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. 12.1, pp. 818820
https://doi.org/10.1107/97809553602060000524 Chapter 12.1. Pointgroup 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 The three main kinds of pointgroup symbols in use today (Schoenflies symbols, Shubnikov symbols and Hermann–Mauguin symbols) are described and listed. International (Hermann–Mauguin) and Shubnikov symbols for symmetry elements and representatives for the lattice symmetry directions in the different crystal families are tabulated. 
For symbolizing space groups, or more correctly types of space groups, different notations have been proposed. The following three are the main ones in use today:
In all three notations, the spacegroup symbol is a modification of a pointgroup symbol.
Symmetry elements occur in lattices, and thus in crystals, only in distinct directions. Pointgroup symbols make use of these discrete directions and their mutual relations.
Most Schoenflies symbols (Table 12.1.4.2, column 1) consist of the basic parts C_{n}, D_{n},^{1} T or O, designating cyclic, dihedral, tetrahedral and octahedral rotation groups, respectively, with . The remaining point groups are described by additional symbols for mirror planes, if present. The subscripts h and v indicate mirror planes perpendicular and parallel to a main axis taken as vertical. For T, the three mutually perpendicular twofold axes and, for O, the three fourfold axes are considered to be the main axes. The index d is used for mirror planes that bisect the angle between two consecutive equivalent rotation axes, i.e. which are diagonal with respect to these axes. For the rotoinversion axes and , which do not fit into the general Schoenflies concept of symbols, other symbols and are in use. The rotoinversion axis is equivalent to and thus designated as .
The Shubnikov symbol is constructed from a minimal set of generators of a point group (for exceptions, see below). Thus, strictly speaking, the symbols represent types of symmetry operations. Since each symmetry operation is related to a symmetry element, the symbols also have a geometrical meaning. The Shubnikov symbols for symmetry operations differ slightly from the international symbols (Table 12.1.3.1). Note that Shubnikov, like Schoenflies, regards symmetry operations of the second kind as rotoreflections rather than as rotoinversions.
^{†}According to a private communication from J. D. H. Donnay, the symbols for elements of the second kind were proposed by M. J. Buerger. Koptsik (1966) used them for the Shubnikov method.

If more than one generator is required, it is not sufficient to give only the types of the symmetry elements; their mutual orientations must be symbolized too. In the Shubnikov symbol, a colon (:), a dot (·) or a slash (/) is used to designate perpendicular, parallel or oblique arrangement of the symmetry elements. For a reflection, the orientation of the actual mirror plane is considered, not that of its normal. The exception mentioned above is the use of instead of in the description of point groups.
The Hermann–Mauguin symbols for finite point groups make use of the fact that the symmetry elements, i.e. proper and improper rotation axes, have definite mutual orientations. If for each point group the symmetry directions are grouped into classes of symmetrical equivalence, at most three classes are obtained. These classes were called Blickrichtungssysteme (Heesch, 1929). If a class contains more than one direction, one of them is chosen as representative.
The Hermann–Mauguin symbols for the crystallographic point groups refer to the symmetry directions of the lattice point groups (holohedries, cf. Part 9 ) and use other representatives than chosen by Heesch [IT (1935), p. 13]. For instance, in the hexagonal case, the primary set of lattice symmetry directions consists of , representative is [001]; the secondary set of lattice symmetry directions consists of [100], [010], and their counterdirections, representative is [100]; the tertiary set of lattice symmetry directions consists of and their counterdirections, representative is . The representatives for the sets of lattice symmetry directions for all lattice point groups are listed in Table 12.1.4.1. The directions are related to the conventional crystallographic basis of each lattice point group (cf. Part 9 ).

The relation between the concept of lattice symmetry directions and group theory is evident. The maximal cyclic subgroups of the maximal rotation group contained in a lattice point group can be divided into, at most, three sets of conjugate subgroups. Each of these sets corresponds to one set of lattice symmetry directions.
After the classification of the directions of rotation axes, the description of the seven maximal rotation subgroups of the lattice point groups is rather simple. For each representative direction, the rotational symmetry element is symbolized by an integer n for an nfold axis, resulting in the symbols of the maximal rotation subgroups 1, 2, 222, 32, 422, 622, 432. The symbol 1 is used for the triclinic case. The complete lattice point group is constructed by multiplying the rotation group by the inversion . For the evenfold axes, 2, 4 and 6, this multiplication results in a mirror plane perpendicular to the rotation axis yielding the symbols . For the oddfold axes 1 and 3, this product leads to the rotoinversion axes and . Thus, for each representative of a set of lattice symmetry directions, the symmetry forms a point group that can be generated by one, or at most two, symmetry operations. The resulting symbols are called full Hermann–Mauguin (or international) symbols. For the lattice point groups they are shown in Table 12.1.4.1.
For the description of a point group of a crystal, we use its lattice symmetry directions. For the representative of each set of lattice symmetry directions, the remaining subgroup is symbolized; if only the primary symmetry direction contains symmetry higher than 1, the symbols `1' for the secondary and tertiary set (if present) can be omitted. For the cubic point groups T and , the representative of the tertiary set would be `1', which is omitted. For the rotoinversion groups and , the remaining subgroups can only be 1 and 3. If the supergroup is , five different types of subgroups can be derived: and m. In the cubic system, for instance, , , 4 or 2 may occur in the primary set. In this case, the symbol m can only occur in the combinations or as can be seen from Table 12.1.4.2.

If the symbols are not only used for the identification of a group but also for its construction, the symbol must contain a list of generating operations and additional relations, if necessary. Following this aspect, the Hermann–Mauguin symbols can be shortened. The choice of generators is not unique; two proposals were presented by Mauguin (1931). In the first proposal, in almost all cases the generators are the same as those of the Shubnikov symbols. In the second proposal, which, apart from some exceptions (see Chapter 12.4 ), is used for the international symbols, Mauguin selected a set of generators and thus a list of short symbols in which reflections have priority (Table 12.1.4.2, column 3). This selection makes the transition from the short pointgroup symbols to the spacegroup symbols fairly simple. These short symbols contain two kinds of notation components:
The generating matrices are uniquely defined by (i) and (ii), if it is assumed that they describe motions with counterclockwise rotational sense about the representative direction looked at end on by the observer. The symbols 2, 4, , 6 and referring to direction [001] are indicators when the pointgroup symbol uses three sets of lattice symmetry directions. For instance, in 4mm the indicator 4 fixes the directions of the mirrors normal to [100] and .
Note: The generation of (a) point group 432 by a rotation 3 around [111] and a rotation 2 and (b) point group by 3 around [111] and a reflection m is only possible if the representative direction of the tertiary set is changed from to [110]; otherwise only the subgroup 32 or 3m of 432 or will be generated.
References
Heesch, H. (1929). Zur systematischen Strukturtheorie II. Z. Kristallogr. 72, 177–201.Hermann, C. (1928). Zur systematischen Strukturtheorie I. Eine neue Raumgruppensymbolik. Z. Kristallogr. 68, 257–287.
Hermann, C. (1931). Bemerkungen zu der vorstehenden Arbeit von Ch. Mauguin. Z. Kristallogr. 76, 559–561.
Internationale Tabellen zur Bestimmung von Kristallstrukturen (1935). I. Band, edited by C. Hermann. Berlin: Borntraeger. [Reprint with corrections: 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. [Revised editions: 1965, 1969 and 1977. Abbreviated as IT (1952).]
Koptsik, V. A. (1966). Shubnikov groups. Moscow University Press. (In Russian.)
Mauguin, Ch. (1931). Sur le symbolisme des groupes de répetition ou de symétrie des assemblages cristallins. Z. Kristallogr. 76, 542–558.
Schoenflies, A. (1891). Krystallsysteme und Krystallstructur. Leipzig: Teubner. [Reprint: Berlin: Springer (1984).]
Schoenflies, A. (1923). Theorie der Kristallstruktur. Berlin: Borntraeger.
Shubnikov, A. V. & Koptsik, V. A. (1972). Symmetry in science and art. Moscow: Nauka. (In Russian.) [Engl. transl: New York: Plenum (1974).]