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, p. 778
Section 3.3.1.4.2. Full Hermann–Mauguin 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 
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 (2n)/m (n = 1, 2, 3). 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 3.3.1.2.
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 T_{h}, 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 (2n)/m, five different types of subgroups can be derived: n/m, 2n, , n and m. In the cubic system, for instance, 4/m, 2/m, , 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.
