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

International Tables for Crystallography (2006). Vol. A, ch. 4.1, p. 56

## Section 4.1.1. Introduction

E. F. Bertauta

aLaboratoire de Cristallographie, CNRS, Grenoble, France

### 4.1.1. Introduction

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The synoptic tables of this section comprise two features:

 (i) Space-group symbols for various settings and choices of the unit cell. Changes of the basis vectors generally cause changes of the Hermann–Mauguin space-group symbol. These axis transformations involve not only permutations of axes, conserving the shape of the cell, but also transformations which lead to different cell shapes and even to multiple cells. (ii) Extended Hermann–Mauguin space-group symbols, in addition to the short and full symbols. The occurrence of `additional symmetry elements' (see below) led to the introduction of `extended space-group symbols' in IT (1952); they are systematically developed in the present section. These additional symmetry elements are displayed in the space-group diagrams and are important for the tabulated `Symmetry operations'.

For each crystal system, the text starts with a historical note on the synoptic tables in the earlier editions of International Tables1 followed by a discussion of points (i) and (ii) above. Finally, those group–subgroup relations (cf. Section 8.3.3 ) are treated that can be recognized from the full and the extended Hermann–Mauguin space-group symbols. This applies mainly to the translationengleiche or t subgroups (type I, cf. Section 2.2.15 ) and to the klassengleiche or k subgroups of type IIa. For the k subgroups of types IIb and IIc, inspection of the synoptic Table 4.3.2.1 provides easy recognition of only those subgroups which originate from the decentring of certain multiple cells: C or F in the tetragonal system (Section 4.3.4 ), R and H in the trigonal and hexagonal systems (Section 4.3.5 ).

### References

International Tables for X-ray Crystallography (1952). Vol. I, edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press. [Abbreviated as IT (1952).]