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. 11.1, pp. 810811
https://doi.org/10.1107/97809553602060000522 Chapter 11.1. Point coordinates, symmetry operations and their symbols^{a}Institut für Mineralogie, Petrologie und Kristallographie, PhilippsUniversität, D35032 Marburg, Germany Chapter 11.1 deals with the different ways of representing symmetry operations, e.g. as linear equations, as pairs of a 3 × 3 and a 3 × 1 matrix, and as coordinate triplets (cf. `General positions' in Part 7 ). The notation used under the heading `Symmetry operations' in Parts 6 and 7 is explained. This kind of symbol contains information on the type of the symmetry operation, on the location of the corresponding symmetry element and, if necessary, on the glide or screw vector. 
The coordinate triplets of a general position, as given in the spacegroup tables, can be taken as a shorthand notation for the symmetry operations of the space group. Each coordinate triplet corresponds to the symmetry operation that maps a point with coordinates x, y, z onto a point with coordinates . The mapping of x, y, z onto is given by the equations: If, as usual, the symmetry operation is represented by a matrix pair, consisting of a matrix W and a column matrix w, the equations read with W is called the rotation part and the translation part; w is the sum of the intrinsic translation part (glide part or screw part) and the location part (due to the location of the symmetry element) of the symmetry operation.
Example
The coordinate triplet stands for the symmetry operation with rotation part and with translation part Matrix multiplication yields
Using the above relation, the assignment of coordinate triplets to symmetry operations given as pairs (W, w) is straightforward.
The information required to describe a symmetry operation by a unique notation depends on the type of the operation (Table 11.1.2.1). The symbols explained below are based on the Hermann–Mauguin symbols (see Chapter 12.2 ), modified and supplemented where necessary. Note that a change of the coordinate basis generally alters the symbol of a given symmetry operation.
