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

International Tables for Crystallography (2016). Vol. A, ch. 3.3, pp. 780-790

## Section 3.3.3. Properties of the international 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

### 3.3.3. Properties of the international symbols

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#### 3.3.3.1. Derivation of the space group from the short symbol

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Because the short international symbol contains a set of generators, it is possible to deduce the space group from it. With the same distinction between generators and indicators as for point groups, the modified point-group symbol directly gives the rotation parts W of the generating operations (W, w).

The modified symbols of the generators determine the glide/screw parts of w. To find the location parts of w, it is necessary to inspect the product relations of the group. The deduction of the set of complete generating operations can be summarized in the following rules:

 (i) The integral translations are included in the set of generators. If the unit cell has centring points, the centring operations are generators. (ii) The location parts of the generators can be set to zero except for the two cases noted under (iii) and (iv). (iii) For non-cubic rotation groups with indicators in the symbol, the location part of the first generator can be set to zero. The location part of the second generator is ; the intersection parameter is derived from the indicator in the [001] direction [cf. example (3) below]. (iv) For cubic rotation groups, the location part of the threefold rotation can be set to zero. For space groups related to the point group 23, the location part of the twofold rotation is derived from the symbol of the twofold operation itself. For space groups related to the point group 432, the location part of the twofold generating rotation is derived from the indicator in the [001] direction [cf. examples (4) and (5) below].

The origin that is selected by these rules is called the origin of the symbol' (Burzlaff & Zimmermann, 1980). It is evident that the reference to the origin of the symbol allows a very short and unique notation of all desirable origins by appending the components of the origin of the symbol to the short space-group symbol, thus yielding the so-called expanded Hermann–Mauguin symbol. The shift of origin can be performed easily, for only the translation parts have to be changed. The components of the transformed translation part can be obtained by [cf. Section 1.5.2.3 and equation (1.5.2.13) ] Applications can be found in Burzlaff & Zimmermann (2002).

#### Examples: Deduction of the generating operations from the short symbol

Some examples for the use of these rules are now described in detail. It is convenient to describe the symmetry operation (W, w) by the corresponding coordinate triplets, i.e. using the so-called shorthand notation, cf. Section 1.2.2 . The coordinate triplets can be interpreted as combinations of two constituents: the first one consists of the coordinates of a point in general position after the application of W on x, y, z, while the second corresponds to the translation part w of the symmetry operation. The coordinate triplets of the symmetry operations are tabulated as the general position in the space-group tables (in some cases a shift of origin is necessary). If preference is given to full matrix notation, Table 1.2.2.1 may be used. The following examples contain, besides the description of the symmetry operations, references to the numbering of the general positions in the space-group tables of this volume; cf. Sections 2.1.3.9 and 2.1.3.11 . Centring translations are written after the numbers, if necessary.

 (1) Besides the integral translations, the generators, as given in the symbol, are according to rule (ii):No shift of origin is necessary. The expanded symbol is . (2) According to rule (i), the I centring is an additional generating translation. Thus, the generators are:To obtain the tabulated general position, a shift of origin by is necessary, the expanded symbol is . (3) Apart from the translations, the generating elements are: According to rule (iii), the location part of the first generator, referring to the secondary set of symmetry directions, is equal to zero. For the second generator, the screw part is equal to zero. The location part is . The expanded symbol gives the tabulated setting. (4) According to rule (iv), the generators are Following rule (iv), the location part of the threefold axis must be set to zero. The screw part of the twofold axis in [001] is , the location part is . No origin shift is necessary. The expanded symbol is . (5) Besides the integral translations, the generators given by the symbol are:The screw part of the twofold axis is zero. According to rule (iv), the location part is . No origin shift is necessary. The expanded symbol is .

#### 3.3.3.2. Derivation of the full symbol from the short symbol

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If the geometrical point of view is again considered, it is possible to derive the full international symbol for a space group. This full symbol can be interpreted as consisting of symmetry elements. It can be generated from the short symbol with the aid of products between symmetry operations. It is possible, however, to derive the glide/screw parts of the elements in the full symbol directly from the glide/screw parts of the short symbol.

The product of operations corresponding to non-parallel glide or mirror planes generates a rotation or screw axis parallel to the intersection line. The screw part of the rotation is equal to the sum of the projections of the glide components of the planes on the axis. The angle between the planes determines the rotation part of the axis. For 90°, we obtain a twofold, for 60° a threefold, for 45° a fourfold and for 30° a sixfold axis.

#### Example:

The product of b and c generates a screw axis in the z direction because the sum of the glide components in the z direction is . The product of c and n generates a screw axis in the x direction and the product between b and n produces a rotation axis 2 in the y direction because the y components for b and n add up to modulo integers.

Thus, the full symbol is

In most cases, the full symbol is identical with the short symbol; differences between full and short symbols can only occur for space groups corresponding to lattice point groups (holohedries) and to the point group . In all these cases, the short symbol is extended to the full symbol by adding the symbol for the maximal purely rotational subgroup. A special procedure is in use for monoclinic space groups. To indicate the choice of coordinate axes, the full symbol is treated like an orthorhombic symbol, in which the directions without symmetry are indicated by 1', even though they do not correspond to lattice symmetry directions in the monoclinic case.

#### 3.3.3.3. Non-symbolized symmetry elements

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Certain symmetry elements are not given explicitly in the full symbol because they can easily be derived. They are:

• (i) Rotoinversion axes that are not used to indicate the lattice symmetry directions.

• (ii) Rotation axes 2 included in the axes 4, and 6 and rotation axes 3 included in the axes , 6 and .

• (iii) Additional symmetry elements occurring in space groups with centred unit cells, cf. Sections 1.4.2.4 and 1.5.4.1 . These types of operation can be deduced from the product of the centring translation (I, g) with a symmetry operation (W, w). The new symmetry operation again has W as rotation part but a different glide/screw part if the component of g parallel to the symmetry element corresponding to W is not a lattice vector; cf. Section 1.5.4.1 .

#### Example

Space group has a twofold axis along b with screw part . The translational part of the centring operation is .

An additional axis parallel to b thus has a translation part . The component indicates a screw axis in the b direction, whereas the component indicates the location of this axis in . Similarly, it can be shown that glide plane c combined with the centring gives a glide plane n.

In the same way, in rhombohedral and cubic space groups, a rotation axis 3 is accompanied by screw axes and .

In space groups with centred unit cells, the location parts of different symmetry elements may coincide. In , for example, the mirror plane m contains simultaneously a non-symbolized glide plane n. The same applies to all mirror planes in Fmmm.

• (iv) Symmetry elements with diagonal orientation always occur with different types of glide/screw parts simultaneously. In space group (111) the translation vector along a can be decomposed asThe diagonal mirror plane with normal along passing through the origin is accompanied by a parallel glide plane with glide part passing through . The same arguments lead to the occurrence of screw axes , and connected with diagonal rotation axes 2 or 3.

• (v) For some investigations connected with klassengleiche subgroups (for subgroups of space groups, cf. Section 1.7.1 ), it is convenient to introduce an extended Hermann–Mauguin symbol that comprises all symmetry elements indicated in (iii) and (iv). The basic concept may be found in papers by Hermann (1929) and in IT (1952). These concepts have been applied by Bertaut (1976) and Zimmermann (1976); cf. Section 1.5.4.1 .

#### Example

The full symbol of space group Imma (74) is

The I-centring operation introduces additional rotation axes and glide planes for all three sets of lattice symmetry directions. The extended Hermann–Mauguin symbol is This symbol shows immediately the eight subgroups with a P lattice corresponding to point group mmm:

#### 3.3.3.4. Standardization rules for short symbols

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The symbols of Bravais lattices and glide planes depend on the choice of basis vectors. As shown in the preceding section, additional translation vectors in centred unit cells produce new symmetry operations with the same rotation but different glide/screw parts. Moreover, it was shown that for diagonal orientations symmetry operations may be represented by different symbols. Thus, different short symbols for the same space group can be derived even if the rules for the selection of the generators and indicators are obeyed.

For the unique designation of a space-group type, a standardization of the short symbol is necessary. Rules for standardization were given first by Hermann (1931) and later in a slightly modified form in IT (1952).

These rules, which are generally followed in the present tables, are given below. Because of the historical development of the symbols (cf. Section 3.3.4), some of the present symbols do not obey the rules, whereas others depending on the crystal class need additional rules for them to be uniquely determined. These exceptions and additions are not explicitly mentioned, but may be discovered from Table 3.3.3.1 in which the short symbols are listed for all space groups. A table for all settings may be found in Section 1.5.4 .

 Table 3.3.3.1| top | pdf | Standard space-group symbols
No.Schoenflies symbolShubnikov symbolSymbols of International TablesComments
1935 EditionPresent Edition
ShortFullShortFull
1 P1 P1 P1 P1
2 (Sh–K)
3 P2 P2 P2 P121
P112
4

5 C2 C2 C2 C121 B2, B112 (IT, 1952)
A112 (Sh–K)
6 Pm Pm Pm P1m1
P11m
7 Pc Pc Pc P1c1 Pb, P11b (IT, 1952)
P11a (Sh–K)
8 Cm Cm Cm C1m1 Bm, B11m (IT, 1952)
A11m (Sh–K)
9 Cc Cc Cc C1c1 Bb, B11b (IT, 1952)
A11a (Sh–K)
10

11

12 (IT, 1952)
(Sh–K)
13 (IT, 1952)
(Sh–K)
14 (IT, 1952)
(Sh–K)
15 (IT, 1952)
A11 2a (Sh–K)
16 P222 P222 P222 P222
17
18
19
20
21 C222 C222 C222 C222
22 F222 F222 F222 F222
23 I222 I222 I222 I222
24
25 Pmm Pmm2 Pmm2 Pmm2
26 Pmc
27 Pcc Pcc2 Pcc2 Pcc2
28 Pma Pma2 Pma2 Pma2
29 Pca
30 Pnc Pnc2 Pnc2 Pnc2 (Sh–K)
31 Pmn
32 Pba Pba2 Pba2 Pba2
33 Pna
34 Pnn Pnn2 Pnn2 Pnn2
35 Cmm Cmm2 Cmm2 Cmm2
36 Cmc
37 Ccc Ccc2 Ccc2 Ccc2
38 Amm Amm2 Amm2 Amm2
39 Abm Abm2 Aem2 Aem2
40 Ama Ama2 Ama2 Ama2
41 Aba Aba2 Aea2 Aea2
42 Fmm Fmm2 Fmm2 Fmm2
43 Fdd Fdd2 Fdd2 Fdd2
44 Imm Imm2 Imm2 Imm2
45 Iba Iba2 Iba2 Iba2 (Sh–K)
46 Ima Ima2 Ima2 Ima2
47 Pmmm Pmmm
48 Pnnn Pnnn
49 Pccm Pccm
50 Pban Pban
51 Pmma Pmma
52 Pnna Pnna
53 Pmna Pmna
54 Pcca Pcca
55 Pbam Pbam
56 Pccn Pccn
57 Pbcm Pbcm
58 Pnnm Pnnm
59 Pmmn Pmmn
60 Pbcn Pbcn
61 Pbca Pbca
62 Pnma Pnma
63 Cmcm Cmcm
64 Cmca Cmce Use former symbol Cmca for generation
65 Cmmm Cmmm
66 Cccm Cccm
67 Cmma Cmme Use former symbol Cmma for generation
68 Ccca Ccce Use former symbol Ccca for generation
69 Fmmm Fmmm
70 Fddd Fddd
71 Immm Immm I2/m 2/m 2/m
72 Ibam Ibam
73 Ibca Ibca (IT, 1952)
74 Imma Imma (IT, 1952)
75 P4 P4 P4 P4
76
77
78
79 I4 I4 I4 I4
80
81
82
83
84
85
86
87
88
89 (c:(a:a)):4:2 P42 P422 P422 P422
90
91
92
93
94
95
96
97 I42 I422 I422 I422
98
99 P4mm P4mm P4mm P4mm
100 P4bm P4bm P4bm P4bm
101 P4cm
102 P4nm
103 P4cc P4cc P4cc P4cc
104 P4nc P4nc P4nc P4nc
105 P4mc
106 P4bc
107 I4mm I4mm I4mm I4mm
108 I4cm I4cm I4cm I4cm
109 I4md
110 I4cd (Sh–K)
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125 (Sh–K)
126
127 (Sh–K)
128
129
130
131
132
133 (Sh–K)
134
135 (Sh–K)
136
137
138
139
140
141
142
143 C3 C3 P3 P3
144
145
146 R3 R3 R3 R3 Hexagonal setting (Sh–K)
Rhombohedral setting (Sh–K)
147
148 Hexagonal setting (Sh–K)
Rhombohedral setting (Sh–K)
149 H32 H321 P312 P312
150 C32 C321 P321 P321
151
152
153
154
155 R32 R32 R32 R32 Hexagonal setting (Sh–K)
Rhombohedral setting (Sh–K)
156 C3m C3m1 P3m1 P3m1
157 H3m H3m1 P31m P31m (Sh–K) with special comment
158 C3c C3c1 P3c1 P3c1
159 H3c H3c1 P31c P31c (Sh–K) with special comment
160 R3m R3m R3m R3m Hexagonal setting (Sh–K)
Rhombohedral setting (Sh–K)
161 R3c R3c R3c R3c Hexagonal setting (Sh–K)
Rhombohedral setting (Sh–K)
162 (Sh–K) with special comment
163 (Sh–K) with special comment
164
165
166 Hexagonal setting (Sh–K)
Rhombohedral setting (Sh–K)
167 Hexagonal setting (Sh–K)
Rhombohedral setting (Sh–K)
168 C6 C6 P6 P6
169
170
171
172
173
174
175
176
177 C62 C622 P622 P622
178
179
180
181
182
183 C6mm C6mm P6mm P6mm
184 C6cc C6cc P6cc P6cc
185 C6cm
186 C6mc
187
188
189
190
191
192
193
194
195 P23 P23 P23 P23
196 F23 F23 F23 F23
197 I23 I23 I23 I23
198
199
200 Pm3 Pm3 (IT, 1952)
201 Pn3 Pn3 (IT, 1952)
202 Fm3 Fm3 (IT, 1952)
203 Fd3 Fd3 (IT, 1952)
204 Im3 Im3 (IT, 1952)
205 Pa3 Pa3 (IT, 1952)
206 Ia3 Ia3 (IT, 1952)
207 P43 P432 P432 P432
208
209 F43 F432 F432 F432
210
211 I43 I432 I432 I432
212
213
214
215
216
217
218
219
220
221 Pm3m Pm3m (IT, 1952)
222 Pn3n Pn3n (IT, 1952)
223 Pm3n Pm3n (IT, 1952)
224 Pn3m Pn3m (IT ,1952)
225 Fm3m Fm3m (IT, 1952)
226 Fm3c Fm3c (IT, 1952)
227 Fd3m Fd3m (IT, 1952)
228 Fd3c Fd3c (IT, 1952)
229 Im3m Im3m (IT, 1952)
230 Ia3d Ia3d (IT, 1952)
Abbreviations used in the column Comments: IT, 1952: International Tables for X-ray Crystallography, Vol. I (1952); Sh–K; Shubnikov & Koptsik (1972). Note that this table contains only one notation for the b-unique setting and one notation for the c-unique setting in the monoclinic case, always referring to cell choice 1 of the space-group tables.

Triclinic symbols are unique if the unit cell is primitive. For the standard setting of monoclinic space groups, the lattice symmetry direction is labelled b. From the three possible centrings A, I and C, the latter one is favoured. If glide components occur in the plane perpendicular to [010], the glide direction c is preferred. In the space groups corresponding to the orthorhombic group mm2, the unique direction of the twofold axis is chosen along c. Accordingly, the face centring C is employed for centrings perpendicular to the privileged direction. For space groups with possible A or B centring, the first one is preferred. For groups 222 and mmm, no privileged symmetry direction exists, so the different possibilities of one-face centring can be reduced to C centring by change of the setting. The choices of unit cell and centring type are fixed by the conventional basis in systems with higher symmetry.

When more than one kind of symmetry elements exist in one representative direction, in most cases the choice for the space-group symbol is made in order of decreasing priority: for reflections and glide reflections m, a, b, c, n, d; for proper rotations and screw rotations ; ; ; [cf. IT (1952), p. 55, and Section 1.4.1 ].

#### 3.3.3.5. Systematic absences

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Hermann (1928a) emphasized that the short symbols permit the derivation of systematic absences of X-ray reflections caused by the glide/screw parts of the symmetry operations. If describes the X-ray reflection and is the matrix representation of a symmetry operation, the matrix can be expanded as follows:The absence of a reflection is governed by the relation (i) = and the scalar product (ii) . A reflection h is absent if condition (i) holds and the scalar product (ii) is not an integer. The calculation must be made for all generators and indicators of the short symbol. Systematic absences, introduced by the further symmetry operations generated, are obtained by the combination of the extinction rules derived for the generators and indicators.

#### Example: Space group

The generators of the space group are the integral translations and the centring translation , the rotation 2 in direction [100]: and the rotation 2 in direction : . The combination of the two generators gives the operation corresponding to the indicator, namely , which represents a fourfold screw rotation in the direction [001].

The integral translations imply no restriction because the scalar product is always an integer. For the centring, condition (i) with holds for all reflections (integral condition), but the scalar product (ii) is an integer only for . Thus, reflections hkl with are absent. The screw rotation 4 has the screw part . Only 00l reflections obey condition (i) (serial extinction). An integral value for the scalar product (ii) requires . The twofold axes in the directions [100] and do not imply further absences because .

Detailed discussion of the theoretical background of conditions for possible general reflections and their derivation is given in Chapter 1.6 .

#### 3.3.3.6. Generalized symmetry

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The international symbols can be suitably modified to describe generalized symmetry, e.g. colour groups, which occur when the symmetry operations are combined with changes of physical properties. For the description of antisymmetry (or `black–white' symmetry), the symbols of the Bravais lattices are supplemented by additional letters for centrings accompanied by a change in colour. For symmetry operations that are not translations, a prime is added to the usual symbol if a change of colour takes place. A complete description of the symbols and a detailed list of references are given by Koptsik (1966). The Shubnikov symbols have not been extended to colour symmetry.

An introduction to the structure, properties and symbols of magnetic subperiodic and magnetic space groups is given in Chapter 3.6 .

### References

International Tables for X-ray Crystallography (1952). Vol. I, edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press. [Revised editions: 1965, 1969 and 1977. Abbreviated as IT (1952).]
Bertaut, E. F. (1976). Study of principal subgroups and their general positions in C and I groups of class mmm – D2h. Acta Cryst. A32, 380–387.
Burzlaff, H. & Zimmermann, H. (1980). On the choice of origins in the description of space groups. Z. Kristallogr. 153, 151–179.
Burzlaff, H. & Zimmermann, H. (2002). On the treatment of settings of space groups and crystal structures by specialized short Hermann–Mauguin space-group symbols. Z. Kristallogr. 217, 135–138.
Hermann, C. (1928a). Zur systematischen Strukturtheorie I. Eine neue Raumgruppensymbolik. Z. Kristallogr. 68, 257–287.
Hermann, C. (1929). Zur systematischen Strukturtheorie IV. Untergruppen. Z. Kristallogr. 69, 533–555.
Hermann, C. (1931). Bemerkungen zu der vorstehenden Arbeit von Ch. Mauguin. Z. Kristallogr. 76, 559–561.
Koptsik, V. A. (1966). Shubnikov Groups. Moscow University Press. (In Russian.)
Zimmermann, H. (1976). Ableitung der Raumgruppen aus ihren klassengleichen Untergruppenbeziehungen. Z. Kristallogr. 143, 485–515.