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. 4.1, pp. 5660
https://doi.org/10.1107/97809553602060000507 Chapter 4.1. Introduction to the synoptic tablesChapters 4.1, 4.2 and 4.3 contain extensive explanations and tabulations of the various types of spacegroup symbols. Chapter 4.1 treats the`'interactions' between symmetry operations and both integral and centring lattice translations. The resulting `additional symmetry elements' and their location in the unit cell are listed. 
The synoptic tables of this section comprise two features:
For each crystal system, the text starts with a historical note on the synoptic tables in the earlier editions of International Tables^{1} 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 spacegroup 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 ).
In space groups, `due to periodicity', symmetry elements occur that are not recorded in the Hermann–Mauguin symbols. These additional symmetry elements are products of a symmetry translation and a symmetry operation . This product is and its geometrical representation is found in the spacegroup diagrams (cf. Sections 8.1.2 and 11.1.1 ).^{2}
Two cases have to be distinguished:

Table 4.1.2.2 lists representative symmetry elements, corresponding to , and their associated glide planes and screw axes, corresponding to . The upper part of the table contains the diagonal twofold axes and symmetry planes that appear as tertiary symmetry elements in tetragonal and cubic space groups and as secondary symmetry elements in rhombohedral space groups (referred to rhombohedral axes). The middle part lists the twofold axes and symmetry planes that are secondary and tertiary symmetry elements in trigonal and hexagonal space groups and secondary symmetry elements in rhombohedral space groups (referred to hexagonal axes). The lower part illustrates the occurrence of threefold screw axes in rhombohedral and cubic space groups for the orientation [111].
Note that integral translations do not produce additional glide or screw components in triclinic, monoclinic and orthorhombic groups.
Example
The operation in a rhombohedral or cubic space group represents a screw rotation with axis along [111]. Indeed, the third power of is the translation t(1, 1, 1), i.e. the periodicity along the threefold axis. The translation t(1, 0, 0) is decomposed uniquely into the screw component parallel to and the location component perpendicular to the threefold axis. The location of the axis is then found to be , which can also be expressed as or .
For 2, m and c, the locations of the symmetry elements at the origin and within the cell can be interchanged.
Example
According to Table 4.1.2.2, the c plane located in x, x, z implies an n plane in . Vice versa, an n plane in x, x, z implies a c plane in .
In the rhombohedral space groups R3c (161) and (167) and in their cubic supergroups, diagonal n planes in x, x, z and, by symmetry, in z, x, x and x, z, x coexist with c planes in , a planes in and b planes in , respectively (cf. Section 4.3.5 ).
Note that the symbol of a glide plane depends on the reference frame. Thus, the abovementioned n planes in the rhombohedral description become c planes in the hexagonal description of R3c and ; similarly, the a, b and c planes become n planes; cf. Sections 1.3.1 and 1.4.4 .
4.1.2.2. Centring translations^{4}
The general rules given under (i) and (ii) remain valid. In lattices C, A, B, I and F, a centring vector t with a component parallel to the symmetry element leads to an additional symmetry element of a different kind. When the centring vector t is perpendicular to the symmetry element or when the symmetry element is an inversion centre or a rotoinversion axis, the additional symmetry element is of the same kind.
The first part of Table 4.1.2.3 contains pairs of symmetry planes related by a centring translation. Each box has three or four entries, which define three or four pairs of `associated' planes; the cell under F contains all the planes under C, A and B. Hence, their locations are not repeated under F. Again, the locations of the two planes can be interchanged.
Example
The product of the Ccentring translation, i.e. , and the reflection through a mirror plane m, located in 0, y, z, is a glide reflection b with glide plane in . Similarly, C centring associates a glide plane c in 0, y, z with a glide plane n in .
Note that the mirror plane and `associated' glide plane coincide geometrically when the centring translation is parallel to the mirror (i.e. no normal component exists); see the first cell under A, the second under B, the third cell under C. Also, two `associated' glide planes (a, b) or (b, c) or (a, c) coincide geometrically. These `double' glide planes are symbolized by `e'; see Table 4.1.2.3 and Section 1.3.2 , Note (x) .
Glide reflections whose square is a pure centring translation are called d; other diagonal glide planes are called g and n; in each case, the glide component is given between parentheses (cf. Sections 2.2.9 and 11.1.2 ).
The second part of Table 4.1.2.3 summarizes pairs of symmetry axes and, in the bottom line, pairs of symmetry centres related by a centring translation. For instance, the Bcentring translation associates a rotation axis 2 along x, 0, 0 with a screw axis along . Here, too, the locations can be interchanged.
Example
The product of the translation with a twofold rotation around x, x, 0 is the operation , which occurs, for instance, in F432 (209). The square of this operation is the fractional translation . The translation is decomposed into a `screw part' and a `location part' perpendicular to it. The location of the additional symmetry element is then found to be which is parallel to that of the axis 2 in x, x, 0.
Inversions. The `midpoint rule' given under (i) for integral translations remains valid. When M occupies successively the eight positions of inversion centres in the primitive cell (cf. Table 4.1.2.1), each of the centrings C, A, B and I creates eight supplementary centres, whereas the F centring produces supplementary centres, leading to a total of 32 inversion centres.
Example
For C centring, add (cf. Table 4.1.2.3) to the eight locations of symmetry centres, given in Table 4.1.2.1, in order to obtain the eight additional symmetry centres ; ; ; ; ; ; ; .
Table 4.1.2.3 contains only representative cases. For 4 and axes, only the standard orientation [001] is given. For diagonal twofold axes, only the orientation [10] is considered. When the locations of all additional symmetry elements of a chosen species are desired, it is sufficient to insert the location of one of the elements into the coordinate triplets of the general position and to remove redundancies.
Example
Insert the location of a axis (see Table 4.1.2.2) into the general position of a cubic space group to obtain four distinct locations of axes in P groups and sixteen in F groups.
When more than one kind of symmetry element occurs for a given symmetry direction, the question of choice arises for defining the appropriate Hermann–Mauguin symbol. This choice is made in order of descending priority:
m, e, a, b, c, n, d; and rotation axes before screw axes.
This priority rule is explicitly stated in IT (1952), pages 55 and 543. It is applied to the spacegroup symbols in IT (1952) and the present edition. There are a few exceptions, however:
References
Internationale Tabellen zur Bestimmung von Kristallstrukturen (1935). 1. Band, edited by C. Hermann. Berlin: Borntraeger. [Revised edition: Ann Arbor: Edwards (1944). Abbreviated as IT (1935).]International Tables for Xray Crystallography (1952). Vol. I, edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press. [Abbreviated as IT (1952).]