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. 777
Section 3.3.1.4.1. Symmetry directions^{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 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 symmetry 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. Sections 1.3.4.3 and 3.1.1.4 ) 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 3.3.1.2. The directions are related to the conventional crystallographic basis of each lattice point group (cf. Section 3.1.1.4 ).

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.
References
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).]Heesch, H. (1929). Zur systematischen Strukturtheorie II. Z. Kristallogr. 72, 177–201.