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. 4.3, pp. 7375

The trigonal and hexagonal crystal systems are considered together, because they form the hexagonal `crystal family', as explained in Chapter 2.1 . Hexagonal lattices occur in both systems, whereas rhombohedral lattices occur only in the trigonal system.
The 1935 edition of International Tables contains the symbols C and H for the hexagonal lattice and R for the rhombohedral lattice. C recalls that the hexagonal lattice can be described by a double rectangular Ccentred cell (orthohexagonal axes); H was used for a hexagonal triple cell (see below); R designates the rhombohedral lattice and is used for both the rhombohedral description (primitive cell) and the hexagonal description (triple cell).
In the 1952 edition the following changes took place (cf. pages x, 51 and 544 of IT 1952): The lattice symbol C was replaced by P for reasons of consistency; the H description was dropped. The symbol R was kept for both descriptions, rhombohedral and hexagonal. The tertiary symmetry element in the short Hermann–Mauguin symbols of class 622, which was omitted in IT (1935), was reestablished.
In the present volume, the use of P and R is the same as in IT (1952). The H cell, however, reappears in the sub and supergroup data of Part 7 and in Table 4.3.2.1 of this section, where short symbols for the H description of trigonal and hexagonal space groups are given. The C cell reappears in the subgroup data for all trigonal and hexagonal space groups having symmetry elements orthogonal to the main axis.
The primitive cells of the hexagonal and the rhombohedral lattice, hP and hR, are defined in Table 2.1.2.1 In Part 7 , the `rhombohedral' description of the hR lattice is designated by `rhombohedral axes'; cf. Chapter 1.2 .
Multiple cells are frequently used to describe both the hexagonal and the rhombohedral lattice.
In the hexagonal crystal classes 62(2), 6m(m) and (m) or (2), where the tertiary symmetry element is between parentheses, the following products hold: or The same relations hold for the corresponding Hermann–Mauguin spacegroup symbols.
Parallel axes 2 and occur perpendicular to the principal symmetry axis. Examples are space groups R32 (155), P321 (150) and P312 (149), where the screw components are (rhombohedral axes) or (hexagonal axes) for R32; for P321; and for P312. Hexagonal examples are P622 (177) and (190).
Likewise, mirror planes m parallel to the main symmetry axis alternate with glide planes, the glide components being perpendicular to the principal axis. Examples are P3m1 (156), P31m (157), R3m (160) and P6mm (183).
Glide planes c parallel to the main axis are interleaved by glide planes n. Examples are P3c1 (158), P31c (159), R3c (161, hexagonal axes), (188). In R3c and , the glide component for hexagonal axes becomes for rhombohedral axes, i.e. the c glide changes to an n glide. Thus, if the space group is referred to rhombohedral axes, diagonal n planes alternate with diagonal a, b or c planes (cf. Section 1.4.4 ).
In R space groups, all additional symmetry elements with glide and screw components have their origin in the action of an integral lattice translation. This is also true for the axes and which appear in all R space groups (cf. Table 4.1.2.2 ). For this reason, the `rhombohedral centring' R is not included in Table 4.1.2.3 , which contains only the centrings A, B, C, I, F.
Maximal k subgroups of index [3] are obtained by `decentring' the triple cells R (hexagonal description), D and H in the trigonal system, H in the hexagonal system. Any one of the three centring points may be taken as origin of the subgroup.
Maximal t subgroups of index [2] are read directly from the full symbol of the space groups of classes 32, 3m, , 622, 6mm, , .
Maximal t subgroups of index [3] follow from the third power of the mainaxis operation. Here the Ccell description is valuable.

References
Internationale Tabellen zur Bestimmung von Kristallstrukturen (1935). 1. Band, edited by C. Hermann. Berlin: Borntraeger. [Revised edition: 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. [Abbreviated as IT (1952).]