , 0, 2 ); 5 + ( 2 , 4, 2 )

a, 3b, c a, 3b, c a, 3b, c

1/4, 0, 1/4 1/4, 1, 1/4 1/4, 2, 1/4

(2; 5) + ( 21 , 12 , 0); 3 + ( 21 , 0, 32 ) 2 + ( 21 , 12 , 0); 3 + ( 21 , 0, 72 ); 5 + ( 21 , 12 , 2) 1 1 2 + ( 21 , 12 , 0); 3 + ( 21 , 0, 11 2 ); 5 + ( 2 , 2 , 4)

a, b, 3c a, b, 3c a, b, 3c

1/4, 1/4, 0 1/4, 1/4, 1 1/4, 1/4, 2

pa, b, c

u, 0, 0

pa, b, c

u, 1/4, 1/4

a, pb, c

0, u, 0

a, pb, c

1/4, u, 1/4

a, b, pc

0, 0, u

a, b, pc

1/4, 1/4, u

• Series of maximal isomorphic subgroups [p] a = pa 3 Fddd (70) (2; 3) + ( 3p 4 − 4 + 2u, 0, 0); 5 + (2u, 0, 0) prime p > 4; 0 ≤ u < p p conjugate subgroups for p = 4n + 1 3p 3 1 3 1 Fddd (70) 2 + ( 3p 4 − 4 + 2u, 2 , 0); 3 + ( 4 − 4 + 2u, 0, 2 ); 5 + (2u, 12 , 12 ) prime p > 2; 0 ≤ u < p p conjugate subgroups for p = 4n − 1 [p] b = pb 3 Fddd (70) 3; 2 + (0, 3p 4 − 4 + 2u, 0); 5 + (0, 2u, 0) prime p > 4; 0 ≤ u < p p conjugate subgroups for p = 4n + 1 3 1 1 1 1 Fddd (70) 2 + ( 21 , 3p 4 − 4 + 2u, 0); 3 + ( 2 , 0, 2 ); 5 + ( 2 , 2u, 2 ) prime p > 2; 0 ≤ u < p p conjugate subgroups for p = 4n − 1 [p] c = pc 3 Fddd (70) 2; 3 + (0, 0, 3p 4 − 4 + 2u); 5 + (0, 0, 2u) prime p > 4; 0 ≤ u < p p conjugate subgroups for p = 4n + 1 3 1 1 Fddd (70) 2 + ( 21 , 12 , 0); 3 + ( 21 , 0, 3p 4 − 4 + 2u); 5 + ( 2 , 2 , 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups for p = 4n − 1 (Continued on the preceding page) 230

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 71, p. 231.

D252h

I 2/m 2/m 2/m

No. 71

I mmm

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t( 21 , 12 , 12 ); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

16

o

(0, 0, 0)+ ( 21 , 12 , 12 )+

1

(1) x, y, z (5) x, ¯ y, ¯ z¯

(2) x, ¯ y, ¯z (6) x, y, z¯

I Maximal translationengleiche subgroups [2] Imm2 (44) (1; 2; 7; 8)+ [2] Im2m (44, Imm2) (1; 3; 6; 8)+ [2] I2mm (44, Imm2) (1; 4; 6; 7)+ [2] I222 (23) (1; 2; 3; 4)+ [2] I112/m (12, A112/m) (1; 2; 5; 6)+ [2] I12/m1 (12, C12/m1) (1; 3; 5; 7)+ [2] I2/m11 (12, C12/m1) (1; 4; 5; 8)+

(3) x, ¯ y, z¯ (7) x, y, ¯z

(4) x, y, ¯ z¯ (8) x, ¯ y, z

c, a, b b, c, a b, −a − b, c −a − c, b, a −b + c, a, b

II Maximal klassengleiche subgroups

• Loss of centring translations [2] [2] [2] [2] [2] [2] [2] [2]

Pmmn (59) Pmnm (59, Pmmn) Pnmm (59, Pmmn) Pnnm (58) Pnmn (58, Pnnm) Pmnn (58, Pnnm) Pnnn (48) Pmmm (47)

• Enlarged unit cell [3] a = 3a Immm (71) Immm (71) Immm (71) [3] b = 3b Immm (71) Immm (71) Immm (71) [3] c = 3c Immm (71) Immm (71) Immm (71)

1; 1; 1; 1; 1; 1; 1; 1;

2; 3; 4; 2; 3; 4; 2; 2;

7; 6; 6; 5; 5; 5; 3; 3;

8; 8; 7; 6; 7; 8; 4; 4;

(3; 4; 5; 6) + ( 21 , 12 , 12 ) (2; 4; 5; 7) + ( 21 , 12 , 12 ) (2; 3; 5; 8) + ( 21 , 12 , 12 ) (3; 4; 7; 8) + ( 21 , 12 , 12 ) (2; 4; 6; 8) + ( 21 , 12 , 12 ) (2; 3; 6; 7) + ( 21 , 12 , 12 ) (5; 6; 7; 8) + ( 21 , 12 , 12 ) 5; 6; 7; 8

c, a, b b, c, a

1/4, 1/4, 1/4 1/4, 1/4, 1/4 1/4, 1/4, 1/4

c, a, b b, c, a 1/4, 1/4, 1/4

2; 3; 5 (2; 3; 5) + (2, 0, 0) (2; 3; 5) + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 3; 5 3; (2; 5) + (0, 2, 0) 3; (2; 5) + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2; 3; 5 2; (3; 5) + (0, 0, 2) 2; (3; 5) + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

(2; 3; 5) + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

pa, b, c

u, 0, 0

3; (2; 5) + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, pb, c

0, u, 0

2; (3; 5) + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, b, pc

0, 0, u

• Series of maximal isomorphic subgroups

[p] a = pa Immm (71)

[p] b = pb Immm (71) [p] c = pc Immm (71)

I Minimal translationengleiche supergroups [2] I4/mmm (139); [3] Im3¯ (204) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations • Decreased unit cell

none

[2] a = 12 a Ammm (65, Cmmm); [2] b = 12 b Bmmm (65, Cmmm); [2] c = 12 c Cmmm (65) Copyright © 2011 International Union of Crystallography

231

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 72, p. 232.

I bam

No. 72

D262h

I 2/b 2/a 2/m

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t( 21 , 12 , 12 ); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

16

k

(0, 0, 0)+ ( 21 , 12 , 12 )+

1

(1) x, y, z (5) x, ¯ y, ¯ z¯

(2) x, ¯ y, ¯z (6) x, y, z¯

(3) x¯ + 12 , y + 12 , z¯ (7) x + 12 , y¯ + 12 , z

I Maximal translationengleiche subgroups [2] Ib2m (46, Ima2) (1; 3; 6; 8)+ [2] I2am (46, Ima2) (1; 4; 6; 7)+ [2] Iba2 (45) (1; 2; 7; 8)+ [2] I222 (23) (1; 2; 3; 4)+ [2] I12/a1 (15, C12/c1) (1; 3; 5; 7)+ [2] I2/b11 (15, C12/c1) (1; 4; 5; 8)+ [2] I112/m (12, A112/m) (1; 2; 5; 6)+

(4) x + 12 , y¯ + 12 , z¯ (8) x¯ + 12 , y + 12 , z c, a, b c, b, −a a − c, b, c −b − c, a, c b, −a − b, c

0, 0, 1/4 0, 0, 1/4 0, 0, 1/4

II Maximal klassengleiche subgroups

• Loss of centring translations [2] [2] [2] [2] [2] [2] [2] [2]

Pcan (60, Pbcn) Pbcn (60) Pbcm (57) Pcam (57, Pbcm) Pccn (56) Pbam (55) Pban (50) Pccm (49)

• Enlarged unit cell [3] a = 3a Ibam (72) Ibam (72) Ibam (72) [3] b = 3b Ibam (72) Ibam (72) Ibam (72) [3] c = 3c Ibam (72) Ibam (72) Ibam (72)

1; 1; 1; 1; 1; 1; 1; 1;

3; 4; 3; 4; 2; 2; 2; 2;

5; 5; 6; 6; 3; 3; 7; 5;

7; 8; 8; 7; 4; 4; 8; 6;

(2; 4; 6; 8) + ( 21 , 12 , 12 ) (2; 3; 6; 7) + ( 21 , 12 , 12 ) (2; 4; 5; 7) + ( 21 , 12 , 12 ) (2; 3; 5; 8) + ( 21 , 12 , 12 ) (5; 6; 7; 8) + ( 21 , 12 , 12 ) 5; 6; 7; 8 (3; 4; 5; 6) + ( 21 , 12 , 12 ) (3; 4; 7; 8) + ( 21 , 12 , 12 )

−b, a, c

−b, a, c

1/4, 1/4, 1/4 1/4, 1/4, 1/4 1/4, 1/4, 1/4 1/4, 1/4, 1/4

2; 5; 3 + (1, 0, 0) (2; 5) + (2, 0, 0); 3 + (3, 0, 0) (2; 5) + (4, 0, 0); 3 + (5, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 5; 3 + (0, 1, 0) (2; 5) + (0, 2, 0); 3 + (0, 1, 0) (2; 5) + (0, 4, 0); 3 + (0, 1, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2; 3; 5 2; (3; 5) + (0, 0, 2) 2; (3; 5) + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

pa, b, c

u, 0, 0

a, pb, c

0, u, 0

a, b, pc

0, 0, u

• Series of maximal isomorphic subgroups [p] a = pa Ibam (72) (2; 5) + (2u, 0, 0); 3 + ( 2p − 12 + 2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] b = pb Ibam (72) (2; 5) + (0, 2u, 0); 3 + (0, 2p − 12 , 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] c = pc Ibam (72) 2; (3; 5) + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups I Minimal translationengleiche supergroups [2] I4/mcm (140) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations • Decreased unit cell

none

[2] c = 12 c Cmmm (65); [2] a = 12 a Aemm (67, Cmme); [2] b = 12 b Bmem (67, Cmme) Copyright © 2011 International Union of Crystallography

232

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 73, pp. 233–234.

D272h

I 21/b 21/c 21/a

No. 73

I bca

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t( 21 , 12 , 12 ); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

16

f

(0, 0, 0)+ ( 21 , 12 , 12 )+

1

(1) x, y, z (5) x, ¯ y, ¯ z¯

¯ z + 12 (2) x¯ + 12 , y, 1 (6) x + 2 , y, z¯ + 12

(3) x, ¯ y + 12 , z¯ + 12 (7) x, y¯ + 12 , z + 12

I Maximal translationengleiche subgroups [2] Ibc2 (45, Iba2) (1; 2; 7; 8)+ [2] Ib2a (45, Iba2) (1; 3; 6; 8)+ [2] I2ca (45, Iba2) (1; 4; 6; 7)+ (1; 2; 3; 4)+ [2] I21 21 21 (24) [2] I112/a (15, A112/a) (1; 2; 5; 6)+ [2] I12/c1 (15, C12/c1) (1; 3; 5; 7)+ [2] I2/b11 (15, C12/c1) (1; 4; 5; 8)+

(4) x + 12 , y¯ + 12 , z¯ (8) x¯ + 12 , y + 12 , z

c, a, b b, c, a

0, 1/4, 0 1/4, 0, 0 0, 0, 1/4

b, −a − b, c −a − c, b, a −b − c, a, c

II Maximal klassengleiche subgroups

• Loss of centring translations [2] [2] [2] [2] [2] [2] [2] [2]

Pbca Pcab Pcaa Pccb Pbab Pbcb Pbaa Pcca

(61) (61, Pbca) (54, Pcca) (54, Pcca) (54, Pcca) (54, Pcca) (54, Pcca) (54)

1; 1; 1; 1; 1; 1; 1; 1;

2; 2; 2; 3; 4; 2; 3; 4;

3; 3; 5; 5; 5; 7; 6; 6;

4; 4; 6; 7; 8; 8; 8; 7;

5; 6; 7; 8 (5; 6; 7; 8) + ( 21 , 12 , 12 ) (3; 4; 7; 8) + ( 21 , 12 , 12 ) (2; 4; 6; 8) + ( 21 , 12 , 12 ) (2; 3; 6; 7) + ( 21 , 12 , 12 ) (3; 4; 5; 6) + ( 21 , 12 , 12 ) (2; 4; 5; 7) + ( 21 , 12 , 12 ) (2; 3; 5; 8) + ( 21 , 12 , 12 )

−b, a, c c, b, −a −b, a, c a, −c, b c, a, b b, c, a

1/4, 1/4, 1/4

1/4, 1/4, 1/4 1/4, 1/4, 1/4 1/4, 1/4, 1/4

• Enlarged unit cell

[3] a = 3a Ibca (73) Ibca (73) Ibca (73) [3] b = 3b Ibca (73) Ibca (73) Ibca (73) [3] c = 3c Ibca (73) Ibca (73) Ibca (73)

3; 5; 2 + (1, 0, 0) 2 + (3, 0, 0); (3; 5) + (2, 0, 0) 2 + (5, 0, 0); (3; 5) + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

2; 5; 3 + (0, 1, 0) (2; 5) + (0, 2, 0); 3 + (0, 1, 0) (2; 5) + (0, 4, 0); 3 + (0, 1, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

5; (2; 3) + (0, 0, 1) 2 + (0, 0, 1); 3 + (0, 0, 3); 5 + (0, 0, 2) 2 + (0, 0, 1); 3 + (0, 0, 5); 5 + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

2 + ( 2p − 12 + 2u, 0, 0); (3; 5) + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

pa, b, c

u, 0, 0

(2; 5) + (0, 2u, 0); 3 + (0, 2p − 12 , 0) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, pb, c

0, u, 0

2 + (0, 0, 2p − 12 ); 3 + (0, 0, 2p − 12 + 2u); 5 + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, b, pc

0, 0, u

• Series of maximal isomorphic subgroups

[p] a = pa Ibca (73)

[p] b = pb Ibca (73) [p] c = pc Ibca (73)

Copyright © 2011 International Union of Crystallography

233

I bca

No. 73

CONTINUED

I Minimal translationengleiche supergroups [2] I41 /acd (142); [3] Ia3¯ (206) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations

none

• Decreased unit cell [2] a = 12 a Aemm (67, Cmme); [2] b = 12 b Bmem (67, Cmme); [2] c = 12 c Cmme (67)

I mma

No. 74

(Continued from the facing page)

I Minimal translationengleiche supergroups [2] I41 /amd (141) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations

none

• Decreased unit cell [2] a = 12 a Ammm (65, Cmmm); [2] b = 12 b Bmmm (65, Cmmm); [2] c = 12 c Cmme (67)

234

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 74, pp. 234–235.

D282h

I 21/m 21/m 21/a

No. 74

I mma

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t( 21 , 12 , 12 ); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

16

j

(0, 0, 0)+ ( 21 , 12 , 12 )+

1

(1) x, y, z (5) x, ¯ y, ¯ z¯

(2) x, ¯ y¯ + 12 , z (6) x, y + 12 , z¯

I Maximal translationengleiche subgroups [2] Im2b (46, Ima2) (1; 3; 6; 8)+ [2] I2mb (46, Ima2) (1; 4; 6; 7)+ [2] Imm2 (44) (1; 2; 7; 8)+ (1; 2; 3; 4)+ [2] I21 21 21 (24) [2] I112/b (15, A112/a) (1; 2; 5; 6)+ [2] I12/m1 (12, C12/m1) (1; 3; 5; 7)+ [2] I2/m11 (12, C12/m1) (1; 4; 5; 8)+

(3) x, ¯ y + 12 , z¯ (7) x, y¯ + 12 , z

(4) x, y, ¯ z¯ (8) x, ¯ y, z

−a, c, b b, c, a b, −a − b, c −a − c, b, a −b − c, a, c

1/4, 0, 1/4 0, 1/4, 0 0, 0, 1/4 1/4, 1/4, 1/4

II Maximal klassengleiche subgroups

• Loss of centring translations [2] [2] [2] [2] [2] [2] [2] [2]

Pnma (62) Pmnb (62, Pnma) Pnmb (53, Pmna) Pmna (53) Pnna (52) Pnnb (52, Pnna) Pmma (51) Pmmb (51, Pmma)

1; 1; 1; 1; 1; 1; 1; 1;

3; 3; 4; 4; 2; 2; 2; 2;

5; 6; 6; 5; 3; 5; 7; 3;

7; 8; 7; 8; 4; 6; 8; 4;

(2; 4; 6; 8) + ( 21 , 12 , 12 ) (2; 4; 5; 7) + ( 21 , 12 , 12 ) (2; 3; 5; 8) + ( 21 , 12 , 12 ) (2; 3; 6; 7) + ( 21 , 12 , 12 ) (5; 6; 7; 8) + ( 21 , 12 , 12 ) (3; 4; 7; 8) + ( 21 , 12 , 12 ) (3; 4; 5; 6) + ( 21 , 12 , 12 ) 5; 6; 7; 8

−b, a, c −b, a, c

1/4, 1/4, 1/4 1/4, 1/4, 1/4 1/4, 1/4, 1/4

−b, a, c 1/4, 1/4, 1/4 −b, a, c

• Enlarged unit cell

[3] a = 3a Imma (74) Imma (74) Imma (74) [3] b = 3b Imma (74) Imma (74) Imma (74) [3] c = 3c Imma (74) Imma (74) Imma (74)

2; 3; 5 (2; 3; 5) + (2, 0, 0) (2; 3; 5) + (4, 0, 0)

3a, b, c 3a, b, c 3a, b, c

1, 0, 0 2, 0, 0

5; (2; 3) + (0, 1, 0) 2 + (0, 3, 0); 3 + (0, 1, 0); 5 + (0, 2, 0) 2 + (0, 5, 0); 3 + (0, 1, 0); 5 + (0, 4, 0)

a, 3b, c a, 3b, c a, 3b, c

0, 1, 0 0, 2, 0

2; 3; 5 2; (3; 5) + (0, 0, 2) 2; (3; 5) + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

pa, b, c

u, 0, 0

a, pb, c

0, u, 0

a, b, pc

0, 0, u

• Series of maximal isomorphic subgroups [p] a = pa Imma (74) (2; 3; 5) + (2u, 0, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] b = pb Imma (74) 2 + (0, 2p − 12 + 2u, 0); 3 + (0, 2p − 12 , 0); 5 + (0, 2u, 0) prime p > 2; 0 ≤ u < p p conjugate subgroups [p] c = pc Imma (74) 2; (3; 5) + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups (Continued on the facing page)

Copyright © 2011 International Union of Crystallography

235

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 75, p. 236.

P4

No. 75

C41

P4

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

4

d

1

(1) x, y, z

(2) x, ¯ y, ¯z

(3) y, ¯ x, z

(4) y, x, ¯z

I Maximal translationengleiche subgroups [2] P2 (3, P112) 1; 2 II Maximal klassengleiche subgroups

• Enlarged unit cell [2] c = 2c P42 (77) P4 (75) [2] a = 2a, b = 2b C4 (75, P4) C4 (75, P4) [2] a = 2a, b = 2b, c = 2c F4 (79, I4) F4 (79, I4) [3] c = 3c P4 (75)

2; 3 + (0, 0, 1) 2; 3

a, b, 2c a, b, 2c

2; 3 2 + (1, 1, 0); 3 + (1, 0, 0)

a − b, a + b, c a − b, a + b, c

1/2, 1/2, 0

2; 3 2; 3 + (0, 0, 1)

a − b, a + b, 2c a − b, a + b, 2c

1/2, 1/2, 0

2; 3

a, b, 3c

• Series of maximal isomorphic subgroups

[p] c = pc P4 (75)

[p2 ] a = pa, b = pb P4 (75)

2; 3 p prime no conjugate subgroups

a, b, pc

2 + (2u, 2v, 0); 3 + (u + v, −u + v, 0) prime p > 2; 0 ≤ u < p; 0 ≤ v < p p2 conjugate subgroups for p = 4n − 1 2 2 [p = q + r ] a = qa − rb, b = ra + qb P4 (75) 2 + (2u, 0, 0); 3 + (u, −u, 0) prime p > 4; q > 0; r > 0; 0 ≤ u < p p conjugate subgroups for p = 4n + 1

pa, pb, c

u, v, 0

qa − rb, ra + qb, c

u, 0, 0

I Minimal translationengleiche supergroups [2] P4/m (83); [2] P4/n (85); [2] P422 (89); [2] P421 2 (90); [2] P4mm (99); [2] P4bm (100); [2] P4cc (103); [2] P4nc (104) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] I4 (79)

• Decreased unit cell

Copyright © 2011 International Union of Crystallography

none

236

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 76, p. 237.

C42

P 41

P 41

No. 76

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

4

a

1

(1) x, y, z

(2) x, ¯ y, ¯ z + 12

(3) y, ¯ x, z + 14

(4) y, x, ¯ z + 34

I Maximal translationengleiche subgroups [2] P21 (4, P1121 ) 1; 2 II Maximal klassengleiche subgroups

• Enlarged unit cell

[2] a = 2a, b = 2b C41 (76, P41 ) C41 (76, P41 ) [3] c = 3c P43 (78)

2; 3 2 + (1, 1, 0); 3 + (1, 0, 0)

a − b, a + b, c a − b, a + b, c

2 + (0, 0, 1); 3 + (0, 0, 2)

a, b, 3c

1/2, 1/2, 0

• Series of maximal isomorphic subgroups

[p] c = pc P43 (78) P41 (76)

[p2 ] a = pa, b = pb P41 (76)

1 2 + (0, 0, 2p − 12 ); 3 + (0, 0, 3p 4 − 4 ) prime p > 2; p = 4n − 1 no conjugate subgroups 2 + (0, 0, 2p − 12 ); 3 + (0, 0, 4p − 14 ) prime p > 4; p = 4n + 1 no conjugate subgroups

2 + (2u, 2v, 0); 3 + (u + v, −u + v, 0) prime p > 2; 0 ≤ u < p; 0 ≤ v < p p2 conjugate subgroups for p = 4n − 1 [p = q2 + r2 ] a = qa − rb, b = ra + qb 2 + (2u, 0, 0); 3 + (u, −u, 0) P41 (76) prime p > 4; q > 0; r > 0; 0 ≤ u < p p conjugate subgroups for p = 4n + 1

I Minimal translationengleiche supergroups [2] P41 22 (91); [2] P41 21 2 (92) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] I41 (80)

• Decreased unit cell [2] c = 12 c P42 (77)

Copyright © 2011 International Union of Crystallography

237

a, b, pc a, b, pc

pa, pb, c

u, v, 0

qa − rb, ra + qb, c

u, 0, 0

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 77, p. 238.

P 42

No. 77

C43

P 42

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

4

d

1

(1) x, y, z

(2) x, ¯ y, ¯z

(3) y, ¯ x, z + 12

(4) y, x, ¯ z + 12

I Maximal translationengleiche subgroups [2] P2 (3, P112) 1; 2 II Maximal klassengleiche subgroups

• Enlarged unit cell

[2] c = 2c P43 (78) P41 (76) [2] a = 2a, b = 2b C42 (77, P42 ) C42 (77, P42 ) [2] a = 2a, b = 2b, c = 2c F41 (80, I41 ) F41 (80, I41 ) [3] c = 3c P42 (77)

(2; 3) + (0, 0, 1) 3; 2 + (0, 0, 1)

a, b, 2c a, b, 2c

2; 3 2 + (1, 1, 0); 3 + (1, 0, 0)

a − b, a + b, c a − b, a + b, c

1/2, 1/2, 0

3; 2 + (0, 0, 1) (2; 3) + (0, 0, 1)

a − b, a + b, 2c a − b, a + b, 2c

0, 1/2, 0 1/2, 0, 0

2; 3 + (0, 0, 1)

a, b, 3c

• Series of maximal isomorphic subgroups [p] c = pc 2; 3 + (0, 0, 2p − 12 ) P42 (77) prime p > 2 no conjugate subgroups [p2 ] a = pa, b = pb P42 (77) 2 + (2u, 2v, 0); 3 + (u + v, −u + v, 0) prime p > 2; 0 ≤ u < p; 0 ≤ v < p p2 conjugate subgroups for p = 4n − 1 2 2 [p = q + r ] a = qa − rb, b = ra + qb 2 + (2u, 0, 0); 3 + (u, −u, 0) P42 (77) prime p > 4; q > 0; r > 0; 0 ≤ u < p p conjugate subgroups for p = 4n + 1

a, b, pc

pa, pb, c

u, v, 0

qa − rb, ra + qb, c

u, 0, 0

I Minimal translationengleiche supergroups [2] P42 /m (84); [2] P42 /n (86); [2] P42 22 (93); [2] P42 21 2 (94); [2] P42 cm (101); [2] P42 nm (102); [2] P42 mc (105); [2] P42 bc (106) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] I4 (79) • Decreased unit cell

[2] c = 12 c P4 (75)

Copyright © 2011 International Union of Crystallography

238

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 78, p. 239.

C44

P 43

P 43

No. 78

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

4

a

1

(1) x, y, z

(2) x, ¯ y, ¯ z + 12

(3) y, ¯ x, z + 34

(4) y, x, ¯ z + 14

I Maximal translationengleiche subgroups [2] P21 (4, P1121 ) 1; 2 II Maximal klassengleiche subgroups

• Enlarged unit cell

[2] a = 2a, b = 2b C43 (78, P43 ) C43 (78, P43 ) [3] c = 3c P41 (76)

2; 3 2 + (1, 1, 0); 3 + (1, 0, 0)

a − b, a + b, c a − b, a + b, c

3; 2 + (0, 0, 1)

a, b, 3c

1/2, 1/2, 0

• Series of maximal isomorphic subgroups

[p] c = pc P43 (78) P41 (76)

[p2 ] a = pa, b = pb P43 (78)

3 2 + (0, 0, 2p − 12 ); 3 + (0, 0, 3p 4 − 4 ) prime p > 4; p = 4n + 1 no conjugate subgroups 2 + (0, 0, 2p − 12 ); 3 + (0, 0, 4p − 34 ) prime p > 2; p = 4n − 1 no conjugate subgroups

2 + (2u, 2v, 0); 3 + (u + v, −u + v, 0) prime p > 2 p2 conjugate subgroups for p = 4n − 1 [p = q2 + r2 ] a = qa − rb, b = ra + qb 2 + (2u, 0, 0); 3 + (u, −u, 0) P43 (78) prime p > 4; q > 0; r > 0; 0 ≤ u < p p conjugate subgroups for p = 4n + 1

I Minimal translationengleiche supergroups [2] P43 22 (95); [2] P43 21 2 (96) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] I41 (80)

• Decreased unit cell [2] c = 12 c P42 (77)

Copyright © 2011 International Union of Crystallography

239

a, b, pc a, b, pc

pa, pb, c

u, v, 0

qa − rb, ra + qb, c

u, 0, 0

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 79, p. 240.

I4

No. 79

C45

I4

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t( 21 , 12 , 12 ); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

c

(0, 0, 0)+ ( 21 , 12 , 12 )+

1

(1) x, y, z

(2) x, ¯ y, ¯z

I Maximal translationengleiche subgroups [2] I2 (5, A112) (1; 2)+

(3) y, ¯ x, z

(4) y, x, ¯z

b, −a − b, c

II Maximal klassengleiche subgroups

• Loss of centring translations [2] P42 (77) [2] P4 (75)

1; 2; (3; 4) + ( 21 , 12 , 12 ) 1; 2; 3; 4

• Enlarged unit cell [3] c = 3c I4 (79)

2; 3

1/2, 0, 0

a, b, 3c

• Series of maximal isomorphic subgroups [p] c = pc I4 (79) 2; 3 prime p > 2 no conjugate subgroups [p2 ] a = pa, b = pb I4 (79) 2 + (2u, 2v, 0); 3 + (u + v, −u + v, 0) prime p > 2; 0 ≤ u < p; 0 ≤ v < p p2 conjugate subgroups for p = 4n − 1 2 2 [p = q + r ] a = qa − rb, b = ra + qb I4 (79) 2 + (2u, 0, 0); 3 + (u, −u, 0) prime p > 4; q > 0; r > 0; 0 ≤ u < p p conjugate subgroups for p = 4n + 1

a, b, pc

pa, pb, c

u, v, 0

qa − rb, ra + qb, c

u, 0, 0

I Minimal translationengleiche supergroups [2] I4/m (87); [2] I422 (97); [2] I4mm (107); [2] I4cm (108) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations

none

• Decreased unit cell [2] c = 12 c C4 (75, P4)

Copyright © 2011 International Union of Crystallography

240

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 80, p. 241.

C46

I 41

I 41

No. 80

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t( 21 , 12 , 12 ); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

b

(0, 0, 0)+ ( 21 , 12 , 12 )+

1

(1) x, y, z

(2) x¯ + 12 , y¯ + 12 , z + 12

(3) y, ¯ x + 12 , z + 14

I Maximal translationengleiche subgroups [2] I2 (5, A112) (1; 2)+

(4) y + 12 , x, ¯ z + 34

b, −a − b, c

II Maximal klassengleiche subgroups

• Loss of centring translations [2] P43 (78) [2] P41 (76)

1; 2; (3; 4) + ( 21 , 12 , 12 ) 1; 2; 3; 4

3/4, 3/4, 0 3/4, 1/4, 0

• Enlarged unit cell

[3] c = 3c I41 (80)

2 + (1, 0, 1); 3 + ( 21 , − 21 , 12 )

a, b, 3c

• Series of maximal isomorphic subgroups [p] c = pc I41 (80) 2 + (0, 0, 2p − 12 ); 3 + (0, 0, 4p − 14 ) prime p > 4; p = 4n + 1 no conjugate subgroups 2 + (1, 0, 2p − 12 ); 3 + ( 21 , − 21 , 4p − 14 ) I41 (80) prime p > 2; p = 4n − 1 no conjugate subgroups [p2 ] a = pa, b = pb 2 + ( 2p − 12 + 2u, 2p − 12 + 2v, 0); I41 (80) 3 + (u + v, 2p − 12 − u + v, 0) prime p > 2; 0 ≤ u < p; 0 ≤ v < p p2 conjugate subgroups for p = 4n − 1 2 2 [p = q + r ] a = qa − rb, b = ra + qb 2 + ( 2q + 2r − 12 + 2u, − 2r + q2 − 12 , 0); I41 (80) 3 + ( 2r + u, q2 − u − 12 , 0) prime p > 4; q > 0; r > 1; q odd; r even; 0 ≤ u < p p conjugate subgroups for p = 4n + 1 I41 (80) 2 + ( 2q + 2r + 12 + 2u, − 2r + q2 − 12 , 0); 3 + ( 2r + 12 + u, q2 − 1 − u, 0) prime p > 4; q > 1; r > 0; q even; r odd; 0 ≤ u < p p conjugate subgroups for p = 4n + 1

a, b, pc a, b, pc

1/2, 0, 0

pa, pb, c

u, v, 0

qa − rb, ra + qb, c

u, 0, 0

qa − rb, ra + qb, c

1/2 + u, 0, 0

I Minimal translationengleiche supergroups [2] I41 /a (88); [2] I41 22 (98); [2] I41 md (109); [2] I41 cd (110) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations

none

• Decreased unit cell [2] c = 12 c C42 (77, P42 )

Copyright © 2011 International Union of Crystallography

241

1/2, 0, 0

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 81, p. 242.

P 4¯

S41

P 4¯

No. 81

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

4

h

1

(1) x, y, z

(2) x, ¯ y, ¯z

(3) y, x, ¯ z¯

(4) y, ¯ x, z¯

I Maximal translationengleiche subgroups [2] P2 (3, P112) 1; 2 II Maximal klassengleiche subgroups

• Enlarged unit cell

[2] c = 2c P4¯ (81) P4¯ (81) [2] a = 2a, b = 2b ¯ C4¯ (81, P4) ¯ ¯ C4 (81, P4) [2] a = 2a, b = 2b, c = 2c ¯ F 4¯ (82, I 4) ¯ ¯ F 4 (82, I 4) = 3c [3] c ⎧ ⎨ P4¯ (81) P4¯ (81) ⎩ ¯ P4 (81)

2; 3 2; 3 + (0, 0, 1)

a, b, 2c a, b, 2c

0, 0, 1/2

2; 3 2 + (1, 1, 0); 3 + (0, 1, 0)

a − b, a + b, c a − b, a + b, c

1/2, 1/2, 0

2; 3 2; 3 + (0, 0, 1)

a − b, a + b, 2c a − b, a + b, 2c

0, 0, 1/2

2; 3 2; 3 + (0, 0, 2) 2; 3 + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

a, b, pc

0, 0, u

pa, pb, c

u, v, 0

qa − rb, ra + qb, c

u, 0, 0

• Series of maximal isomorphic subgroups [p] c = pc P4¯ (81) 2; 3 + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups [p2 ] a = pa, b = pb P4¯ (81) 2 + (2u, 2v, 0); 3 + (u − v, u + v, 0) prime p > 2; 0 ≤ u < p; 0 ≤ v < p p2 conjugate subgroups for p = 4n − 1 [p = q2 + r2 ] a = qa − rb, b = ra + qb P4¯ (81) 2 + (2u, 0, 0); 3 + (u, u, 0) prime p > 4; q > 0; r > 0; 0 ≤ u < p p conjugate subgroups for p = 4n + 1

I Minimal translationengleiche supergroups ¯ ¯ (112); [2] P42 ¯ 1 m (113); [2] P42 ¯ 1 c (114); [2] P4/m (83); [2] P42 /m (84); [2] P4/n (85); [2] P42 /n (86); [2] P42m (111); [2] P42c ¯ ¯ ¯ ¯ [2] P4m2 (115); [2] P4c2 (116); [2] P4b2 (117); [2] P4n2 (118) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] I 4¯ (82)

• Decreased unit cell

Copyright © 2011 International Union of Crystallography

none

242

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 82, p. 243.

S42

I 4¯

I 4¯

No. 82

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t( 21 , 12 , 12 ); (2); (3) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

g

(0, 0, 0)+ ( 21 , 12 , 12 )+

1

(1) x, y, z

(2) x, ¯ y, ¯z

I Maximal translationengleiche subgroups [2] I2 (5, A112) (1; 2)+

(3) y, x, ¯ z¯

(4) y, ¯ x, z¯

b, −a − b, c

II Maximal klassengleiche subgroups

• Loss of centring translations [2] P4¯ (81) 1; 2; 3; 4 [2] P4¯ (81) 1; 2; (3; 4) + ( 21 , 12 , 12 )

1/2, 0, 1/4

• Enlarged unit cell

[3] ⎧ c = 3c ⎨ I 4¯ (82) I 4¯ (82) ⎩ ¯ I 4 (82)

2; 3 2; 3 + (0, 0, 2) 2; 3 + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

a, b, pc

0, 0, u

pa, pb, c

u, v, 0

qa − rb, ra + qb, c

u, 0, 0

• Series of maximal isomorphic subgroups

[p] c = pc I 4¯ (82)

[p2 ] a = pa, b = pb I 4¯ (82)

2; 3 + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups

2 + (2u, 2v, 0); 3 + (u − v, u + v, 0) prime p > 2; 0 ≤ u < p; 0 ≤ v < p p2 conjugate subgroups for p = 4n − 1 2 2 [p = q + r ] a = qa − rb, b = ra + qb I 4¯ (82) 2 + (2u, 0, 0); 3 + (u, u, 0) prime p > 4; q > 0; r > 0; 0 ≤ u < p p conjugate subgroups for p = 4n + 1

I Minimal translationengleiche supergroups ¯ ¯ (120); [2] I 42m ¯ ¯ (122) [2] I4/m (87); [2] I41 /a (88); [2] I 4m2 (119); [2] I 4c2 (121); [2] I 42d II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations

none

• Decreased unit cell ¯ [2] c = 12 c C4¯ (81, P4)

Copyright © 2011 International Union of Crystallography

243

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 83, p. 244.

P 4/m

No. 83

C4h1

P 4/m

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

l

1

(1) x, y, z (5) x, ¯ y, ¯ z¯

(2) x, ¯ y, ¯z (6) x, y, z¯

(3) y, ¯ x, z (7) y, x, ¯ z¯

(4) y, x, ¯z (8) y, ¯ x, z¯

I Maximal translationengleiche subgroups [2] P4¯ (81) 1; 2; 7; 8 [2] P4 (75) 1; 2; 3; 4 [2] P2/m (10, P112/m) 1; 2; 5; 6 II Maximal klassengleiche subgroups

• Enlarged unit cell [2] c = 2c P42 /m (84) P42 /m (84) P4/m (83) P4/m (83) [2] a = 2a, b = 2b C4/e (85, P4/n) C4/e (85, P4/n) C4/m (83, P4/m) C4/m (83, P4/m) [2] a = 2a, b = 2b, c = 2c F4/m (87, I4/m) F4/m (87, I4/m) F4/m (87, I4/m) F4/m (87, I4/m) [3] c = 3c P4/m (83) P4/m (83) P4/m (83)

2; 2; 2; 2;

5; 3 + (0, 0, 1) (3; 5) + (0, 0, 1) 3; 5 3; 5 + (0, 0, 1)

a, b, 2c a, b, 2c a, b, 2c a, b, 2c

0, 0, 1/2 0, 0, 1/2

2; 3; 5 + (1, 0, 0) 2 + (1, 1, 0); 3 + (1, 0, 0); 5 + (0, 1, 0) 2; 3; 5 (2; 5) + (1, 1, 0); 3 + (1, 0, 0)

a − b, a + b, c a − b, a + b, c a − b, a + b, c a − b, a + b, c

2; 3; 5 2; 3; 5 + (0, 0, 1) (2; 5) + (1, 1, 0); 3 + (1, 0, 0) 2 + (1, 1, 0); 3 + (1, 0, 0); 5 + (1, 1, 1)

a − b, a + b, 2c a − b, a + b, 2c a − b, a + b, 2c a − b, a + b, 2c

0, 0, 1/2 1/2, 1/2, 0 1/2, 1/2, 1/2

2; 3; 5 2; 3; 5 + (0, 0, 2) 2; 3; 5 + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

a, b, pc

0, 0, u

pa, pb, c

u, v, 0

qa − rb, ra + qb, c

u, 0, 0

• Series of maximal isomorphic subgroups [p] c = pc P4/m (83) 2; 3; 5 + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups [p2 ] a = pa, b = pb P4/m (83) (2; 5) + (2u, 2v, 0); 3 + (u + v, −u + v, 0) prime p > 2; 0 ≤ u < p; 0 ≤ v < p p2 conjugate subgroups for p = 4n − 1 [p = q2 + r2 ] a = qa − rb, b = ra + qb P4/m (83) (2; 5) + (2u, 0, 0); 3 + (u, −u, 0) prime p > 4; q > 0; r > 0; 0 ≤ u < p p conjugate subgroups for p = 4n + 1 I Minimal translationengleiche supergroups [2] P4/mmm (123); [2] P4/mcc (124); [2] P4/mbm (127); [2] P4/mnc (128) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] I4/m (87)

• Decreased unit cell

Copyright © 2011 International Union of Crystallography

none

244

1/2, 0, 0 0, 1/2, 0 1/2, 1/2, 0

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 84, p. 245.

C4h2

P 42/m

P 42/m

No. 84

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

k

1

(1) x, y, z (5) x, ¯ y, ¯ z¯

(2) x, ¯ y, ¯z (6) x, y, z¯

(3) y, ¯ x, z + 12 (7) y, x, ¯ z¯ + 12

(4) y, x, ¯ z + 12 (8) y, ¯ x, z¯ + 12

I Maximal translationengleiche subgroups [2] P4¯ (81) 1; 2; 7; 8 [2] P42 (77) 1; 2; 3; 4 [2] P2/m (10, P112/m) 1; 2; 5; 6

0, 0, 1/4

II Maximal klassengleiche subgroups

• Enlarged unit cell

[2] a = 2a, b = 2b C42 /e (86, P42 /n) C42 /e (86, P42 /n) C42 /m (84, P42 /m) C42 /m (84, P42 /m) [3] c = 3c P42 /m (84) P42 /m (84) P42 /m (84)

2; 3; 5 + (0, 1, 0) 2 + (1, 1, 0); (3; 5) + (1, 0, 0) 2; 3; 5 (2; 5) + (1, 1, 0); 3 + (1, 0, 0)

a − b, a + b, c a − b, a + b, c a − b, a + b, c a − b, a + b, c

2; 5; 3 + (0, 0, 1) 2; 3 + (0, 0, 1); 5 + (0, 0, 2) 2; 3 + (0, 0, 1); 5 + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

a, b, pc

0, 0, u

pa, pb, c

u, v, 0

qa − rb, ra + qb, c

u, 0, 0

0, 1/2, 0 1/2, 0, 0 1/2, 1/2, 0

• Series of maximal isomorphic subgroups

[p] c = pc P42 /m (84)

[p2 ] a = pa, b = pb P42 /m (84)

2; 3 + (0, 0, 2p − 12 ); 5 + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups

(2; 5) + (2u, 2v, 0); 3 + (u + v, −u + v, 0) prime p > 2; 0 ≤ u < p; 0 ≤ v < p p2 conjugate subgroups for p = 4n − 1 2 2 [p = q + r ] a = qa − rb, b = ra + qb (2; 5) + (u, u, 0); 3 + (u, 0, 0) P42 /m (84) prime p > 4; q > 0; r > 0; 0 ≤ u < p p conjugate subgroups for p = 4n + 1

I Minimal translationengleiche supergroups [2] P42 /mmc (131); [2] P42 /mcm (132); [2] P42 /mbc (135); [2] P42 /mnm (136) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] I4/m (87) • Decreased unit cell

[2] c = 12 c P4/m (83)

Copyright © 2011 International Union of Crystallography

245

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 85, pp. 246–247.

P 4/n ORIGIN CHOICE

No. 85

C4h3

P 4/n

1, Origin at 4¯ on n, at − 14 , 14 , 0 from 1¯

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

g

1

(1) x, y, z (5) x¯ + 12 , y¯ + 12 , z¯

(2) x, ¯ y, ¯z (6) x + 12 , y + 12 , z¯

I Maximal translationengleiche subgroups [2] P4¯ (81) 1; 2; 7; 8 [2] P4 (75) 1; 2; 3; 4 [2] P2/n (13, P112/a) 1; 2; 5; 6

(3) y¯ + 12 , x + 12 , z (7) y, x, ¯ z¯

(4) y + 12 , x¯ + 12 , z (8) y, ¯ x, z¯

−a − b, a, c

1/2, 0, 0 1/4, 1/4, 0

II Maximal klassengleiche subgroups

• Enlarged unit cell

[2] c = 2c P42 /n (86) P42 /n (86) P4/n (85) P4/n (85) [3] c = 3c P4/n (85) P4/n (85) P4/n (85)

2; 2; 2; 2;

5; 3 + (0, 0, 1) (3; 5) + (0, 0, 1) 3; 5 3; 5 + (0, 0, 1)

a, b, 2c a, b, 2c a, b, 2c a, b, 2c

2; 3; 5 2; 3; 5 + (0, 0, 2) 2; 3; 5 + (0, 0, 4)

0, 0, 1/2 0, 0, 1/2

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

a, b, pc

0, 0, u

• Series of maximal isomorphic subgroups

[p] c = pc P4/n (85)

[p2 ] a = pa, b = pb P4/n (85)

2; 3; 5 + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups

2 + (2u, 2v, 0); 3 + ( 2p − 12 + u + v, 2p − 12 − u + v, 0); pa, pb, c 5 + ( 2p − 12 + 2u, 2p − 12 + 2v, 0) prime p > 2; 0 ≤ u < p; 0 ≤ v < p p2 conjugate subgroups for p = 4n − 1 2 2 [p = q + r ] a = qa − rb, b = ra + qb P4/n (85) 2 + (2u, 0, 0); 3 + ( 2q + 2r − 12 + u, q2 − 2r − 12 − u, 0); qa − rb, ra + qb, c 5 + ( 2q + 2r − 12 + 2u, q2 − 2r − 12 , 0) prime p > 4; q > 0; r > 0; 0 ≤ u < p p conjugate subgroups for p = 4n + 1

I Minimal translationengleiche supergroups [2] P4/nbm (125); [2] P4/nnc (126); [2] P4/nmm (129); [2] P4/ncc (130) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] C4/m (83, P4/m); [2] I4/m (87)

• Decreased unit cell

Copyright © 2011 International Union of Crystallography

none

246

u, v, 0

u, 0, 0

C4h3

P 4/n

ORIGIN CHOICE

P 4/n

No. 85

2, Origin at 1¯ on n, at 14 , − 14 , 0 from 4¯

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

g

1

(1) x, y, z (5) x, ¯ y, ¯ z¯

(2) x¯ + 12 , y¯ + 12 , z (6) x + 12 , y + 12 , z¯

(3) y¯ + 12 , x, z (7) y + 12 , x, ¯ z¯

I Maximal translationengleiche subgroups [2] P4¯ (81) 1; 2; 7; 8 [2] P4 (75) 1; 2; 3; 4 [2] P2/n (13, P112/a) 1; 2; 5; 6

(4) y, x¯ + 12 , z (8) y, ¯ x + 12 , z¯

−a − b, a, c

1/4, 3/4, 0 1/4, 1/4, 0 0, 1/2, 0

II Maximal klassengleiche subgroups

• Enlarged unit cell

[2] c = 2c P42 /n (86) P42 /n (86) P4/n (85) P4/n (85) [3] c = 3c P4/n (85) P4/n (85) P4/n (85)

2; 2; 2; 2;

5; 3 + (0, 0, 1) (3; 5) + (0, 0, 1) 3; 5 3; 5 + (0, 0, 1)

a, b, 2c a, b, 2c a, b, 2c a, b, 2c

2; 3; 5 2; 3; 5 + (0, 0, 2) 2; 3; 5 + (0, 0, 4)

0, 1/2, 0 0, 1/2, 1/2 0, 0, 1/2

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

a, b, pc

0, 0, u

pa, pb, c

u, v, 0

qa − rb, ra + qb, c

u, 0, 0

qa − rb, ra + qb, c

1/2 + u, 0, 0

• Series of maximal isomorphic subgroups

[p] c = pc P4/n (85)

[p2 ] a = pa, b = pb P4/n (85)

2; 3; 5 + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups

2 + ( 2p − 12 + 2u, 2p − 12 + 2v, 0); 3 + ( 2p − 12 + u + v, −u + v, 0); 5 + (2u, 2v, 0) prime p > 2; 0 ≤ u < p; 0 ≤ v < p p2 conjugate subgroups for p = 4n − 1 2 2 [p = q + r ] a = qa − rb, b = ra + qb P4/n (85) 2 + ( 2q + 2r − 12 + 2u, q2 − 2r − 12 , 0); 3 + ( 2q − 12 + u, − 2r − u, 0); 5 + (2u, 0, 0) prime p = 4n + 1; q odd; r > 1; r even; 0 ≤ u < p p conjugate subgroups for each pair of q and r P4/n (85) 2 + ( 2q + 2r + 12 + 2u, q2 − 2r − 12 , 0); 3 + ( 2q + u, − 2r − 12 − u, 0); 5 + (1 + 2u, 0, 0) prime p = 4n + 1; q > 1; q even; r odd; 0 ≤ u < p p conjugate subgroups for each pair of q and r

I Minimal translationengleiche supergroups [2] P4/nbm (125); [2] P4/nnc (126); [2] P4/nmm (129); [2] P4/ncc (130) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] C4/m (83, P4/m); [2] I4/m (87) • Decreased unit cell

none

247

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 86, pp. 248–249.

P 42/n ORIGIN CHOICE

C4h4

P 42/n

No. 86 ¯ at − 1 , − 1 , − 1 from 1¯ 1, Origin at 4, 4 4 4

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

g

1

(1) x, y, z (5) x¯ + 12 , y¯ + 12 , z¯ + 12

(2) x, ¯ y, ¯z (6) x + 12 , y + 12 , z¯ + 12

(3) y¯ + 12 , x + 12 , z + 12 (7) y, x, ¯ z¯

I Maximal translationengleiche subgroups [2] P4¯ (81) 1; 2; 7; 8 [2] P42 (77) 1; 2; 3; 4 [2] P2/n (13, P112/a) 1; 2; 5; 6

(4) y + 12 , x¯ + 12 , z + 12 (8) y, ¯ x, z¯

−a − b, a, c

1/2, 0, 0 1/4, 1/4, 1/4

a − b, a + b, 2c a − b, a + b, 2c a − b, a + b, 2c a − b, a + b, 2c

0, 0, 1/2 1/2, 1/2, 0 1/2, 1/2, 1/2

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

a, b, pc

0, 0, u

II Maximal klassengleiche subgroups

• Enlarged unit cell

[2] a = 2a, b = 2b, c = 2c F41 /d (88, I41 /a) F41 /d (88, I41 /a) F41 /d (88, I41 /a) F41 /d (88, I41 /a) [3] c = 3c P42 /n (86) P42 /n (86) P42 /n (86)

2; 2; 2; 2;

3; 5 3; 5 + (0, 0, 1) 5; 3 + (0, 0, 1) (3; 5) + (0, 0, 1)

2; (3; 5) + (0, 0, 1) 2; 3 + (0, 0, 1); 5 + (0, 0, 3) 2; 3 + (0, 0, 1); 5 + (0, 0, 5)

• Series of maximal isomorphic subgroups

[p] c = pc P42 /n (86)

[p2 ] a = pa, b = pb P42 /n (86)

2; 3 + (0, 0, 2p − 12 ); 5 + (0, 0, 2p − 12 + 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups

2 + (2u, 2v, 0); 3 + ( 2p − 12 + u + v, 2p − 12 − u + v, 0); pa, pb, c 5 + ( 2p − 12 + 2u, 2p − 12 + 2v, 0) prime p > 2; 0 ≤ u < p; 0 ≤ v < p p2 conjugate subgroups for p = 4n − 1 2 2 [p = q + r ] a = qa − rb, b = ra + qb 2 + (2u, 0, 0); 3 + ( 2q + 2r − 12 + u, q2 − 2r − 12 − u, 0); qa − rb, ra + qb, c P42 /n (86) 5 + ( 2q + 2r − 12 + 2u, q2 − 2r − 12 , 0) prime p > 4; q > 0; r > 0; 0 ≤ u < p p conjugate subgroups for p = 4n + 1

I Minimal translationengleiche supergroups [2] P42 /nbc (133); [2] P42 /nnm (134); [2] P42 /nmc (137); [2] P42 /ncm (138) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] C42 /m (84, P42 /m); [2] I4/m (87)

• Decreased unit cell [2] c = 12 c P4/n (85)

Copyright © 2011 International Union of Crystallography

248

u, v, 0

u, 0, 0

C4h4

P 42/n

ORIGIN CHOICE

P 42/n

No. 86

2, Origin at 1¯ on n, at 14 , 14 , 14 from 4¯

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

g

1

(1) x, y, z (5) x, ¯ y, ¯ z¯

(2) x¯ + 12 , y¯ + 12 , z (6) x + 12 , y + 12 , z¯

(3) y, ¯ x + 12 , z + 12 (7) y, x¯ + 12 , z¯ + 12

I Maximal translationengleiche subgroups [2] P4¯ (81) 1; 2; 7; 8 [2] P42 (77) 1; 2; 3; 4 [2] P2/n (13, P112/a) 1; 2; 5; 6

(4) y + 12 , x, ¯ z + 12 1 (8) y¯ + 2 , x, z¯ + 12

−a − b, a, c

1/4, 1/4, 1/4 3/4, 1/4, 0

II Maximal klassengleiche subgroups

• Enlarged unit cell

[2] a = 2a, b = 2b, c = 2c F41 /d (88, I41 /a) F41 /d (88, I41 /a) F41 /d (88, I41 /a) F41 /d (88, I41 /a) [3] c = 3c P42 /n (86) P42 /n (86) P42 /n (86)

2; 2; 2; 2;

3; 5 3; 5 + (0, 0, 1) 5; 3 + (0, 0, 1) (3; 5) + (0, 0, 1)

2; 5; 3 + (0, 0, 1) 2; 3 + (0, 0, 1); 5 + (0, 0, 2) 2; 3 + (0, 0, 1); 5 + (0, 0, 4)

a − b, a + b, 2c a − b, a + b, 2c a − b, a + b, 2c a − b, a + b, 2c

0, 0, 1/2 1/2, 1/2, 0 1/2, 1/2, 1/2

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

a, b, pc

0, 0, u

pa, pb, c

u, v, 0

qa − rb, ra + qb, c

u, 0, 0

qa − rb, ra + qb, c

1/2 + u, 0, 0

• Series of maximal isomorphic subgroups

[p] c = pc P42 /n (86)

[p2 ] a = pa, b = pb P42 /n (86)

2; 3 + (0, 0, 2p − 12 ); 5 + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups

2 + ( 2p − 12 + 2u, 2p − 12 + 2v, 0); 3 + (u + v, 2p − 12 − u + v, 0); 5 + (2u, 2v, 0) prime p > 2; 0 ≤ u < p; 0 ≤ v < p p2 conjugate subgroups for p = 4n − 1 2 2 [p = q + r ] a = qa − rb, b = ra + qb 2 + ( 2q + 2r − 12 + 2u, q2 − 2r − 12 , 0); P42 /n (86) 3 + ( 2r + u, q2 − 12 − u, 0); 5 + (2u, 0, 0) prime p = 4n + 1; q odd; r > 1; r even; 0 ≤ u < p p conjugate subgroups for each pair of q and r 2 + ( 2q + 2r + 12 + 2u, q2 − 2r − 12 , 0); P42 /n (86) 3 + ( 2r + 12 + u, q2 − 1 − u, 0); 5 + (1 + 2u, 0, 0) prime p = 4n + 1; q > 1; q even; r odd; 0 ≤ u < p p conjugate subgroups for each pair of q and r

I Minimal translationengleiche supergroups [2] P42 /nbc (133); [2] P42 /nnm (134); [2] P42 /nmc (137); [2] P42 /ncm (138) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] C42 /m (84, P42 /m); [2] I4/m (87) • Decreased unit cell

[2] c = 12 c P4/n (85)

249

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 87, p. 250.

I 4/m

No. 87

C4h5

I 4/m

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t( 21 , 12 , 12 ); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

16

i

(0, 0, 0)+ ( 21 , 12 , 12 )+

1

(1) x, y, z (5) x, ¯ y, ¯ z¯

(2) x, ¯ y, ¯z (6) x, y, z¯

I Maximal translationengleiche subgroups [2] I 4¯ (82) (1; 2; 7; 8)+ [2] I4 (79) (1; 2; 3; 4)+ [2] I2/m (12, A112/m) (1; 2; 5; 6)+

(3) y, ¯ x, z (7) y, x, ¯ z¯

(4) y, x, ¯z (8) y, ¯ x, z¯

b, −a − b, c

II Maximal klassengleiche subgroups

• Loss of centring translations [2] [2] [2] [2]

P42 /n (86) P4/n (85) P42 /m (84) P4/m (83)

1; 1; 1; 1;

2; 2; 2; 2;

7; 3; 5; 3;

8; 4; 6; 4;

(3; 4; 5; 6) + ( 21 , 12 , 12 ) (5; 6; 7; 8) + ( 21 , 12 , 12 ) (3; 4; 7; 8) + ( 21 , 12 , 12 ) 5; 6; 7; 8

1/4, 1/4, 1/4 1/4, 1/4, 1/4 0, 1/2, 0

• Enlarged unit cell

[3] c = 3c I4/m (87) I4/m (87) I4/m (87)

2; 3; 5 2; 3; 5 + (0, 0, 2) 2; 3; 5 + (0, 0, 4)

• Series of maximal isomorphic subgroups [p] c = pc I4/m (87) 2; 3; 5 + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups [p2 ] a = pa, b = pb I4/m (87) (2; 5) + (2u, 2v, 0); 3 + (u + v, −u + v, 0) prime p > 2; 0 ≤ u < p; 0 ≤ v < p p2 conjugate subgroups for p = 4n − 1 2 2 [p = q + r ] a = qa − rb, b = ra + qb I4/m (87) (2; 5) + (2u, 0, 0); 3 + (u, −u, 0) prime p > 4; q > 0; r > 0; 0 ≤ u < p p conjugate subgroups for p = 4n + 1

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

a, b, pc

0, 0, u

pa, pb, c

u, v, 0

qa − rb, ra + qb, c

u, 0, 0

I Minimal translationengleiche supergroups [2] I4/mmm (139); [2] I4/mcm (140) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations

none

• Decreased unit cell [2] c = 12 c C4/m (83, P4/m)

Copyright © 2011 International Union of Crystallography

250

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 88, pp. 251–252.

C4h6

I 41/a

ORIGIN CHOICE

I 41/a

No. 88

¯ at 0, − 1 , − 1 from 1¯ 1, Origin at 4, 4 8

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t( 21 , 12 , 12 ); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

16

f

(0, 0, 0)+ ( 21 , 12 , 12 )+

1

(1) x, y, z (5) x, ¯ y¯ + 12 , z¯ + 14

(2) x¯ + 12 , y¯ + 12 , z + 12 (6) x + 12 , y, z¯ + 34

(3) y, ¯ x + 12 , z + 14 (7) y, x, ¯ z¯

I Maximal translationengleiche subgroups [2] I 4¯ (82) (1; 2; 7; 8)+ [2] I41 (80) (1; 2; 3; 4)+ [2] I2/a (15, A112/a) (1; 2; 5; 6)+

(4) y + 12 , x, ¯ z + 34 (8) y¯ + 12 , x + 12 , z¯ + 12

b, −a − b, c

0, 1/4, 1/8

II Maximal klassengleiche subgroups

• Loss of centring translations • Enlarged unit cell [3] c = 3c ⎧ ⎨ I41 /a (88) I4 /a (88) ⎩ 1 I41 /a (88)

none

(2; 5) + (1, 0, 1); 3 + ( 21 , − 21 , 12 ) 2 + (1, 0, 1); 3 + ( 21 , − 21 , 12 ); 5 + (1, 0, 3) 2 + (1, 0, 1); 3 + ( 21 , − 21 , 12 ); 5 + (1, 0, 5)

a, b, 3c a, b, 3c a, b, 3c

1/2, 0, 1/4 1/2, 0, 5/4 1/2, 0, 9/4

a, b, pc

0, 0, u

a, b, pc

1/2, 0, 1/4 + u

pa, pb, c

u, v, 0

qa − rb, ra + qb, c

u, 0, 0

qa − rb, ra + qb, c

1/2 + u, 0, 1/4

• Series of maximal isomorphic subgroups

[p] c = pc I41 /a (88) I41 /a (88)

[p2 ] a = pa, b = pb I41 /a (88)

2 + (0, 0, 2p − 12 ); 3 + (0, 0, 4p − 14 ); 5 + (0, 0, 4p − 14 + 2u) prime p > 4; 0 ≤ u < p p conjugate subgroups for p = 4n + 1 2 + (1, 0, 2p − 12 ); 3 + ( 21 , − 21 , 4p − 14 ); 5 + (1, 0, 4p + 14 + 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups for p = 4n − 1

2 + ( 2p − 12 + 2u, 2p − 12 + 2v, 0); 3 + (u + v, 2p − 12 − u + v, 0); 5 + (2u, 2p − 12 + 2v, 0) prime p > 2; 0 ≤ u < p; 0 ≤ v < p p2 conjugate subgroups for p = 4n − 1 [p = q2 + r2 ] a = qa − rb, b = ra + qb 2 + ( 2q + 2r − 12 + 2u, q2 − 2r − 12 , 0); I41 /a (88) 3 + ( 2r + u, q2 − 12 − u, 0); 5 + ( 2r + 2u, q2 − 12 , 0) prime p > 4; q > 0; r > 1; 0 ≤ u < p p conjugate subgroups for odd q and p = 4n + 1 2 + ( 2q + 2r + 12 + 2u, q2 − 2r − 12 , 0); I41 /a (88) 3 + ( 2r + 12 + u, q2 − 1 − u, 0); 5 + ( 2r + 1 + 2u, q2 − 12 , 12 ) prime p > 4; q > 1; r > 0; 0 ≤ u < p p conjugate subgroups for even q and p = 4n + 1 I Minimal translationengleiche supergroups [2] I41 /amd (141); [2] I41 /acd (142) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations

none

• Decreased unit cell [2] c = 12 c C42 /e (86, P42 /n)

Copyright © 2011 International Union of Crystallography

251

I 41/a ORIGIN CHOICE

C4h6

I 41/a

No. 88 2, Origin at 1¯ on glide plane b, at 0, 14 , 18 from 4¯

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t( 21 , 12 , 12 ); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

16

f

(0, 0, 0)+ ( 21 , 12 , 12 )+

1

(1) x, y, z (5) x, ¯ y, ¯ z¯

(2) x¯ + 12 , y, ¯ z + 12 1 (6) x + 2 , y, z¯ + 12

(3) y¯ + 34 , x + 14 , z + 14 (7) y + 14 , x¯ + 34 , z¯ + 34

I Maximal translationengleiche subgroups [2] I 4¯ (82) (1; 2; 7; 8)+ (1; 2; 3; 4)+ [2] I41 (80) [2] I2/a (15, A112/a) (1; 2; 5; 6)+

(4) y + 34 , x¯ + 34 , z + 34 (8) y¯ + 14 , x + 14 , z¯ + 14

b, −a − b, c

0, 1/4, 5/8 1/2, 1/4, 0

II Maximal klassengleiche subgroups

• Loss of centring translations • Enlarged unit cell

[3] ⎧ c = 3c ⎨ I41 /a (88) I4 /a (88) ⎩ 1 I41 /a (88)

none

2 + (1, 0, 1); 3 + ( 21 , − 21 , 12 ); 5 + (1, 0, 0) 2 + (1, 0, 1); 3 + ( 21 , − 21 , 12 ); 5 + (1, 0, 2) 2 + (1, 0, 1); 3 + ( 21 , − 21 , 12 ); 5 + (1, 0, 4)

• Series of maximal isomorphic subgroups [p] c = pc I41 /a (88) 2 + (0, 0, 2p − 12 ); 3 + (0, 0, 4p − 14 ); 5 + (0, 0, 2u) prime p > 4; 0 ≤ u < p p conjugate subgroups for p = 4n + 1 2 + (1, 0, 2p − 12 ); 3 + ( 21 , − 21 , 4p − 14 ); 5 + (1, 0, 2u) I41 /a (88) prime p > 2; 0 ≤ u < p p conjugate subgroups for p = 4n − 1 [p2 ] a = pa, b = pb I41 /a (88) 2 + ( 2p + 12 + 2u, 2v, 0); p 1 3 3 + ( 3p 4 − 4 + u + v, 4 − 4 − u + v, 0); 5 + (1 + 2u, 2v, 0) prime p > 2; 0 ≤ u < p; 0 ≤ v < p p2 conjugate subgroups for p = 4n − 1 2 2 [p = q + r ] a = qa − rb, b = ra + qb I41 /a (88) 2 + ( 2q − 12 + 2u, − 2r , 0); q r 3 3r 1 3 + ( 3q 4 + 4 − 4 + u, 4 − 4 − 4 − u, 0); 5 + (2u, 0, 0) prime p = 4n + 1; 0 ≤ u < p; q > 0; r = 2n ; q + r = 4n + 1 p > 12; p conjugate subgroups for each triplet of q, r and p I41 /a (88) 2 + ( 2q + 12 + 2u, − 2r , 0); q r 1 3r 3 3 + ( 3q 4 + 4 − 4 + u, 4 − 4 − 4 − u, 0); 5 + (1 + 2u, 0, 0) prime p = 4n + 1; 0 ≤ u < p; q > 0; r = 2n ; q + r = 4n − 1 p > 4; p conjugate subgroups for each triplet of q, r and p I41 /a (88) 2 + ( 2q + 2u, − 2r + 12 , 0); q r 1 3r 1 1 1 1 3 + ( 3q 4 + 4 − 4 + u, 4 − 4 − 4 − u, 0); 5 + ( 2 + 2u, 2 , 2 ) prime p = 4n + 1; 0 ≤ u < p; q = 2n ; r > 0; q + r = 4n + 1 p > 12; p conjugate subgroups for each triplet of q, r and p I41 /a (88) 2 + ( 2q + 1 + 2u, − 2r + 12 , 0); q r 1 3r 3 3 1 1 3 + ( 3q 4 + 4 + 4 + u, 4 − 4 − 4 − u, 0); 5 + ( 2 + 2u, 2 , 2 ) prime p = 4n + 1; 0 ≤ u < p; q = 2n ; r > 0; q + r = 4n − 1 p > 4; p conjugate subgroups for each triplet of q, r and p

a, b, 3c a, b, 3c a, b, 3c

1/2, 0, 0 1/2, 0, 1 1/2, 0, 2

a, b, pc

0, 0, u

a, b, pc

1/2, 0, u

pa, pb, c

1/2 + u, v, 0

qa − rb, ra + qb, c

u, 0, 0

qa − rb, ra + qb, c

1/2 + u, 0, 0

qa − rb, ra + qb, c

1/4 + u, 1/4, 1/4

qa − rb, ra + qb, c

3/4 + u, 1/4, 1/4

I Minimal translationengleiche supergroups [2] I41 /amd (141); [2] I41 /acd (142) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations • Decreased unit cell

none

[2] c = 12 c C42 /e (86, P42 /n)

252

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 89, p. 253.

D14

P422

P422

No. 89

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

p

1

(1) x, y, z (5) x, ¯ y, z¯

(2) x, ¯ y, ¯z (6) x, y, ¯ z¯

I Maximal translationengleiche subgroups [2] P411 (75, P4) 1; 2; 3; 4 [2] P212 (21, C222) 1; 2; 7; 8 [2] P221 (16, P222) 1; 2; 5; 6

(3) y, ¯ x, z (7) y, x, z¯

(4) y, x, ¯z (8) y, ¯ x, ¯ z¯

a − b, a + b, c

II Maximal klassengleiche subgroups

• Enlarged unit cell [2] c = 2c P42 22 (93) P42 22 (93) P422 (89) P422 (89) [2] a = 2a, b = 2b C4221 (90, P421 2) C4221 (90, P421 2) C422 (89, P422) C422 (89, P422) [2] a = 2a, b = 2b, c = 2c F422 (97, I422) F422 (97, I422) F422 (97, I422) F422 (97, I422) [3] c = 3c P422 (89) P422 (89) P422 (89)

2; 2; 2; 2;

5; 3 + (0, 0, 1) (3; 5) + (0, 0, 1) 3; 5 3; 5 + (0, 0, 1)

a, b, 2c a, b, 2c a, b, 2c a, b, 2c

0, 0, 1/2 0, 0, 1/2

2; 3; 5 + (1, 0, 0) 2; 5; 3 + (1, 0, 0) 2; 3; 5 2 + (1, 1, 0); (3; 5) + (1, 0, 0)

a − b, a + b, c a − b, a + b, c a − b, a + b, c a − b, a + b, c

2; 2; 2; 2;

a − b, a + b, 2c a − b, a + b, 2c a − b, a + b, 2c a − b, a + b, 2c

0, 0, 1/2 1/2, 1/2, 0 1/2, 1/2, 1/2

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

3; 5 3; 5 + (0, 0, 1) (3; 5) + (1, 0, 0) 5; 3 + (1, 0, 0)

2; 3; 5 2; 3; 5 + (0, 0, 2) 2; 3; 5 + (0, 0, 4)

• Series of maximal isomorphic subgroups [p] c = pc a, b, pc P422 (89) 2; 3; 5 + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups [p2 ] a = pa, b = pb P422 (89) 2 + (2u, 2v, 0); 3 + (u + v, −u + v, 0); 5 + (2u, 0, 0) pa, pb, c prime p > 2; 0 ≤ u < p; 0 ≤ v < p p2 conjugate subgroups I Minimal translationengleiche supergroups [2] P4/mmm (123); [2] P4/mcc (124); [2] P4/nbm (125); [2] P4/nnc (126); [3] P432 (207) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] I422 (97)

• Decreased unit cell

Copyright © 2011 International Union of Crystallography

none

253

1/2, 1/2, 0 1/2, 1/2, 0

0, 0, u

u, v, 0

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 90, p. 254.

P 4 21 2

No. 90

D24

P 4 21 2

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

g

1

(1) x, y, z (5) x¯ + 12 , y + 12 , z¯

(2) x, ¯ y, ¯z (6) x + 12 , y¯ + 12 , z¯

(3) y¯ + 12 , x + 12 , z (7) y, x, z¯

I Maximal translationengleiche subgroups [2] P411 (75, P4) 1; 2; 3; 4 [2] P212 (21, C222) 1; 2; 7; 8 [2] P221 1 (18, P21 21 2) 1; 2; 5; 6

(4) y + 12 , x¯ + 12 , z (8) y, ¯ x, ¯ z¯

a − b, a + b, c

0, 1/2, 0

II Maximal klassengleiche subgroups

• Enlarged unit cell

[2] c = 2c P42 21 2 (94) P42 21 2 (94) P421 2 (90) P421 2 (90) [3] c = 3c P421 2 (90) P421 2 (90) P421 2 (90)

2; 2; 2; 2;

5; 3 + (0, 0, 1) (3; 5) + (0, 0, 1) 3; 5 3; 5 + (0, 0, 1)

a, b, 2c a, b, 2c a, b, 2c a, b, 2c

2; 3; 5 2; 3; 5 + (0, 0, 2) 2; 3; 5 + (0, 0, 4)

0, 0, 1/2 0, 0, 1/2

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

2; 3; 5 + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, b, pc

0, 0, u

2 + (2u, 2v, 0); 3 + ( 2p − 12 + u + v, 2p − 12 − u + v, 0); 5 + ( 2p − 12 + 2u, 2p − 12 , 0) prime p > 2; 0 ≤ u < p; 0 ≤ v < p p2 conjugate subgroups

pa, pb, c

u, v, 0

• Series of maximal isomorphic subgroups

[p] c = pc P421 2 (90)

[p2 ] a = pa, b = pb P421 2 (90)

I Minimal translationengleiche supergroups [2] P4/mbm (127); [2] P4/mnc (128); [2] P4/nmm (129); [2] P4/ncc (130) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] C422 (89, P422); [2] I422 (97)

• Decreased unit cell

Copyright © 2011 International Union of Crystallography

none

254

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 91, p. 255.

D34

P 41 2 2

P 41 2 2

No. 91

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

d

1

(1) x, y, z (5) x, ¯ y, z¯

(2) x, ¯ y, ¯ z + 12 (6) x, y, ¯ z¯ + 12

(3) y, ¯ x, z + 14 (7) y, x, z¯ + 34

I Maximal translationengleiche subgroups [2] P41 11 (76, P41 ) 1; 2; 3; 4 1; 2; 7; 8 [2] P21 12 (20, C2221 ) 1; 2; 5; 6 [2] P21 21 (17, P2221 )

(4) y, x, ¯ z + 34 (8) y, ¯ x, ¯ z¯ + 14

a − b, a + b, c

0, 0, 1/8 0, 0, 1/4

2; 3; 5 + (1, 0, 0) 2; 5; 3 + (1, 0, 0) 2; 3; 5 + (0, 0, 1) 2 + (1, 1, 0); 3 + (1, 0, 0); 5 + (1, 0, 1)

a − b, a + b, c a − b, a + b, c a − b, a + b, c a − b, a + b, c

1/2, 1/2, 1/4 0, 0, 1/4 0, 0, 3/8 1/2, 1/2, 3/8

5; 2 + (0, 0, 1); 3 + (0, 0, 2) 2 + (0, 0, 1); (3; 5) + (0, 0, 2) 2 + (0, 0, 1); 3 + (0, 0, 2); 5 + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

1 2 + (0, 0, 2p − 12 ); 3 + (0, 0, 3p 4 − 4 ); 5 + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups for p = 4n − 1 2 + (0, 0, 2p − 12 ); 3 + (0, 0, 4p − 14 ); 5 + (0, 0, 2u) prime p > 4; 0 ≤ u < p p conjugate subgroups for p = 4n + 1

a, b, pc

0, 0, u

a, b, pc

0, 0, u

2 + (2u, 2v, 0); 3 + (u + v, −u + v, 0); 5 + (2u, 0, 0) prime p > 2; 0 ≤ u < p; 0 ≤ v < p p2 conjugate subgroups

pa, pb, c

u, v, 0

II Maximal klassengleiche subgroups

• Enlarged unit cell

[2] a = 2a, b = 2b C41 221 (92, P41 21 2) C41 221 (92, P41 21 2) C41 22 (91, P41 22) C41 22 (91, P41 22) [3] c = 3c P43 22 (95) P43 22 (95) P43 22 (95)

• Series of maximal isomorphic subgroups

[p] c = pc P43 22 (95) P41 22 (91)

[p2 ] a = pa, b = pb P41 22 (91)

I Minimal translationengleiche supergroups

none

II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] I41 22 (98) • Decreased unit cell [2] c = 12 c P42 22 (93)

Copyright © 2011 International Union of Crystallography

255

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 92, p. 256.

P 41 21 2

No. 92

D44

P 4 1 21 2

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

b

1

(1) x, y, z (5) x¯ + 12 , y + 12 , z¯ + 14

(2) x, ¯ y, ¯ z + 12 (6) x + 12 , y¯ + 12 , z¯ + 34

(3) y¯ + 12 , x + 12 , z + 14 (7) y, x, z¯

I Maximal translationengleiche subgroups [2] P41 11 (76, P41 ) 1; 2; 3; 4 1; 2; 7; 8 [2] P21 12 (20, C2221 ) 1; 2; 5; 6 [2] P21 21 1 (19, P21 21 21 )

(4) y + 12 , x¯ + 12 , z + 34 (8) y, ¯ x, ¯ z¯ + 12

a − b, a + b, c

0, 1/2, 0 0, 0, 1/4 1/4, 0, 3/8

II Maximal klassengleiche subgroups

• Enlarged unit cell

[3] c = 3c P43 21 2 (96) P43 21 2 (96) P43 21 2 (96)

2 + (0, 0, 1); (3; 5) + (0, 0, 2) 2 + (0, 0, 1); 3 + (0, 0, 2); 5 + (0, 0, 4) 2 + (0, 0, 1); 3 + (0, 0, 2); 5 + (0, 0, 6)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

1 2 + (0, 0, 2p − 12 ); 3 + (0, 0, 3p 4 − 4 ); 1 5 + (0, 0, 3p 4 − 4 + 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups for p = 4n − 1 2 + (0, 0, 2p − 12 ); 3 + (0, 0, 4p − 14 ); 5 + (0, 0, 4p − 14 + 2u) prime p > 4; 0 ≤ u < p p conjugate subgroups for p = 4n + 1

a, b, pc

0, 0, u

a, b, pc

0, 0, u

2 + (2u, 2v, 0); 3 + ( 2p − 12 + u + v, 2p − 12 − u + v, 0); 5 + ( 2p − 12 + 2u, 2p − 12 , 0) prime p > 2; 0 ≤ u < p; 0 ≤ v < p p2 conjugate subgroups

pa, pb, c

u, v, 0

• Series of maximal isomorphic subgroups

[p] c = pc P43 21 2 (96)

P41 21 2 (92)

[p2 ] a = pa, b = pb P41 21 2 (92)

I Minimal translationengleiche supergroups [3] P41 32 (213) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] C41 22 (91, P41 22); [2] I41 22 (98) • Decreased unit cell

[2] c = 12 c P42 21 2 (94)

Copyright © 2011 International Union of Crystallography

256

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 93, p. 257.

D54

P 42 2 2

P 42 2 2

No. 93

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

p

1

(1) x, y, z (5) x, ¯ y, z¯

(2) x, ¯ y, ¯z (6) x, y, ¯ z¯

(3) y, ¯ x, z + 12 (7) y, x, z¯ + 12

I Maximal translationengleiche subgroups [2] P42 11 (77, P42 ) 1; 2; 3; 4 [2] P212 (21, C222) 1; 2; 7; 8 [2] P221 (16, P222) 1; 2; 5; 6

(4) y, x, ¯ z + 12 (8) y, ¯ x, ¯ z¯ + 12

a − b, a + b, c

0, 0, 1/4

II Maximal klassengleiche subgroups

• Enlarged unit cell

[2] c = 2c P43 22 (95) P43 22 (95) P41 22 (91) P41 22 (91) [2] a = 2a, b = 2b C42 221 (94, P42 21 2) C42 221 (94, P42 21 2) C42 22 (93, P42 22) C42 22 (93, P42 22) [2] a = 2a, b = 2b, c = 2c F41 22 (98, I41 22) F41 22 (98, I41 22) F41 22 (98, I41 22) F41 22 (98, I41 22) [3] c = 3c P42 22 (93) P42 22 (93) P42 22 (93)

5; (2; 3) + (0, 0, 1) (2; 3; 5) + (0, 0, 1) 3; 5; 2 + (0, 0, 1) 3; (2; 5) + (0, 0, 1)

a, b, 2c a, b, 2c a, b, 2c a, b, 2c

2; 5; 3 + (1, 0, 0) 2; 3; 5 + (1, 0, 0) 2; 3; 5 + (0, 0, 1) 2 + (1, 1, 0); 3 + (1, 0, 0); 5 + (1, 0, 1)

a − b, a + b, c a − b, a + b, c a − b, a + b, c a − b, a + b, c

1/2, 1/2, 0 0, 0, 1/4 1/2, 1/2, 1/4

3; 5; 2 + (0, 0, 1) 3; (2; 5) + (0, 0, 1) (2; 3; 5) + (1, 0, 0) (2; 3) + (1, 0, 0); 5 + (1, 0, 1)

a − b, a + b, 2c a − b, a + b, 2c a − b, a + b, 2c a − b, a + b, 2c

0, 1/2, 0 0, 1/2, 1/2 1/2, 0, 0 1/2, 0, 1/2

2; 5; 3 + (0, 0, 1) 2; 3 + (0, 0, 1); 5 + (0, 0, 2) 2; 3 + (0, 0, 1); 5 + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

2; 3 + (0, 0, 2p − 12 ); 5 + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, b, pc

0, 0, u

2 + (2u, 2v, 0); 3 + (u + v, −u + v, 0); 5 + (2u, 0, 0) prime p > 2; 0 ≤ u < p; 0 ≤ v < p p2 conjugate subgroups

pa, pb, c

u, v, 0

0, 0, 1/2 0, 0, 1/2

• Series of maximal isomorphic subgroups

[p] c = pc P42 22 (93)

[p2 ] a = pa, b = pb P42 22 (93)

I Minimal translationengleiche supergroups [2] P42 /mmc (131); [2] P42 /mcm (132); [2] P42 /nbc (133); [2] P42 /nnm (134); [3] P42 32 (208) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] I422 (97)

• Decreased unit cell [2] c = 12 c P422 (89)

Copyright © 2011 International Union of Crystallography

257

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 94, p. 258.

P 42 21 2

No. 94

D64

P 4 2 21 2

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

g

1

(1) x, y, z (5) x¯ + 12 , y + 12 , z¯ + 12

(2) x, ¯ y, ¯z (6) x + 12 , y¯ + 12 , z¯ + 12

(3) y¯ + 12 , x + 12 , z + 12 (7) y, x, z¯

I Maximal translationengleiche subgroups 1; 2; 3; 4 [2] P42 11 (77, P42 ) [2] P212 (21, C222) 1; 2; 7; 8 1; 2; 5; 6 [2] P221 1 (18, P21 21 2)

(4) y + 12 , x¯ + 12 , z + 12 (8) y, ¯ x, ¯ z¯

a − b, a + b, c

0, 1/2, 0 0, 0, 1/4

II Maximal klassengleiche subgroups

• Enlarged unit cell

[2] c = 2c P43 21 2 P43 21 2 P41 21 2 P41 21 2 [3] c = 3c P42 21 2 P42 21 2 P42 21 2

(96) (96) (92) (92)

5; (2; 3) + (0, 0, 1) (2; 3; 5) + (0, 0, 1) 3; 5; 2 + (0, 0, 1) 3; (2; 5) + (0, 0, 1)

a, b, 2c a, b, 2c a, b, 2c a, b, 2c

(94) (94) (94)

2; (3; 5) + (0, 0, 1) 2; 3 + (0, 0, 1); 5 + (0, 0, 3) 2; 3 + (0, 0, 1); 5 + (0, 0, 5)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

2; 3 + (0, 0, 2p − 12 ); 5 + (0, 0, 2p − 12 + 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, b, pc

0, 0, u

2 + (2u, 2v, 0); 3 + ( 2p − 12 + u + v, 2p − 12 − u + v, 0); 5 + ( 2p − 12 + 2u, 2p − 12 , 0) prime p > 2; 0 ≤ u < p; 0 ≤ v < p p2 conjugate subgroups

pa, pb, c

u, v, 0

0, 0, 1/2 0, 0, 1/2

• Series of maximal isomorphic subgroups

[p] c = pc P42 21 2 (94)

[p2 ] a = pa, b = pb P42 21 2 (94)

I Minimal translationengleiche supergroups [2] P42 /mbc (135); [2] P42 /mnm (136); [2] P42 /nmc (137); [2] P42 /ncm (138) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] C42 22 (93, P42 22); [2] I422 (97)

• Decreased unit cell

[2] c = 12 c P421 2 (90)

Copyright © 2011 International Union of Crystallography

258

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 95, p. 259.

D74

P 43 2 2

P 43 2 2

No. 95

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

d

1

(1) x, y, z (5) x, ¯ y, z¯

(2) x, ¯ y, ¯ z + 12 (6) x, y, ¯ z¯ + 12

(3) y, ¯ x, z + 34 (7) y, x, z¯ + 14

I Maximal translationengleiche subgroups [2] P43 11 (78, P43 ) 1; 2; 3; 4 1; 2; 7; 8 [2] P21 12 (20, C2221 ) 1; 2; 5; 6 [2] P21 21 (17, P2221 )

(4) y, x, ¯ z + 14 (8) y, ¯ x, ¯ z¯ + 34

a − b, a + b, c

0, 0, 3/8 0, 0, 1/4

2; 5; 3 + (1, 0, 0) 2; 3; 5 + (1, 0, 0) 2; 3; 5 + (0, 0, 1) 2 + (1, 1, 0); 3 + (1, 0, 0); 5 + (1, 0, 1)

a − b, a + b, c a − b, a + b, c a − b, a + b, c a − b, a + b, c

0, 0, 1/4 1/2, 1/2, 1/4 0, 0, 1/8 1/2, 1/2, 1/8

3; 5; 2 + (0, 0, 1) 3; 2 + (0, 0, 1); 5 + (0, 0, 2) 3; 2 + (0, 0, 1); 5 + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

3 2 + (0, 0, 2p − 12 ); 3 + (0, 0, 3p 4 − 4 ); 5 + (0, 0, 2u) prime p > 4; 0 ≤ u < p p conjugate subgroups for p = 4n + 1 2 + (0, 0, 2p − 12 ); 3 + (0, 0, 4p − 34 ); 5 + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups for p = 4n − 1

a, b, pc

0, 0, u

a, b, pc

0, 0, u

2 + (2u, 2v, 0); 3 + (u + v, −u + v, 0); 5 + (2u, 0, 0) prime p > 2; 0 ≤ u < p; 0 ≤ v < p p2 conjugate subgroups

pa, pb, c

u, v, 0

II Maximal klassengleiche subgroups

• Enlarged unit cell

[2] a = 2a, b = 2b C43 221 (96, P43 21 2) C43 221 (96, P43 21 2) C43 22 (95, P43 22) C43 22 (95, P43 22) [3] c = 3c P41 22 (91) P41 22 (91) P41 22 (91)

• Series of maximal isomorphic subgroups

[p] c = pc P43 22 (95) P41 22 (91)

[p2 ] a = pa, b = pb P43 22 (95)

I Minimal translationengleiche supergroups

none

II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] I41 22 (98) • Decreased unit cell [2] c = 12 c P42 22 (93)

Copyright © 2011 International Union of Crystallography

259

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 96, p. 260.

P 43 21 2

No. 96

D84

P 4 3 21 2

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

b

1

(1) x, y, z (5) x¯ + 12 , y + 12 , z¯ + 34

(2) x, ¯ y, ¯ z + 12 (6) x + 12 , y¯ + 12 , z¯ + 14

(3) y¯ + 12 , x + 12 , z + 34 (7) y, x, z¯

I Maximal translationengleiche subgroups [2] P43 11 (78, P43 ) 1; 2; 3; 4 1; 2; 7; 8 [2] P21 12 (20, C2221 ) 1; 2; 5; 6 [2] P21 21 1 (19, P21 21 21 )

(4) y + 12 , x¯ + 12 , z + 14 (8) y, ¯ x, ¯ z¯ + 12

a − b, a + b, c

0, 1/2, 0 0, 0, 1/4 1/4, 0, 1/8

II Maximal klassengleiche subgroups

• Enlarged unit cell

[3] c = 3c P41 21 2 (92) P41 21 2 (92) P41 21 2 (92)

3; 5; 2 + (0, 0, 1) 3; 2 + (0, 0, 1); 5 + (0, 0, 2) 3; 2 + (0, 0, 1); 5 + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

3 2 + (0, 0, 2p − 12 ); 3 + (0, 0, 3p 4 − 4 ); 3 5 + (0, 0, 3p 4 − 4 + 2u) prime p > 4; 0 ≤ u < p p conjugate subgroups for p = 4n + 1 2 + (0, 0, 2p − 12 ); 3 + (0, 0, 4p − 34 ); 5 + (0, 0, 4p − 34 + 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups for p = 4n − 1

a, b, pc

0, 0, u

a, b, pc

0, 0, u

2 + (2u, 2v, 0); 3 + ( 2p − 12 + u + v, 2p − 12 − u + v, 0); 5 + ( 2p − 12 + 2u, 2p − 12 , 0) prime p > 2; 0 ≤ u < p; 0 ≤ v < p p2 conjugate subgroups

pa, pb, c

u, v, 0

• Series of maximal isomorphic subgroups

[p] c = pc P43 21 2 (96)

P41 21 2 (92)

[p2 ] a = pa, b = pb P43 21 2 (96)

I Minimal translationengleiche supergroups [3] P43 32 (212) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] C43 22 (95, P43 22); [2] I41 22 (98) • Decreased unit cell

[2] c = 12 c P42 21 2 (94)

Copyright © 2011 International Union of Crystallography

260

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 97, p. 261.

D94

I 422

I 422

No. 97

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t( 21 , 12 , 12 ); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

16

k

(0, 0, 0)+ ( 21 , 12 , 12 )+

1

(1) x, y, z (5) x, ¯ y, z¯

(2) x, ¯ y, ¯z (6) x, y, ¯ z¯

(3) y, ¯ x, z (7) y, x, z¯

I Maximal translationengleiche subgroups [2] I411 (79, I4) (1; 2; 3; 4)+ [2] I221 (23, I222) (1; 2; 5; 6)+ [2] I212 (22, F222) (1; 2; 7; 8)+

(4) y, x, ¯z (8) y, ¯ x, ¯ z¯

a − b, a + b, c

II Maximal klassengleiche subgroups

• Loss of centring translations [2] [2] [2] [2]

P42 21 2 (94) P42 22 (93) P421 2 (90) P422 (89)

1; 1; 1; 1;

2; 2; 2; 2;

7; 5; 3; 3;

8; 6; 4; 4;

(3; 4; 5; 6) + ( 21 , 12 , 12 ) (3; 4; 7; 8) + ( 21 , 12 , 12 ) (5; 6; 7; 8) + ( 21 , 12 , 12 ) 5; 6; 7; 8

0, 1/2, 0 0, 1/2, 1/4

• Enlarged unit cell

[3] c = 3c I422 (97) I422 (97) I422 (97)

2; 3; 5 2; 3; 5 + (0, 0, 2) 2; 3; 5 + (0, 0, 4)

a, b, 3c a, b, 3c a, b, 3c

0, 0, 1 0, 0, 2

2; 3; 5 + (0, 0, 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups

a, b, pc

0, 0, u

2 + (2u, 2v, 0); 3 + (u + v, −u + v, 0); 5 + (2u, 0, 0) prime p > 2; 0 ≤ u < p; 0 ≤ v < p p2 conjugate subgroups

pa, pb, c

u, v, 0

• Series of maximal isomorphic subgroups

[p] c = pc I422 (97)

[p2 ] a = pa, b = pb I422 (97)

I Minimal translationengleiche supergroups [2] I4/mmm (139); [2] I4/mcm (140); [3] F432 (209); [3] I432 (211) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations

none

• Decreased unit cell

[2] c = 12 c C422 (89, P422)

Copyright © 2011 International Union of Crystallography

261

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 98, p. 262.

I 41 2 2

No. 98

D104

I 41 2 2

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t( 21 , 12 , 12 ); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

16

g

1

(0, 0, 0)+ ( 21 , 12 , 12 )+ (1) x, y, z (5) x¯ + 12 , y, z¯ + 34

(2) x¯ + 12 , y¯ + 12 , z + 12 (6) x, y¯ + 12 , z¯ + 14

I Maximal translationengleiche subgroups [2] I41 11 (80, I41 ) (1; 2; 3; 4)+ (1; 2; 5; 6)+ [2] I21 21 (24, I21 21 21 ) [2] I21 12 (22, F222) (1; 2; 7; 8)+

(3) y, ¯ x + 12 , z + 14 (7) y + 12 , x + 12 , z¯ + 12

a − b, a + b, c

(4) y + 12 , x, ¯ z + 34 (8) y, ¯ x, ¯ z¯

0, 1/4, 3/8

II Maximal klassengleiche subgroups

• Loss of centring translations [2] [2] [2] [2]

P43 21 2 (96) P43 22 (95) P41 21 2 (92) P41 22 (91)

• Enlarged unit cell [3] c = 3c ⎧ ⎨ I41 22 (98) I4 22 (98) ⎩ 1 I41 22 (98)

1; 1; 1; 1;

2; 2; 2; 2;

7; 5; 3; 3;

8; 6; 4; 4;

(3; 4; 5; 6) + ( 21 , 12 , 12 ) (3; 4; 7; 8) + ( 21 , 12 , 12 ) (5; 6; 7; 8) + ( 21 , 12 , 12 ) 5; 6; 7; 8

2 + (1, 0, 1); 3 + ( 21 , − 21 , 12 ); 5 + (1, 0, 2) 2 + (1, 0, 1); 3 + ( 21 , − 21 , 12 ); 5 + (1, 0, 4) 2 + (1, 0, 1); 3 + ( 21 , − 21 , 12 ); 5 + (1, 0, 6)

1/4, 3/4, 1/4 1/4, 1/4, 3/8 1/4, 1/4, 0 3/4, 1/4, 3/8

a, b, 3c a, b, 3c a, b, 3c

1/2, 0, 1/4 1/2, 0, 5/4 1/2, 0, 9/4

2 + (0, 0, 2p − 12 ); 3 + (0, 0, 4p − 14 ); 3 5 + (0, 0, 3p 4 − 4 + 2u) prime p > 4; 0 ≤ u < p p conjugate subgroups for p = 4n + 1 2 + (1, 0, 2p − 12 ); 3 + ( 21 , − 21 , 4p − 14 ); 1 5 + (1, 0, 3p 4 − 4 + 2u) prime p > 2; 0 ≤ u < p p conjugate subgroups for p = 4n − 1

a, b, pc

0, 0, u

a, b, pc

1/2, 0, 1/4 + u

2 + ( 2p − 12 + 2u, 2p − 12 + 2v, 0); 3 + (u + v, 2p − 12 − u + v, 0); 5 + ( 2p − 12 + 2u, 0, 0) prime p > 2; 0 ≤ u < p; 0 ≤ v < p p2 conjugate subgroups

pa, pb, c

u, v, 0

• Series of maximal isomorphic subgroups

[p] c = pc I41 22 (98)

I41 22 (98)

[p2 ] a = pa, b = pb I41 22 (98)

I Minimal translationengleiche supergroups [2] I41 /amd (141); [2] I41 /acd (142); [3] F41 32 (210); [3] I41 32 (214) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations

none

• Decreased unit cell [2] c = 12 c C42 22 (93, P42 22)

Copyright © 2011 International Union of Crystallography

262

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 99, p. 263.

C4v1

P4mm

P4mm

No. 99

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

g

1

(1) x, y, z (5) x, y, ¯z

(2) x, ¯ y, ¯z (6) x, ¯ y, z

(3) y, ¯ x, z (7) y, ¯ x, ¯z

I Maximal translationengleiche subgroups [2] P411 (75, P4) 1; 2; 3; 4 [2] P21m (35, Cmm2) 1; 2; 7; 8 [2] P2m1 (25, Pmm2) 1; 2; 5; 6

(4) y, x, ¯z (8) y, x, z

a − b, a + b, c

II Maximal klassengleiche subgroups

• Enlarged unit cell [2] c = 2c P42 mc (105) P4cc (103) P42 cm (101) P4mm (99) [2] a = 2a, b = 2b C4md (100, P4bm) C4md (100, P4bm) C4mm (99, P4mm) C4mm (99, P4mm) [2] a = 2a, b = 2b, c = 2c F4mc (108, I4cm) F4mc (108, I4cm) F4mm (107, I4mm) F4mm (107, I4mm) [3] c = 3c P4mm (99)

2; 2; 2; 2;

5; 3 + (0, 0, 1) 3; 5 + (0, 0, 1) (3; 5) + (0, 0, 1) 3; 5

a, b, 2c a, b, 2c a, b, 2c a, b, 2c

2; 3; 5 + (0, 1, 0) 2; 5; 3 + (1, 0, 0) 2; 3; 5 2 + (1, 1, 0); 3 + (1, 0, 0); 5 + (0, 1, 0)

a − b, a + b, c a − b, a + b, c a − b, a + b, c a − b, a + b, c

2; 2; 2; 2;

a − b, a + b, 2c a − b, a + b, 2c a − b, a + b, 2c a − b, a + b, 2c

3; 5 + (0, 0, 1) 3 + (1, 0, 0); 5 + (0, 1, 1) 3; 5 3 + (1, 0, 0); 5 + (0, 1, 0)

2; 3; 5

1/2, 1/2, 0

1/2, 1/2, 0 1/2, 1/2, 0

a, b, 3c

• Series of maximal isomorphic subgroups [p] c = pc P4mm (99) 2; 3; 5 p prime no conjugate subgroups [p2 ] a = pa, b = pb P4mm (99) 2 + (2u, 2v, 0); 3 + (u + v, −u + v, 0); 5 + (0, 2v, 0) prime p > 2; 0 ≤ u < p; 0 ≤ v < p p2 conjugate subgroups

a, b, pc

pa, pb, c

I Minimal translationengleiche supergroups [2] P4/mmm (123); [2] P4/nmm (129) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] I4mm (107)

• Decreased unit cell

Copyright © 2011 International Union of Crystallography

1/2, 1/2, 0

none

263

u, v, 0

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 100, p. 264.

P4bm

No. 100

C4v2

P4bm

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

d

1

(1) x, y, z (5) x + 12 , y¯ + 12 , z

(2) x, ¯ y, ¯z (6) x¯ + 12 , y + 12 , z

I Maximal translationengleiche subgroups [2] P411 (75, P4) 1; 2; 3; 4 [2] P21m (35, Cmm2) 1; 2; 7; 8 [2] P2b1 (32, Pba2) 1; 2; 5; 6

(3) y, ¯ x, z (7) y¯ + 12 , x¯ + 12 , z

(4) y, x, ¯z (8) y + 12 , x + 12 , z

a − b, a + b, c

0, 1/2, 0

II Maximal klassengleiche subgroups

• Enlarged unit cell [2] c = 2c P42 bc (106) P4nc (104) P42 nm (102) P4bm (100) [3] c = 3c P4bm (100)

2; 2; 2; 2;

5; 3 + (0, 0, 1) 3; 5 + (0, 0, 1) (3; 5) + (0, 0, 1) 3; 5

a, b, 2c a, b, 2c a, b, 2c a, b, 2c

2; 3; 5

a, b, 3c

• Series of maximal isomorphic subgroups [p] c = pc P4bm (100) 2; 3; 5 p prime no conjugate subgroups [p2 ] a = pa, b = pb P4bm (100) 2 + (2u, 2v, 0); 3 + (u + v, −u + v, 0); 5 + ( 2p − 12 , 2p − 12 + 2v, 0) prime p > 2; 0 ≤ u < p; 0 ≤ v < p p2 conjugate subgroups

a, b, pc

pa, pb, c

I Minimal translationengleiche supergroups [2] P4/nbm (125); [2] P4/mbm (127) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] C4mm (99, P4mm); [2] I4cm (108)

• Decreased unit cell

Copyright © 2011 International Union of Crystallography

0, 1/2, 0

none

264

u, v, 0

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 101, p. 265.

C4v3

P 42 c m

P 42 c m

No. 101

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

e

1

(1) x, y, z (5) x, y, ¯ z + 12

(2) x, ¯ y, ¯z (6) x, ¯ y, z + 12

(3) y, ¯ x, z + 12 (7) y, ¯ x, ¯z

I Maximal translationengleiche subgroups [2] P42 11 (77, P42 ) 1; 2; 3; 4 [2] P21m (35, Cmm2) 1; 2; 7; 8 [2] P2c1 (27, Pcc2) 1; 2; 5; 6

(4) y, x, ¯ z + 12 (8) y, x, z

a − b, a + b, c

II Maximal klassengleiche subgroups

• Enlarged unit cell

[2] a = 2a, b = 2b C42 cd (106, P42 bc) C42 cd (106, P42 bc) C42 cm (105, P42 mc) C42 cm (105, P42 mc) [3] c = 3c P42 cm (101)

2; 2; 2; 2;

3; 5 + (0, 1, 0) 5; 3 + (1, 0, 0) 3; 5 3 + (1, 0, 0); 5 + (0, 1, 0)

a − b, a + b, c a − b, a + b, c a − b, a + b, c a − b, a + b, c

2; (3; 5) + (0, 0, 1)

1/2, 1/2, 0

a, b, 3c

• Series of maximal isomorphic subgroups [p] c = pc 2; (3; 5) + (0, 0, 2p − 12 ) P42 cm (101) prime p > 2 no conjugate subgroups [p2 ] a = pa, b = pb 2 + (2u, 2v, 0); 3 + (u + v, −u + v, 0); 5 + (0, 2v, 0) P42 cm (101) prime p > 2; 0 ≤ u < p; 0 ≤ v < p p2 conjugate subgroups I Minimal translationengleiche supergroups [2] P42 /mcm (132); [2] P42 /ncm (138) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] C42 cm (105, P42 mc); [2] I4cm (108) • Decreased unit cell

[2] c = 12 c P4mm (99)

Copyright © 2011 International Union of Crystallography

1/2, 1/2, 0

265

a, b, pc

pa, pb, c

u, v, 0

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 102, p. 266.

P 42 n m

No. 102

C4v4

P 42 n m

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

d

1

(1) x, y, z (5) x + 12 , y¯ + 12 , z + 12

(2) x, ¯ y, ¯z (6) x¯ + 12 , y + 12 , z + 12

(3) y¯ + 12 , x + 12 , z + 12 (7) y, ¯ x, ¯z

I Maximal translationengleiche subgroups [2] P42 11 (77, P42 ) 1; 2; 3; 4 [2] P21m (35, Cmm2) 1; 2; 7; 8 [2] P2n1 (34, Pnn2) 1; 2; 5; 6

(4) y + 12 , x¯ + 12 , z + 12 (8) y, x, z

a − b, a + b, c

0, 1/2, 0

II Maximal klassengleiche subgroups

• Enlarged unit cell

[2] a = 2a, b = 2b, c = 2c F41 dc (110, I41 cd) F41 dc (110, I41 cd) F41 dm (109, I41 md) F41 dm (109, I41 md) [3] c = 3c P42 nm (102)

2; 2; 2; 2;

3; 5 + (0, 0, 1) 5; 3 + (0, 0, 1) 3; 5 (3; 5) + (0, 0, 1)

a − b, a + b, 2c a − b, a + b, 2c a − b, a + b, 2c a − b, a + b, 2c

2; (3; 5) + (0, 0, 1)

1/2, 1/2, 0

a, b, 3c

• Series of maximal isomorphic subgroups [p] c = pc 2; (3; 5) + (0, 0, 2p − 12 ) P42 nm (102) prime p > 2 no conjugate subgroups [p2 ] a = pa, b = pb 2 + (2u, 2v, 0); 3 + ( 2p − 12 + u + v, 2p − 12 − u + v, 0); P42 nm (102) 5 + ( 2p − 12 , 2p − 12 + 2v, 0) prime p > 2; 0 ≤ u < p; 0 ≤ v < p p2 conjugate subgroups I Minimal translationengleiche supergroups [2] P42 /nnm (134); [2] P42 /mnm (136) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] C42 cm (105, P42 mc); [2] I4mm (107) • Decreased unit cell [2] c = 12 c P4bm (100)

Copyright © 2011 International Union of Crystallography

1/2, 1/2, 0

266

a, b, pc

pa, pb, c

u, v, 0

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 103, p. 267.

C4v5

P4cc

P4cc

No. 103

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

d

1

(1) x, y, z (5) x, y, ¯ z + 12

(2) x, ¯ y, ¯z (6) x, ¯ y, z + 12

(3) y, ¯ x, z (7) y, ¯ x, ¯ z + 12

I Maximal translationengleiche subgroups [2] P411 (75, P4) 1; 2; 3; 4 [2] P21c (37, Ccc2) 1; 2; 7; 8 [2] P2c1 (27, Pcc2) 1; 2; 5; 6

(4) y, x, ¯z (8) y, x, z + 12

a − b, a + b, c

II Maximal klassengleiche subgroups

• Enlarged unit cell [2] a = 2a, b = 2b C4cd (104, P4nc) C4cd (104, P4nc) C4cc (103, P4cc) C4cc (103, P4cc) [3] c = 3c P4cc (103)

2; 3; 5 + (0, 1, 0) 2; 5; 3 + (1, 0, 0) 2; 3; 5 2 + (1, 1, 0); 3 + (1, 0, 0); 5 + (0, 1, 0)

a − b, a + b, c a − b, a + b, c a − b, a + b, c a − b, a + b, c

2; 3; 5 + (0, 0, 1)

a, b, 3c

• Series of maximal isomorphic subgroups [p] c = pc P4cc (103) 2; 3; 5 + (0, 0, 2p − 12 ) prime p > 2 no conjugate subgroups [p2 ] a = pa, b = pb P4cc (103) 2 + (2u, 2v, 0); 3 + (u + v, −u + v, 0); 5 + (0, 2v, 0) prime p > 2; 0 ≤ u < p; 0 ≤ v < p p2 conjugate subgroups I Minimal translationengleiche supergroups [2] P4/mcc (124); [2] P4/ncc (130) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] I4cm (108)

• Decreased unit cell

[2] c = 12 c P4mm (99)

Copyright © 2011 International Union of Crystallography

267

1/2, 1/2, 0 1/2, 1/2, 0

a, b, pc

pa, pb, c

u, v, 0

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 104, p. 268.

P4nc

No. 104

C4v6

P4nc

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

c

1

(1) x, y, z (5) x + 12 , y¯ + 12 , z + 12

(2) x, ¯ y, ¯z (6) x¯ + 12 , y + 12 , z + 12

I Maximal translationengleiche subgroups [2] P411 (75, P4) 1; 2; 3; 4 [2] P21c (37, Ccc2) 1; 2; 7; 8 [2] P2n1 (34, Pnn2) 1; 2; 5; 6

(3) y, ¯ x, z (7) y¯ + 12 , x¯ + 12 , z + 12

a − b, a + b, c

(4) y, x, ¯z (8) y + 12 , x + 12 , z + 12

0, 1/2, 0

II Maximal klassengleiche subgroups

• Enlarged unit cell [3] c = 3c P4nc (104)

2; 3; 5 + (0, 0, 1)

a, b, 3c

• Series of maximal isomorphic subgroups [p] c = pc P4nc (104) 2; 3; 5 + (0, 0, 2p − 12 ) prime p > 2 no conjugate subgroups [p2 ] a = pa, b = pb P4nc (104) 2 + (2u, 2v, 0); 3 + (u + v, −u + v, 0); 5 + ( 2p − 12 , 2p − 12 + 2v, 0) prime p > 2; 0 ≤ u < p; 0 ≤ v < p p2 conjugate subgroups I Minimal translationengleiche supergroups [2] P4/nnc (126); [2] P4/mnc (128) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] C4cc (103, P4cc); [2] I4mm (107)

• Decreased unit cell

[2] c = 12 c P4bm (100)

Copyright © 2011 International Union of Crystallography

268

a, b, pc

pa, pb, c

u, v, 0

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 105, p. 269.

C4v7

P 42 m c

P 42 m c

No. 105

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

f

1

(1) x, y, z (5) x, y, ¯z

(2) x, ¯ y, ¯z (6) x, ¯ y, z

(3) y, ¯ x, z + 12 (7) y, ¯ x, ¯ z + 12

I Maximal translationengleiche subgroups [2] P42 11 (77, P42 ) 1; 2; 3; 4 [2] P21c (37, Ccc2) 1; 2; 7; 8 [2] P2m1 (25, Pmm2) 1; 2; 5; 6

(4) y, x, ¯ z + 12 (8) y, x, z + 12

a − b, a + b, c

II Maximal klassengleiche subgroups

• Enlarged unit cell

[2] a = 2a, b = 2b C42 md (102, P42 nm) C42 md (102, P42 nm) C42 mc (101, P42 cm) C42 mc (101, P42 cm) [3] c = 3c P42 mc (105)

2; 2; 2; 2;

3; 5 + (0, 1, 0) 5; 3 + (1, 0, 0) 3; 5 3 + (1, 0, 0); 5 + (0, 1, 0)

a − b, a + b, c a − b, a + b, c a − b, a + b, c a − b, a + b, c

2; 5; 3 + (0, 0, 1)

1/2, 1/2, 0

a, b, 3c

• Series of maximal isomorphic subgroups [p] c = pc 2; 5; 3 + (0, 0, 2p − 12 ) P42 mc (105) prime p > 2 no conjugate subgroups [p2 ] a = pa, b = pb 2 + (2u, 2v, 0); 3 + (u + v, −u + v, 0); 5 + (0, 2v, 0) P42 mc (105) prime p > 2; 0 ≤ u < p; 0 ≤ v < p p2 conjugate subgroups I Minimal translationengleiche supergroups [2] P42 /mmc (131); [2] P42 /nmc (137) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] C42 mc (101, P42 cm); [2] I4mm (107) • Decreased unit cell

[2] c = 12 c P4mm (99)

Copyright © 2011 International Union of Crystallography

1/2, 1/2, 0

269

a, b, pc

pa, pb, c

u, v, 0

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 106, p. 270.

P 42 b c

No. 106

C4v8

P 42 b c

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

8

c

1

(1) x, y, z (5) x + 12 , y¯ + 12 , z

(2) x, ¯ y, ¯z (6) x¯ + 12 , y + 12 , z

I Maximal translationengleiche subgroups [2] P42 11 (77, P42 ) 1; 2; 3; 4 [2] P21c (37, Ccc2) 1; 2; 7; 8 [2] P2b1 (32, Pba2) 1; 2; 5; 6

(3) y, ¯ x, z + 12 (7) y¯ + 12 , x¯ + 12 , z + 12

(4) y, x, ¯ z + 12 (8) y + 12 , x + 12 , z + 12

a − b, a + b, c

0, 1/2, 0

II Maximal klassengleiche subgroups

• Enlarged unit cell

[3] c = 3c P42 bc (106)

2; 5; 3 + (0, 0, 1)

a, b, 3c

• Series of maximal isomorphic subgroups [p] c = pc P42 bc (106) 2; 5; 3 + (0, 0, 2p − 12 ) prime p > 2 no conjugate subgroups [p2 ] a = pa, b = pb 2 + (2u, 2v, 0); 3 + (u + v, −u + v, 0); P42 bc (106) 5 + ( 2p − 12 , 2p − 12 + 2v, 0) prime p > 2; 0 ≤ u < p; 0 ≤ v < p p2 conjugate subgroups I Minimal translationengleiche supergroups [2] P42 /nbc (133); [2] P42 /mbc (135) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations [2] C42 mc (101, P42 cm); [2] I4cm (108) • Decreased unit cell

[2] c = 12 c P4bm (100)

Copyright © 2011 International Union of Crystallography

270

a, b, pc

pa, pb, c

u, v, 0

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 107, p. 271.

C4v9

I 4mm

I 4mm

No. 107

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t( 21 , 12 , 12 ); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

16

e

(0, 0, 0)+ ( 21 , 12 , 12 )+

1

(1) x, y, z (5) x, y, ¯z

(2) x, ¯ y, ¯z (6) x, ¯ y, z

(3) y, ¯ x, z (7) y, ¯ x, ¯z

I Maximal translationengleiche subgroups [2] I411 (79, I4) (1; 2; 3; 4)+ [2] I2m1 (44, Imm2) (1; 2; 5; 6)+ [2] I21m (42, Fmm2) (1; 2; 7; 8)+

(4) y, x, ¯z (8) y, x, z

a − b, a + b, c

II Maximal klassengleiche subgroups

• Loss of centring translations [2] [2] [2] [2]

P42 mc (105) P4nc (104) P42 nm (102) P4mm (99)

1; 1; 1; 1;

2; 2; 2; 2;

5; 3; 7; 3;

6; 4; 8; 4;

(3; 4; 7; 8) + ( 21 , 12 , 12 ) (5; 6; 7; 8) + ( 21 , 12 , 12 ) (3; 4; 5; 6) + ( 21 , 12 , 12 ) 5; 6; 7; 8

0, 1/2, 0

• Enlarged unit cell

[3] c = 3c I4mm (107)

2; 3; 5

a, b, 3c

• Series of maximal isomorphic subgroups

[p] c = pc I4mm (107)

[p2 ] a = pa, b = pb I4mm (107)

2; 3; 5 prime p > 2 no conjugate subgroups

a, b, pc

2 + (2u, 2v, 0); 3 + (u + v, −u + v, 0); 5 + (0, 2v, 0) prime p > 2; 0 ≤ u < p; 0 ≤ v < p p2 conjugate subgroups

pa, pb, c

I Minimal translationengleiche supergroups [2] I4/mmm (139) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations

none

• Decreased unit cell [2] c = 12 c C4mm (99, P4mm)

Copyright © 2011 International Union of Crystallography

271

u, v, 0

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 108, p. 272.

I 4cm

No. 108

C4v10

I 4cm

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t( 21 , 12 , 12 ); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

16

d

(0, 0, 0)+ ( 21 , 12 , 12 )+

1

(1) x, y, z (5) x, y, ¯ z + 12

(2) x, ¯ y, ¯z (6) x, ¯ y, z + 12

(3) y, ¯ x, z (7) y, ¯ x, ¯ z + 12

I Maximal translationengleiche subgroups [2] I411 (79, I4) (1; 2; 3; 4)+ [2] I2c1 (45, Iba2) (1; 2; 5; 6)+ [2] I21m (42, Fmm2) (1; 2; 7; 8)+

(4) y, x, ¯z (8) y, x, z + 12

a − b, a + b, c

0, 1/2, 0

II Maximal klassengleiche subgroups

• Loss of centring translations [2] [2] [2] [2]

P42 bc (106) P4cc (103) P42 cm (101) P4bm (100)

• Enlarged unit cell [3] c = 3c I4cm (108)

1; 1; 1; 1;

2; 2; 2; 2;

7; 3; 5; 3;

8; 4; 6; 4;

(3; 4; 5; 6) + ( 21 , 12 , 12 ) 5; 6; 7; 8 (3; 4; 7; 8) + ( 21 , 12 , 12 ) (5; 6; 7; 8) + ( 21 , 12 , 12 )

2; 3; 5 + (0, 0, 1)

0, 1/2, 0 0, 1/2, 0

a, b, 3c

• Series of maximal isomorphic subgroups

[p] c = pc I4cm (108)

[p2 ] a = pa, b = pb I4cm (108)

2; 3; 5 + (0, 0, 2p − 12 ) prime p > 2 no conjugate subgroups

a, b, pc

2 + (2u, 2v, 0); 3 + (u + v, −u + v, 0); 5 + (0, 2v, 0) prime p > 2; 0 ≤ u < p; 0 ≤ v < p p2 conjugate subgroups

pa, pb, c

I Minimal translationengleiche supergroups [2] I4/mcm (140) II Minimal non-isomorphic klassengleiche supergroups

• Additional centring translations

none

• Decreased unit cell [2] c = 12 c C4mm (99, P4mm)

Copyright © 2011 International Union of Crystallography

272

u, v, 0

International Tables for Crystallography (2011). Vol. A1, Maximal subgroups of space group 109, p. 273.

C4v11

I 41 m d

I 41 m d

No. 109

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t( 21 , 12 , 12 ); (2); (3); (5) General position

Coordinates

Multiplicity, Wyckoff letter, Site symmetry

16

c

(0, 0, 0)+ ( 21 , 12 , 12 )+

1

(1) x, y, z (5) x, y, ¯z

(2) x¯ + 12 , y¯ + 12 , z + 12 (6) x¯ + 12 , y + 12 , z + 12

I Maximal translationengleiche subgroups [2] I41 11 (80, I41 ) (1; 2; 3; 4)+ [2] I2m1 (44, Imm2) (1; 2; 5; 6)+ [2] I21d (43, Fdd2) (1; 2; 7; 8)+

(3) y, ¯ x + 12 , z + 14 (7) y, ¯ x¯ + 12 , z + 14

(4) y + 12 , x, ¯ z + 34 (8) y + 12 , x, z + 34

a − b, a + b, c

0, 1/2, 0

II Maximal klassengleiche subgroups

• Loss of centring translations • Enlarged unit cell [3] c = 3c I41 md (109)

none

5; 2 + (1, 0, 1); 3 + ( 21 , − 21 , 12 )

a, b, 3c

1/2, 0, 0

• Series of maximal isomorphic subgroups

[p] c = pc I41 md (109) I41 md (109)

[p2 ] a = pa, b = pb I41 md (109)

5; 2 + (0, 0, 2p − 12 ); 3 + (0, 0, 4p − 14 ) prime p > 4; p = 4n + 1 no conjugate subgroups 5; 2 + (1, 0, 2p − 12 ); 3 + ( 21 , − 21 , 4p − 14 ) prime p > 2; p = 4n − 1 no conjugate subgroups

a, b, pc a, b, pc

1/2, 0, 0

2 + ( 2p − 12 + 2u, 2p − 12 + 2v, 0); 3 + (u + v, 2p − 12 − u + v,