International
Tables for Crystallography Volume A Spacegroup symmetry Edited by Th. Hahn © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. A, ch. 2.1, pp. 1416
doi: 10.1107/97809553602060000504 Chapter 2.1. Classification and coordinate systems of space groups^{a}Institut für Kristallographie, RheinischWestfälische Technische Hochschule, Aachen, Germany, and ^{b}Laboratorium voor Chemische Fysica, Rijksuniversiteit Groningen, The Netherlands This chapter and Chapter 2.2 form the main guide to understanding and using the planegroup and spacegroup tables in Parts 6 and 7 . Chapter 2.1 displays, with the help of an extensive synoptic table, the classification of the 17 plane groups and 230 space groups into geometric crystal classes, Bravais lattices, crystal systems and crystal families. This is followed by a characterization of the conventional crystallographic coordinate systems. 
The present volume is a computerbased extension and complete revision of the symmetry tables of the two previous series of International Tables, the Internationale Tabellen zur Bestimmung von Kristallstrukturen (1935) and the International Tables for Xray Crystallography (1952).^{1}
The main part of the volume consists of tables and diagrams for the 17 types of plane groups (Part 6 ) and the 230 types of space groups (Part 7 ). The two types of line groups are treated separately in Section 2.2.17 , because of their simplicity. For the history of the Tables and a comparison of the various editions, reference is made to the Preface of this volume. Attention is drawn to Part 1 where the symbols and terms used in this volume are defined.
The present part forms a guide to the entries in the spacegroup tables with instructions for their practical use. Only a minimum of theory is provided, and the emphasis is on practical aspects. For the theoretical background the reader is referred to Parts 8–15 , which include also suitable references. A textbook version of spacegroup symmetry and the use of these tables (with exercises) is provided by Hahn & Wondratschek (1994).
In this volume, the plane groups and space groups are classified according to three criteria:

For three dimensions, this applies to the triclinic, monoclinic, orthorhombic, tetragonal and cubic systems. The only complication exists in the hexagonal crystal family for which several subdivisions into systems have been proposed in the literature. In this volume, as well as in IT (1952), the space groups of the hexagonal crystal family are grouped into two `crystal systems' as follows: all space groups belonging to the five crystal classes 3, , 32, 3m and , i.e. having 3, , or as principal axis, form the trigonal crystal system, irrespective of whether the Bravais lattice is hP or hR; all space groups belonging to the seven crystal classes 6, , 622, 6mm, 2m and , i.e. having 6, , , , , or as principal axis, form the hexagonal crystal system; here the lattice is always hP (cf. Section 8.2.8 ). The crystal systems, as defined above, are listed in column 3 of Table 2.1.2.1.
A different subdivision of the hexagonal crystal family is in use, mainly in the French literature. It consists of grouping all space groups based on the hexagonal Bravais lattice hP (lattice point symmetry ) into the `hexagonal' system and all space groups based on the rhombohedral Bravais lattice hR (lattice point symmetry ) into the `rhombohedral' system. In Section 8.2.8 , these systems are called `Lattice systems'. They were called `Bravais systems' in earlier editions of this volume.
The theoretical background for the classification of space groups is provided in Chapter 8.2 .
A plane group or space group usually is described by means of a crystallographic coordinate system, consisting of a crystallographic basis (basis vectors are lattice vectors) and a crystallographic origin (origin at a centre of symmetry or at a point of high site symmetry). The choice of such a coordinate system is not mandatory since in principle a crystal structure can be referred to any coordinate system; cf. Section 8.1.4 .
The selection of a crystallographic coordinate system is not unique. Conventionally, a righthanded set of basis vectors is taken such that the symmetry of the plane or space group is displayed best. With this convention, which is followed in the present volume, the specific restrictions imposed on the cell parameters by each crystal family become particularly simple. They are listed in columns 6 and 7 of Table 2.1.2.1. If within these restrictions the smallest cell is chosen, a conventional (crystallographic) basis results. Together with the selection of an appropriate conventional (crystallographic) origin (cf. Sections 2.2.2 and 2.2.7 ), such a basis defines a conventional (crystallographic) coordinate system and a conventional cell. The conventional cell of a point lattice or a space group, obtained in this way, turns out to be either primitive or to exhibit one of the centring types listed in Chapter 1.2 . The centring type of a conventional cell is transferred to the lattice which is described by this cell; hence, we speak of primitive, facecentred, bodycentred etc. lattices. Similarly, the cell parameters are often called lattice parameters; cf. Section 8.3.1 and Chapter 9.1 for further details.
In the triclinic, monoclinic and orthorhombic crystal systems, additional conventions (for instance cell reduction or metrical conventions based on the lengths of the cell edges) are needed to determine the choice and the labelling of the axes. Reduced bases are treated in Chapters 9.1 and 9.2 , orthorhombic settings in Section 2.2.6.4 , and monoclinic settings and cell choices in Section 2.2.16 .
In this volume, all space groups within a crystal family are referred to the same kind of conventional coordinate system with the exception of the hexagonal crystal family in three dimensions. Here, two kinds of coordinate systems are used, the hexagonal and the rhombohedral systems. In accordance with common crystallographic practice, all space groups based on the hexagonal Bravais lattice hP (18 trigonal and 27 hexagonal space groups) are described only with a hexagonal coordinate system (primitive cell),^{2} whereas the seven space groups based on the rhombohedral Bravais lattice hR are treated in two versions, one referred to `hexagonal axes' (triple obverse cell) and one to `rhombohedral axes' (primitive cell); cf. Chapter 1.2 . In practice, hexagonal axes are preferred because they are easier to visualize.
Note: For convenience, the relations between the cell parameters a, c of the triple hexagonal cell and the cell parameters , of the primitive rhombohedral cell (cf. Table 2.1.2.1) are listed:
References
Hahn, Th. & Wondratschek, H. (1994). Symmetry of crystals. Sofia: Heron Press.Internationale Tabellen zur Bestimmung von Kristallstrukturen (1935). 1. Band, edited by C. Hermann. Berlin: Borntraeger. [Revised edition: Ann Arbor: Edwards (1944). Abbreviated as IT (1935).]
International Tables for Crystallography (2004). Vol. C, 3rd ed., edited by A. J. C. Wilson & E. Prince. Dordrecht: Kluwer Academic Publishers.
International Tables for Xray Crystallography (1952). Vol. I, edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press. [Revised editions: 1965, 1969 and 1977. Abbreviated as IT (1952).]
Wolff, P. M. de, Belov, N. V., Bertaut, E. F., Buerger, M. J., Donnay, J. D. H., Fischer, W., Hahn, Th., Koptsik, V. A., Mackay, A. L., Wondratschek, H., Wilson, A. J. C. & Abrahams, S. C. (1985). Nomenclature for crystal families, Bravaislattice types and arithmetic classes. Report of the International Union of Crystallography Adhoc Committee on the Nomenclature of Symmetry. Acta Cryst. A41, 278–280.