International
Tables for Crystallography Volume A Spacegroup symmetry Edited by M. I. Aroyo © International Union of Crystallography 2016 
International Tables for Crystallography (2016). Vol. A, ch. 3.3, pp. 780790
Section 3.3.3. Properties of the international symbols^{a}Universität Erlangen–Nürnberg, RobertKochStrasse 4a, D91080 Uttenreuth, Germany, and ^{b}Institut für Angewandte Physik, Lehrstuhl für Kristallographie und Strukturphysik, Universität Erlangen–Nürnberg, Bismarckstrasse 10, D91054 Erlangen, Germany 
Because the short international symbol contains a set of generators, it is possible to deduce the space group from it. With the same distinction between generators and indicators as for point groups, the modified pointgroup symbol directly gives the rotation parts W of the generating operations (W, w).
The modified symbols of the generators determine the glide/screw parts of w. To find the location parts of w, it is necessary to inspect the product relations of the group. The deduction of the set of complete generating operations can be summarized in the following rules:
The origin that is selected by these rules is called the `origin of the symbol' (Burzlaff & Zimmermann, 1980). It is evident that the reference to the origin of the symbol allows a very short and unique notation of all desirable origins by appending the components of the origin of the symbol to the short spacegroup symbol, thus yielding the socalled expanded Hermann–Mauguin symbol. The shift of origin can be performed easily, for only the translation parts have to be changed. The components of the transformed translation part can be obtained by [cf. Section 1.5.2.3 and equation (1.5.2.13) ] Applications can be found in Burzlaff & Zimmermann (2002).
Examples: Deduction of the generating operations from the short symbol
Some examples for the use of these rules are now described in detail. It is convenient to describe the symmetry operation (W, w) by the corresponding coordinate triplets, i.e. using the socalled shorthand notation, cf. Section 1.2.2 . The coordinate triplets can be interpreted as combinations of two constituents: the first one consists of the coordinates of a point in general position after the application of W on x, y, z, while the second corresponds to the translation part w of the symmetry operation. The coordinate triplets of the symmetry operations are tabulated as the general position in the spacegroup tables (in some cases a shift of origin is necessary). If preference is given to full matrix notation, Table 1.2.2.1 may be used. The following examples contain, besides the description of the symmetry operations, references to the numbering of the general positions in the spacegroup tables of this volume; cf. Sections 2.1.3.9 and 2.1.3.11 . Centring translations are written after the numbers, if necessary.
If the geometrical point of view is again considered, it is possible to derive the full international symbol for a space group. This full symbol can be interpreted as consisting of symmetry elements. It can be generated from the short symbol with the aid of products between symmetry operations. It is possible, however, to derive the glide/screw parts of the elements in the full symbol directly from the glide/screw parts of the short symbol.
The product of operations corresponding to nonparallel glide or mirror planes generates a rotation or screw axis parallel to the intersection line. The screw part of the rotation is equal to the sum of the projections of the glide components of the planes on the axis. The angle between the planes determines the rotation part of the axis. For 90°, we obtain a twofold, for 60° a threefold, for 45° a fourfold and for 30° a sixfold axis.
Example:
The product of b and c generates a screw axis in the z direction because the sum of the glide components in the z direction is . The product of c and n generates a screw axis in the x direction and the product between b and n produces a rotation axis 2 in the y direction because the y components for b and n add up to modulo integers.
In most cases, the full symbol is identical with the short symbol; differences between full and short symbols can only occur for space groups corresponding to lattice point groups (holohedries) and to the point group . In all these cases, the short symbol is extended to the full symbol by adding the symbol for the maximal purely rotational subgroup. A special procedure is in use for monoclinic space groups. To indicate the choice of coordinate axes, the full symbol is treated like an orthorhombic symbol, in which the directions without symmetry are indicated by `1', even though they do not correspond to lattice symmetry directions in the monoclinic case.
Certain symmetry elements are not given explicitly in the full symbol because they can easily be derived. They are:
Example
The full symbol of space group Imma (74) is
The Icentring operation introduces additional rotation axes and glide planes for all three sets of lattice symmetry directions. The extended Hermann–Mauguin symbol is This symbol shows immediately the eight subgroups with a P lattice corresponding to point group mmm:
The symbols of Bravais lattices and glide planes depend on the choice of basis vectors. As shown in the preceding section, additional translation vectors in centred unit cells produce new symmetry operations with the same rotation but different glide/screw parts. Moreover, it was shown that for diagonal orientations symmetry operations may be represented by different symbols. Thus, different short symbols for the same space group can be derived even if the rules for the selection of the generators and indicators are obeyed.
For the unique designation of a spacegroup type, a standardization of the short symbol is necessary. Rules for standardization were given first by Hermann (1931) and later in a slightly modified form in IT (1952).
These rules, which are generally followed in the present tables, are given below. Because of the historical development of the symbols (cf. Section 3.3.4), some of the present symbols do not obey the rules, whereas others depending on the crystal class need additional rules for them to be uniquely determined. These exceptions and additions are not explicitly mentioned, but may be discovered from Table 3.3.3.1 in which the short symbols are listed for all space groups. A table for all settings may be found in Section 1.5.4 .
^{†}Abbreviations used in the column Comments: IT, 1952: International Tables for Xray Crystallography, Vol. I (1952); Sh–K; Shubnikov & Koptsik (1972). Note that this table contains only one notation for the bunique setting and one notation for the cunique setting in the monoclinic case, always referring to cell choice 1 of the spacegroup tables.

Triclinic symbols are unique if the unit cell is primitive. For the standard setting of monoclinic space groups, the lattice symmetry direction is labelled b. From the three possible centrings A, I and C, the latter one is favoured. If glide components occur in the plane perpendicular to [010], the glide direction c is preferred. In the space groups corresponding to the orthorhombic group mm2, the unique direction of the twofold axis is chosen along c. Accordingly, the face centring C is employed for centrings perpendicular to the privileged direction. For space groups with possible A or B centring, the first one is preferred. For groups 222 and mmm, no privileged symmetry direction exists, so the different possibilities of oneface centring can be reduced to C centring by change of the setting. The choices of unit cell and centring type are fixed by the conventional basis in systems with higher symmetry.
When more than one kind of symmetry elements exist in one representative direction, in most cases the choice for the spacegroup symbol is made in order of decreasing priority: for reflections and glide reflections m, a, b, c, n, d; for proper rotations and screw rotations ; ; ; [cf. IT (1952), p. 55, and Section 1.4.1 ].
Hermann (1928a) emphasized that the short symbols permit the derivation of systematic absences of Xray reflections caused by the glide/screw parts of the symmetry operations. If describes the Xray reflection and is the matrix representation of a symmetry operation, the matrix can be expanded as follows:The absence of a reflection is governed by the relation (i) = and the scalar product (ii) . A reflection h is absent if condition (i) holds and the scalar product (ii) is not an integer. The calculation must be made for all generators and indicators of the short symbol. Systematic absences, introduced by the further symmetry operations generated, are obtained by the combination of the extinction rules derived for the generators and indicators.
Example: Space group
The generators of the space group are the integral translations and the centring translation , the rotation 2 in direction [100]: and the rotation 2 in direction : . The combination of the two generators gives the operation corresponding to the indicator, namely , which represents a fourfold screw rotation in the direction [001].
The integral translations imply no restriction because the scalar product is always an integer. For the centring, condition (i) with holds for all reflections (integral condition), but the scalar product (ii) is an integer only for . Thus, reflections hkl with are absent. The screw rotation 4 has the screw part . Only 00l reflections obey condition (i) (serial extinction). An integral value for the scalar product (ii) requires . The twofold axes in the directions [100] and do not imply further absences because .
Detailed discussion of the theoretical background of conditions for possible general reflections and their derivation is given in Chapter 1.6 .
The international symbols can be suitably modified to describe generalized symmetry, e.g. colour groups, which occur when the symmetry operations are combined with changes of physical properties. For the description of antisymmetry (or `black–white' symmetry), the symbols of the Bravais lattices are supplemented by additional letters for centrings accompanied by a change in colour. For symmetry operations that are not translations, a prime is added to the usual symbol if a change of colour takes place. A complete description of the symbols and a detailed list of references are given by Koptsik (1966). The Shubnikov symbols have not been extended to colour symmetry.
An introduction to the structure, properties and symbols of magnetic subperiodic and magnetic space groups is given in Chapter 3.6 .
References
International Tables for Xray Crystallography (1952). Vol. I, edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press. [Revised editions: 1965, 1969 and 1977. Abbreviated as IT (1952).]Bertaut, E. F. (1976). Study of principal subgroups and their general positions in C and I groups of class mmm – D_{2h}. Acta Cryst. A32, 380–387.
Burzlaff, H. & Zimmermann, H. (1980). On the choice of origins in the description of space groups. Z. Kristallogr. 153, 151–179.
Burzlaff, H. & Zimmermann, H. (2002). On the treatment of settings of space groups and crystal structures by specialized short Hermann–Mauguin spacegroup symbols. Z. Kristallogr. 217, 135–138.
Hermann, C. (1928a). Zur systematischen Strukturtheorie I. Eine neue Raumgruppensymbolik. Z. Kristallogr. 68, 257–287.
Hermann, C. (1929). Zur systematischen Strukturtheorie IV. Untergruppen. Z. Kristallogr. 69, 533–555.
Hermann, C. (1931). Bemerkungen zu der vorstehenden Arbeit von Ch. Mauguin. Z. Kristallogr. 76, 559–561.
Koptsik, V. A. (1966). Shubnikov Groups. Moscow University Press. (In Russian.)
Zimmermann, H. (1976). Ableitung der Raumgruppen aus ihren klassengleichen Untergruppenbeziehungen. Z. Kristallogr. 143, 485–515.