International
Tables for Crystallography Volume A Spacegroup symmetry Edited by M. I. Aroyo © International Union of Crystallography 2015 
International Tables for Crystallography (2015). Vol. A, ch. 1.4, pp. 5556

The symmetry operations of a space group are conveniently partitioned into the cosets with respect to the translation subgroup. All operations which belong to the same coset have the same linear part and, if a single operation from a coset is given, all other operations in this coset are obtained by composition with a translation. However, not all symmetry operations in a coset with respect to the translation subgroup are operations of the same type and, furthermore, they may belong to element sets of different symmetry elements. In general, one can distinguish the following cases:
In order to distinguish the different cases, a closer analysis of the type of a symmetry operation and its symmetry element is required. These types, however, might be obscured by two obstacles:
These issues can be overcome by decomposing the translation part w of a symmetry operation into an intrinsic translation part which is fixed by the linear part W of and thus parallel to the geometric element of , and a location part , which is perpendicular to the intrinsic translation part. Note that the subspace of vectors fixed by W and the subspace perpendicular to this space of fixed vectors are complementary subspaces, i.e. their dimensions add up to 3, therefore this decomposition is always possible.
The procedure for determining the intrinsic translation part of a symmetry operation is described in Section 1.2.2.4 , and is based on the fact that the kth power of a symmetry operation with linear part W of order k must be a pure translation, i.e. for some lattice translation . The intrinsic translation part of is then defined as .
The difference is perpendicular to and it is called the location part of w. This terminology is justified by the fact that the location part can be reduced to o by an origin shift, i.e. the location part indicates whether the origin of the chosen coordinate system lies on the geometric element of .
The transformation of point coordinates and matrix–column pairs under an origin shift is explained in detail in Sections 1.5.1.3 and 1.5.2.3 , and the complete procedure for determining the additional symmetry operations will be discussed in the context of the synoptic tables in Section 1.5.4 . In this section we will restrict ourselves to a detailed discussion of two examples which illustrate typical phenomena.
Example 1
Consider a space group of type Fmm2 (42). The information on the general position and on the symmetry operations given in the spacegroup tables are reproduced in Fig. 1.4.2.2. From this information one deduces that coset representatives with respect to the translation subgroup are the identity element , a rotation with the c axis as geometric element, a reflection with the plane as geometric element and a reflection with the plane as geometric element (with the indices following the numbering in the table).
Composing these coset representatives with the centring translations , and gives rise to elements in the same cosets, but with different types of symmetry operations and symmetry elements in several cases.
In this example, all additional symmetry operations are listed in the symmetryoperations block of the spacegroup tables of Fmm2 because they are due to compositions of the coset representatives with centring translations.
The additional symmetry operations can easily be recognized in the symmetryelement diagrams (cf. Section 1.4.2.5). Fig. 1.4.2.3 shows the symmetryelement diagram of Fmm2 for the projection along the c axis. One sees that twofold rotation axes alternate with twofold screw axes and that mirror planes alternate with `double' or eglide planes, i.e. glide planes with two glide vectors. For example, the dot–dashed lines at and in Fig. 1.4.2.3 represent the b and c glides with normal vector along the a axis [for a discussion of eglide notation, see Sections 1.2.3 and 2.1.2 , and de Wolff et al., 1992].
Example 2
In a space group of type P4mm (99), representatives of the space group with respect to the translation subgroup are the powers of a fourfold rotation and reflections with normal vectors along the a and the b axis and along the diagonals [110] and (cf. Fig. 1.4.2.4).

Generalposition and symmetryoperations blocks as given in the spacegroup tables for space group P4mm (99). 
In this case, additional symmetry operations occur although there are no centring translations. Consider for example the reflection with the plane as geometric element. Composing this reflection with the translation gives rise to the symmetry operation represented by . This operation maps a point with coordinates to and is thus a glide reflection with the plane as geometric element and as glide vector. In a similar way, composing the other diagonal reflection with translations yields further glide reflections.
These glide reflections are symmetry operations which are not listed in the symmetryoperations block, although they are clearly of a different type to the operations given there. However, in the symmetryelement diagram as shown in Fig. 1.4.2.5, the corresponding symmetry elements are displayed as diagonal dashed lines which alternate with the solid diagonal lines representing the diagonal reflections.
References
Wolff, P. M. de, Billiet, Y., Donnay, J. D. H., Fischer, W., Galiulin, R. B., Glazer, A. M., Hahn, Th., Senechal, M., Shoemaker, D. P., Wondratschek, H., Wilson, A. J. C. & Abrahams, S. C. (1992). Symbols for symmetry elements and symmetry operations. Final report of the International Union of Crystallography AdHoc Committee on the Nomenclature of Symmetry. Acta Cryst. A48, 727–732.