International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by Th. Hahn

© International Union of Crystallography 2006
International Tables for Crystallography (2006). Vol. A. ch. 2.2, pp. 17-41
https://doi.org/10.1107/97809553602060000505

Chapter 2.2. Contents and arrangement of the tables

Th. Hahna* and A. Looijenga-Vosb‡

a Institut für Kristallographie, Rheinisch-Westfälische Technische Hochschule, Aachen, Germany, and bLaboratorium voor Chemische Fysica, Rijksuniversiteit Groningen, The Netherlands
Correspondence e-mail:  hahn@xtl.rwth-aachen.de

Footnotes

‡ Present address: Roland Holstlaan 908, 2624 JK Delft, The Netherlands.

1 A space group is called `symmorphic' if, apart from the lattice translations, all generating symmetry operations leave one common point fixed. Permitted as generators are thus only the point-group operations: rotations, reflections, inversions and rotoinversions (cf. Section 8.1.6[link] ).
2 A space-group symbol is invariant under sign changes of the axes; i.e. the same symbol applies to the right-handed coordinate systems abc, [{\bf a}\overline{{\bf b}}\overline{{\bf c}}, \overline{{\bf a}}{\bf b}\overline{{\bf c}}, \overline{{\bf a}}\overline{{\bf b}}{\bf c}] and the left-handed systems [\overline{{\bf a}}{\bf bc}, {\bf a}\overline{{\bf b}}{\bf c}, {\bf ab}\overline{{\bf c}}, \overline{{\bf a}}\overline{{\bf b}}\overline{{\bf c}}].
3 The term Position (singular) is defined as a set of symmetrically equivalent points, in agreement with IT (1935)[link]: Point position; Punktlage (German); Position (French). Note that in IT (1952)[link] the plural, equivalent positions, was used.
4 Often called point symmetry: Punktsymmetrie or Lagesymmetrie (German): symétrie ponctuelle (French).
5 The reflection conditions were called Auslöschungen (German), missing spectra (English) and extinctions (French) in IT (1935)[link] and `Conditions limiting possible reflections' in IT (1952)[link]; they are often referred to as `Systematic or space-group absences' (cf. Chapter 12.3[link] ).
6 Space groups with different space-group numbers are non-isomorphic, except for the members of the 11 pairs of enantiomorphic space groups which are isomorphic.
7 Subgroups belonging to the enantiomorphic space-group type of [{\cal G}] are isomorphic to [{\cal G}] and, therefore, are listed under IIc and not under IIb.
8 Unconventional Hermann–Mauguin symbols may include unconventional cells like c centring in quadratic plane groups, F centring in monoclinic, or C and F centring in tetragonal space groups. Furthermore, the triple hexagonal cells h and H are used for certain sub- and supergroups of the hexagonal plane groups and of the trigonal and hexagonal P space groups, respectively. The cells h and H are defined in Chapter 1.2[link] . Examples are subgroups of plane groups p3 (13) and p6mm (17) and of space groups P3 (143) and [P6/mcc] (192).
9 Without this restriction, the amount of data would be excessive. For instance, space group Pmmm (47) has 63 maximal subgroups of index [2], of which seven are t subgroups and listed explicitly under I. The 16 entries under IIb refer to 50 actual subgroups and the one entry under IIc stands for the remaining 6 subgroups.
10 For normalizers of space groups, see Section 8.3.6[link] and Part 15[link] , where also references to automorphisms are given.
11 These three vectors obey the `closed-triangle' condition [{\bf e} + {\bf f} + {\bf g} = {\bf 0}]; they can be considered as two-dimensional homogeneous axes.
12 In IT (1952)[link], the terms `1st setting' and `2nd setting' were used for `unique axis c' and `unique axis b'. In the present volume, these terms have been dropped in favour of the latter names, which are unambiguous.