International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2016). Vol. A, ch. 3.3, p. 778

Section 3.3.1.4.2. Full Hermann–Mauguin symbols

H. Burzlaffa and H. Zimmermannb*

aUniversität Erlangen–Nürnberg, Robert-Koch-Strasse 4a, D-91080 Uttenreuth, Germany, and bInstitut für Angewandte Physik, Lehrstuhl für Kristallographie und Strukturphysik, Universität Erlangen–Nürnberg, Bismarckstrasse 10, D-91054 Erlangen, Germany
Correspondence e-mail:  helmuth.zimmermann@krist.uni-erlangen.de

3.3.1.4.2. Full Hermann–Mauguin symbols

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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 n-fold 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 [\overline{1}]. For the even-fold axes, 2, 4 and 6, this multiplication results in a mirror plane perpendicular to the rotation axis yielding the symbols (2n)/m (n = 1, 2, 3). For the odd-fold axes 1 and 3, this product leads to the rotoinversion axes [\overline{1}] and [\overline{3}]. 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 3.3.1.2[link].

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 Th, the representative of the tertiary set would be `1', which is omitted. For the rotoinversion groups [\overline{1}] and [\overline{3}], the remaining subgroups can only be 1 and 3. If the supergroup is (2n)/m, five different types of subgroups can be derived: n/m, 2n, [\overline{2n}], n and m. In the cubic system, for instance, 4/m, 2/m, [\bar{4}], 4 or 2 may occur in the primary set. In this case, the symbol m can only occur in the combinations 2/m or 4/m as can be seen from Table 3.3.1.3[link].

Table 3.3.1.3| top | pdf |
Point-group symbols

SchoenfliesShubnikovInternational Tables, short symbolInternational Tables, full symbol
[C_{1}] 1 1 1
[C_{i}] [\tilde{2}] [\bar{1}] [\bar{1}]
[C_{2}] 2 2 2
[C_{s}] m m m
[C_{2h}] [2:m] [2/m] [2/m]
[D_{2}] [2:2] 222 222
[C_{2v}] [2 \cdot m] mm2 mm2
[D_{2h}] [m \cdot 2:m] mmm [2/m\ 2/m\ 2/m]
[C_{4}] 4 4 4
[S_{4}] [\widetilde{4}] [\bar{4}] [\bar{4}]
[C_{4h}] [4:m] [4/m] [4/m]
[D_{4}] [4:2] 422 422
[C_{4v}] [4 \cdot m] 4mm 4mm
[D_{2d}] [\widetilde{4}:2] [\bar{4}2m] or [\bar{4}m2] [\bar{4}2m] or [\bar{4}m2]
[D_{4h}] [m \cdot 4:m] [4/mmm] [4/m\ 2/m\ 2/m]
[C_{3}] 3 3 3
[C_{3i}] [\widetilde{6}] [\bar{3}] [\bar{3}]
[D_{3}] [3:2] 32 or 321 or 312 32 or 321 or 312
[C_{3v}] [3 \cdot m] 3m or 3m1 or 31m 3m or 3m1 or 31m
[D_{3d}] [\widetilde{6} \cdot m] [\bar{3}m] or [\bar{3}m1] or [\bar{3}1m] [\bar{3}\; 2/m] or [\bar{3}\; {2/m} 1] or [\bar{3} 1 {2/m}]
[C_{6}] 6 6 6
[C_{3h}] [3:m] [\bar{6}] [\bar{6}]
[C_{6h}] [6:m] [6/m] [6/m]
[D_{6}] [6:2] 622 622
[C_{6v}] [6\cdot m] 6mm 6mm
[D_{3h}] [m\cdot 3:m] [\bar{6}m2] or [\bar{6}2m] [\bar{6}m2] or [\bar{6}2m]
[D_{6h}] [m\cdot 6:m] [6/mmm] [6/m\; 2/m\; 2/m]
T [3/2] 23 23
[T_{h}] [\widetilde{6}/2] [m\bar{3}] [{2/m} \bar{3}]
O [3/4] 432 432
[T_{d}] [3/\widetilde{4}] [\bar{4}3m] [\bar{4}3m]
[O_{h}] [\widetilde{6}/4] [m\bar{3}m] [4/m\; \bar{3}\; 2/m]








































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