International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C, ch. 1.4, pp. 20-21

## Section 1.4.2. Classification of space groups

A. J. C. Wilsona

aSt John's College, Cambridge CB2 1TP, England

### 1.4.2. Classification of space groups

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Arithmetic crystal classes may be used to classify space groups on a scale somewhat finer than that given by the geometric crystal classes. Space groups are members of the same arithmetic crystal class if they belong to the same geometric crystal class, have the same Bravais lattice, and (when relevant) have the same orientation of the lattice relative to the point group. Each one-dimensional arithmetic crystal class contains a single space group, symbolized by and , respectively. Most two-dimensional arithmetic crystal classes contain only a single space group; only 2mmp has as many as three.

The space groups belonging to each geometric and arithmetic crystal class in two and three dimensions are indicated in Tables 1.4.1.1 and 1.4.2.1, and some statistics for the three-dimensional classes are given in Table 1.4.3.1. 12 three-dimensional classes contain only a single space group, whereas two contain 16 each. Certain arithmetic crystal classes (3P, 312P, 321P, 422P, 6P, 622P, 432P) contain enantiomorphous pairs of space groups, so that the number of members of these classes depends on whether the enantiomorphs are combined or distinguished. Such classes occur twice in Table 1.4.3.1, as indicated by the footnotes.

 Table 1.4.2.1| top | pdf | The three-dimensional space groups, arranged by arithmetic crystal class; in a few geometric crystal classes this differs somewhat from the conventional numerical order; see International Tables Volume A, Table 8.3.4.1
Crystal systemCrystal classSpace group
GeometricArithmetic
NumberSymbolNumberSymbol
Triclinic 1 1 1P 1 P1
2 2
Monoclinic 2 3 2P 3
4
4 2C 5 C2
m 5 mP 6 Pm
7 Pc
6 mC 8 Cm
9 Cc
2/m 7 2/mP 10
11
13
14
8 2/mC 12
15
Orthorhombic 222 9 222P 16 P222
17
18
19
10 222C 20
21 C222
11 222F 22 F222
12 222I 23 I222
24
mm 13 mm2P 25 Pmm2
26
27 Pcc2
28 Pma2
29
30 Pnc2
31
32 Pba2
33
34 Pnn2
14 mm2C 35 Cmm2
36
37 Ccc2
15 2mmC
(Amm2)
38 C2mm
(Amm2)
39
(Aem2)
40
41
(Aea2)
16 mm2F 42 Fmm2
43 Fdd2
17 mm2I 44 Imm2
45 Iba2
46 Ima2
mmm 18 mmmP 47 Pmmm
48 Pnnn
49 Pccm
50 Pban
51 Pmma
52 Pnna
53 Pmna
54 Pcca
55 Pbam
56 Pccn
57 Pbcm
58 Pnnm
59 Pmmn
60 Pbcn
61 Pbca
62 Pnma
19 mmmC 63 Cmcm
64 Cmce
65 Cmmm
66 Cccm
67 Cmme
68 Ccce
20 mmmF 69 Fmmm
70 Fddd
21 mmmI 71 Immm
72 Ibam
73 Ibca
74 Imma
Tetragonal 4 22 4P 75 P4
76
77
78
23 4I 79 I4
80
24 81
25 82
4/m 26 4/mP 83
84
85
86
27 4/mI 87
88
422 28 422P 89 P422
90
91
92
93
94
95
96
29 422I 97
98
4mm 30 4mmP 99
100
101
102
103
104
105
106
31 4mmI 107
108
109
110
32 111
112
113
114
33 115
116
117
118
34 119
120
35 121
122
4/mmm 36 4/mmmP 123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
37 4/mmmI 139 I4/mmm
140 I4/mcm
141
142
Trigonal 3 38 3P 143 P3
144
145
39 3R 146 R3
40 147
41 148
32 42 312P 149 P312
151
153
43 321P 150 P321
152
154
44 32R 155 R32
3m 45 3m1P 156
158
46 31mP 157
159
47 3mR 160
161
48 162
163
49 164
165
50 166
167
Hexagonal 6 51 6P 168 P6
169
170
171
172
173
52 174
6/m 53 6/mP 175
176
622 54 622P 177
178
179
180
181
182
6mm 55 6mmP 183 P6mm
184
185
186
56 187
188
57 189
190
6/mmm 58 6/mmmP 191
192
193
194
Cubic 23 59 23P 195 P23
198
60 23F 196 F23
61 23I 197 I23
199
62 200
201
205
63 202
203
64 204
206
432 65 432P 207 P432
208
213
212
66 432F 209
210
67 432I 211 I432
214
68 215
218
69 216
219
70 217
220
71 221
222
223
224
72 225
226
227
228
73 229
230
 Table 1.4.3.1| top | pdf | Arithmetic crystal classes classified by the number of space groups that they contain
Number of space groups in the classSymbols of the arithmetic crystal classes
1 1P
2C
222F

3R 32R

23F
2 2P mP mC
222C 222I mm2F mmmF
4I 422I
3P 312P 321P 31mP 3mR

23P 23I 432F 432I

3
3P 312P 321P
4P

4 2/mP
222P 2mmC mmmI
4P 4mmI
6P 622P 6mmP
432P
6 mmmC
422P
6P 622P
8 422P 4mmP
10 mm2P
16 mmmP
4/mmmP
Enantiomorphs combined.
Enantiomorphs distinguished.

The space groups in Table 1.4.2.1 are listed in the order of the arithmetic crystal class to which they belong. It will be noticed that arrangement according to the conventional space-group numbering would separate members of the same arithmetic crystal class in the geometric classes 2/m, 3m, 23, , 432, and . This point is discussed in detail in Volume A of International Tables, Section 8.3.4 . The symbols of five space groups [ (Aem2), C2ce (Aea2), Cmce, Cmme, Ccce] have been conformed to those recommended in the fourth, revised edition of Volume A of International Tables.

#### 1.4.2.1. Symmorphic space groups

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The 73 space groups known as symmorphic' are in one-to-one correspondence with the arithmetic crystal classes, and their standard short' symbols (Bertaut, 2005) are obtained by interchanging the order of the geometric crystal class and the Bravais cell in the symbol for the arithmetic space group. In fact, conventional crystallographic symbolism did not distinguish between arithmetic crystal classes and symmorphic space groups until recently (de Wolff et al., 1985); the symbol of the symmorphic group was used also for the arithmetic class.

This relationship between the symbols, and the equivalent rule-of-thumb symmorphic space groups are those whose standard (short) symbols do not contain glide planes or screw axes, reveal nothing fundamental about the nature of symmorphism; they are simply a consequence of the conventions governing the construction of symbols in International Tables for Crystallography.3

Although the standard symbols of the symmorphic space groups do not contain screw axes or glide planes, this is a result of the manner in which the space-group symbols have been devised. Most symmorphic space groups do in fact contain screw axes and/or glide planes. This is immediately obvious for the symmorphic space groups based on centred cells; C2 contains equal numbers of diad rotation axes and diad screw axes, and Cm contains equal numbers of reflection planes and glide planes. This is recognized in the `extended' space-group symbols (Bertaut, 2005), but these are clumsy and not commonly used; those for C2 and Cm are and , respectively. In the more symmetric crystal systems, even symmorphic space groups with primitive cells contain screw axes and/or glide planes; () contains many diad screw axes and P4/mmm () contains both screw axes and glide planes.

The balance of symmetry elements within the symmorphic space groups is discussed in more detail in Subsection 9.7.1.2 .

### References

Bertaut, E. F. (2005). Synoptic tables of space-group symbols. International tables for crystallography, Vol. A, edited by Th. Hahn, Part 4. Heidelberg: Springer.
International Tables for Crystallography (2005). Vol. A. Space-group symmetry, fifth ed., edited by Th. Hahn. Heidelberg: Springer.
Wolff, P. M. de, Belov, N. V., Bertaut, E. F., Buerger, M. J., Donnay, J. D. H., Fischer, W., Hahn, Th., Koptsik, V. A., Mackay, A. L., Wondratschek, H., Wilson, A. J. C. & Abrahams, S. C. (1985). Nomenclature for crystal families, Bravais-lattice types and arithmetic classes. Acta Cryst. A41, 278–280.