International
Tables for
Crystallography
Volume A1
Symmetry relations between space groups
Edited by Hans Wondratschek and Ulrich Müller

© International Union of Crystallography 2006
International Tables for Crystallography (2006). Vol. A1. ch. 2.1, pp. 42-58   | 1 | 2 |
https://doi.org/10.1107/97809553602060000542

Chapter 2.1. Guide to the subgroup tables and graphs

Hans Wondratscheka* and Mois I. Aroyob

a Institut für Kristallographie, Universität, D-76128 Karlsruhe, Germany, and bDepartamento de Física de la Materia Condensada, Facultad de Ciencias, Universidad del País Vasco, Apartado 644, E-48080 Bilbao, Spain
Correspondence e-mail:  wondra@physik.uni-karlsruhe.de

Footnotes

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'.
2 The system of equations (2.1.3.1)[link] is similar but not identical to the system of equations (2.1.3.5)[link], which describes a symmetry operation [\sf W] by the matrix W and the column w. Both W and w are listed as the general position in the space-group tables of IT A, cf. Part 5[link] and Chapter 11.2[link] of IT A. The essential difference is that in equation (2.1.3.6)[link] the matrix W is multiplied by the column x from the right-hand side whereas in equation (2.1.3.3)[link] the matrix P is multiplied by the row [({\bi a})^{\rm T}] from the left-hand side. Therefore, the running index in W is the second one, whereas in [{\bi P}] it is the first one.

c Département de Physique, UFR des Sciences et Techniques, Université de Bretagne Occidentale, F-29200 Brest, France

3 F. Gähler (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 [{\cal G}].
4 For the term `crystal family' cf. Section 1.2.5.2[link] , or, for more details, IT A, Section 8.2.7[link] .
5 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[link] .
6 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.