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. 6276
doi: 10.1107/97809553602060000509 Chapter 4.3. Symbols for space groupsChapters 4.1 , 4.2 and 4.3 contain extensive explanations and tabulations of the various types of spacegroup symbols. Chapter 4.3 contains for each crystal system extensive tables of the short, full and extended spacegroup symbols for various settings and cells. The explanatory text treats, system by system, the `additional symmetry elements' of the space groups. In addition, many examples of those group–subgroup relations are presented which can be read directly from the extended spacegroup symbols. 
There are only two triclinic space groups, and . P1 is quite outstanding because all its subgroups are also P1. They are listed in Table 13.2.2.1 for indices up to [7]. has subgroups , isomorphic, and P1, nonisomorphic.
In the triclinic system, a primitive unit cell can always be selected. In some cases, however, it may be advantageous to select a larger cell, with A, B, C, I or F centring.
The two types of reduced bases (reduced cells) are discussed in Section 9.2.2.
In IT (1935) only the b axis was considered as the unique axis. In IT (1952) two choices were given: the caxis setting was called the `first setting' and the baxis setting was designated the `second setting'.
To avoid the presence of two standard spacegroup symbols side by side, in the present tables only one standard short symbol has been chosen, that conforming to the longlasting tradition of the baxis unique (cf. Sections 2.2.4 and 2.2.16 ). However, for reasons of rigour and completeness, in Table 4.3.2.1 the full symbols are given not only for the caxis and the baxis settings but also for the aaxis setting. Thus, Table 4.3.2.1 has six columns which in pairs refer to these three settings. In the headline, the unique axis of each setting is underlined.
^{†}For the five space groups Aem2 (39), Aea2 (41), Cmce (64), Cmme (67) and Ccce (68), the `new' spacegroup symbols, containing the symbol `e' for the `double' glide plane, are given for all settings. These symbols were first introduced in the Fourth Edition of this volume (IT 1995); cf. Foreword to the Fourth Edition. For further explanations, see Section 1.3.2, Note (x)
and the spacegroup diagrams.
^{‡}For space groups Cmca (64), Cmma (67) and Imma (74), the first lines of the extended symbols, as tabulated here, correspond with the symbols for the six settings in the diagrams of these space groups (Part 7). An alternative formulation which corresponds with the coordinate triplets is given in Section 4.3.3. ^{§}Axes and parallel to axes 3 are not indicated in the extended symbols: cf. Chapter 4.1 . For the glideplane symbol `e', see the Foreword to the Fourth Edition (IT 1995) and Section 1.3.2, Note (x) . 
Additional complications arise from the presence of fractional translations due to glide planes in the primitive cell [groups (7), (13), (14)], due to centred cells [ (5), (8), (12)], or due to both [ (9), (15)]. For these groups, three different choices of the two oblique axes are possible which are called `cell choices' 1, 2 and 3 (see Section 2.2.16 ). If this is combined with the three choices of the unique axis, symbols result. If we add the effect of the permutation of the two oblique axes (and simultaneously reversing the sense of the unique axis to keep the system righthanded, as in abc and ), we arrive at the symbols listed in Table 4.3.2.1 for each of the eight space groups mentioned above.
The spacegroup symbols (3), (4), (6), (10) and (11) do not depend on the cell choice: in these cases, one line of six spacegroup symbols is sufficient.
For space groups with centred lattices (A, B, C, I), extended symbols are given; the `additional symmetry elements' (due to the centring) are printed in the half line below the spacegroup symbol.
The use of the present tabulation is illustrated by two examples, Pm, which does not depend on the cell choice, and , which does.
Examples
How does a monoclinic spacegroup symbol transform for the various settings of the same unit cell? This can be easily recognized with the help of the headline of Table 4.3.2.1, completed to the following scheme: The use of this threeline scheme is illustrated by the following examples.
Examples

It is easy to read all monoclinic maximal t and k subgroups of types I and IIa directly from the extended full symbols of a space group. Maximal subgroups of types IIb and IIc cannot be recognized by simple inspection of the synoptic Table 4.3.2.1
Example: (15, unique axis a)
The t subgroups of index [2] (type I) are B211(C2); Bb11(Cc); .
The k subgroups of index [2] (type IIa) are P2b11(P2c): (); P2n11(P2c); ().
Some subgroups of index [4] (not maximal) are P211(P2); (); Pb11(Pc); Pn11(Pc); ; B1(P1).
The synoptic table of IT (1935) contained spacegroup symbols for the six orthorhombic `settings', corresponding to the six permutations of the basis vectors a, b, c. In IT (1952), lefthanded systems like were changed to righthanded systems by reversing the orientation of the c axis, as in . Note that reversal of two axes does not change the handedness of a coordinate system, so that the settings , , and are equivalent in this respect. The tabulation thus deals with the possible righthanded settings. For further details see Section 2.2.6.4.
An important innovation of IT (1952) was the introduction of extended symbols for the centred groups A, B, C, I, F. These symbols are systematically developed in Table 4.3.2.1. Settings which permute the two axes a and b are listed side by side so that the two C settings appear together, followed by the two A and the two B settings.
In crystal classes mm2 and 222, the last symmetry element is the product of the first two and thus is not independent. It was omitted in the short Hermann–Mauguin symbols of IT (1935) for all space groups of class mm2, but was restored in IT (1952). In space groups of class 222, the last symmetry element cannot be omitted (see examples below).
For the new `double' glide plane symbol `e', see the Foreword to the Fourth Edition (IT 1995) and Section 1.3.2, Note (x) .
The present section emphasizes the use of the extended and full symbols for the derivation of maximal subgroups of types I and IIa; maximal orthorhombic subgroups of types IIb and IIc cannot be recognized by inspection of the synoptic Table 4.3.2.1.


In the 1935 edition of International Tables, for each tetragonal P and I space group an additional Ccell and Fcell description was given. In the corresponding spacegroup symbols, secondary and tertiary symmetry elements were simply interchanged. Coordinate triplets for these larger cells were not printed, except for the space groups of class . In IT (1952), the C and F cells were dropped from the spacegroup tables but kept in the comparative tables.
In the present edition, the C and F cells reappear in the sub and supergroup tabulations of Part 7 , as well as in the synoptic Table 4.3.2.1, where short and extended (twoline) symbols are given for P and C cells, as well as for I and F cells.
In the crystal classes 42(2), 4m(m), or , , where the tertiary symmetry elements are between parentheses, one finds Analogous relations hold for the space groups. In order to have the symmetry direction of the tertiary symmetry elements along [] (cf. Table 2.2.4.1 ), one has to choose the primary and secondary symmetry elements in the product rule along [001] and [010].
Example
In , one has so that would be the short symbol. In fact, in IT (1935), the tertiary symmetry element was suppressed for all groups of class 422, but reestablished in IT (1952), the main reason being the generation of the fourfold rotation as the product of the secondary and tertiary symmetry operations: etc.
As a result of periodicity, in all space groups of classes 422, and , the two tertiary diagonal axes 2, along [] and [110], alternate with axes , the screw component being , 0 (cf. Table 4.1.2.2 ).
Likewise, tertiary diagonal mirrors m in x, x, z and in space groups of classes 4mm, and alternate with glide planes called g,^{1} the glide components being , , 0. The same glide components produce also an alternation of diagonal glide planes c and n (cf. Table 4.1.2.2 ).
The transformations from the P to the two C cells, or from the I to the two F cells, are (cf. Fig. 5.1.3.5 ). The secondary and tertiary symmetry directions are interchanged in the double cells. It is important to know how primary, secondary and tertiary symmetry elements change in the new cells .

Examples are given for maximal k subgroups of P groups (i), of I groups (ii), and for maximal tetragonal, orthorhombic and monoclinic t subgroups.


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.

In the synoptic tables of IT (1935) and IT (1952), for cubic space groups short and full Hermann–Mauguin symbols were listed. They agree, except that in IT (1935) the tertiary symmetry element of the space groups of class 432 was omitted; it was reestablished in IT (1952).
In the present edition, the symbols of IT (1952) are retained, with one exception. In the space groups of crystal classes and , the short symbols contain instead of 3 (cf. Section 2.2.4 ). In Table 4.3.2.1, short and full symbols for all cubic space groups are given. In addition, for centred groups F and I and for P groups with tertiary symmetry elements, extended spacegroup symbols are listed. In space groups of classes 432 and , the product rule (as defined below) is applied in the first line of the extended symbol.
Conventionally, the representative directions of the primary, secondary and tertiary symmetry elements are chosen as [001], [111], and [] (cf. Table 2.2.4.1 for the equivalent directions). As in tetragonal and hexagonal space groups, tertiary symmetry elements are not independent. In classes 432, and , there are product rules where the tertiary symmetry element is in parentheses; analogous rules hold for the space groups belonging to these classes. When the symmetry directions of the primary and secondary symmetry elements are chosen along [001] and [111], respectively, the tertiary symmetry direction is [011], according to the product rule. In order to have the tertiary symmetry direction along [], one has to choose the somewhat awkward primary and secondary symmetry directions [010] and [].
Examples

Owing to periodicity, the tertiary symmetry elements alternate; diagonal axes 2 alternate with parallel screw axes ; diagonal planes m alternate with parallel glide planes g; diagonal n planes, i.e. planes with glide components , alternate with glide planes a, b or c (cf. Chapter 4.1 and Tables 4.1.2.2 and 4.1.2.3 ). For the meaning of the various glide planes g, see Section 11.1.2 and the entries Symmetry operations in Part 7 .
The extended symbol of (202) shows clearly that , , and are maximal subgroups. , , and are maximal subgroups of (229). Space groups with d glide planes have no k subgroup of lattice P.

References
Bertaut, E. F. (1976). Study of principal subgroups and of their general positions in C and I groups of class mmm–D_{zh}. Acta Cryst. A32, 380–387.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 Crystallography (1995). Vol. A, fourth, revised ed., edited by Th. Hahn. Dordrecht: Kluwer Academic Publishers. [Abbreviated as IT (1995).]
International Tables for Xray Crystallography (1952). Vol. I, edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press. [Abbreviated as IT (1952).]