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 [{\scr p}1] and [{\scr p}m], 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[link] and 1.4.2.1[link], and some statistics for the three-dimensional classes are given in Table 1.4.3.1[link]. 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[link], 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[link], Table 8.3.4.1[link]

Crystal systemCrystal classSpace group
GeometricArithmetic
NumberSymbolNumberSymbol
Triclinic 1 1 1P 1 P1
[\bar1] 2 [\bar1P] 2 [P\bar1]
Monoclinic 2 3 2P 3 [P2]
4 [P2_1]
4 2C 5 C2
m 5 mP 6 Pm
7 Pc
6 mC 8 Cm
9 Cc
2/m 7 2/mP 10 [P2/m]
11 [P2_1/m]
13 [P2/c]
14 [P2_1/c]
8 2/mC 12 [C2/m]
15 [C2/c]
Orthorhombic 222 9 222P 16 P222
17 [P222_1]
18 [P2_12_12]
19 [P2_12_12_1]
10 222C 20 [C222_1]
21 C222
11 222F 22 F222
12 222I 23 I222
24 [I2_12_12_1]
mm 13 mm2P 25 Pmm2
26 [Pmc2_1]
27 Pcc2
28 Pma2
29 [Pca2_1]
30 Pnc2
31 [Pmn2_1]
32 Pba2
33 [Pna2_1]
34 Pnn2
14 mm2C 35 Cmm2
36 [Cmc2_1]
37 Ccc2
15 2mmC
(Amm2)
38 C2mm
(Amm2)
39 [C2me]
(Aem2)
40 [C2cm]
[(Ama2)]
41 [C2ce]
(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 [P4_1]
77 [P4_2]
78 [P4_3]
23 4I 79 I4
80 [I4_1]
[\bar4] 24 [\bar4P] 81 [P\bar4]
25 [\bar4I] 82 [I\bar4]
4/m 26 4/mP 83 [P4/m]
84 [P4_2/m]
85 [P4/n]
86 [P4_2/n]
27 4/mI 87 [I4/m]
88 [I4_1/a]
422 28 422P 89 P422
90 [P42_12]
91 [P4_122]
92 [P4_12_12]
93 [P4_222]
94 [P4_22_12]
95 [P4_322]
96 [P4_32_12]
29 422I 97 [I422]
98 [I4_122]
4mm 30 4mmP 99 [P4mm]
100 [P4bm]
101 [P4_2cm]
102 [P4_2nm]
103 [P4cc]
104 [P4nc]
105 [P4_2mc]
106 [P4_2bc]
31 4mmI 107 [I4mm]
108 [I4cm]
109 [I4_1md]
110 [I4_1cd]
[\bar4m] 32 [\bar42mP] 111 [P\bar42m]
112 [P\bar42c]
113 [P\bar42_1m]
114 [P\bar42_1c]
33 [\bar4m2P] 115 [P\bar4m2]
116 [P\bar4c2]
117 [P\bar4b2]
118 [P\bar4n2]
34 [\bar4m2I] 119 [I\bar4m2]
120 [I\bar4c2]
35 [\bar42mI] 121 [I\bar42m]
122 [I\bar42d]
4/mmm 36 4/mmmP 123 [P4/mmm]
124 [P4/mcc]
125 [P4/nbm]
126 [P4/nnc]
127 [P4/mbm]
128 [P4/mnc]
129 [P4/nmm]
130 [P4/ncc]
131 [P4_2/mmc]
132 [P4_2/mcm]
133 [P4_2/nbc]
134 [P4_2/nnm]
135 [P4_2/mbc]
136 [P4_2/mnm]
137 [P4_2/nmc]
138 [P4_2/ncm]
37 4/mmmI 139 I4/mmm
140 I4/mcm
141 [I4_1/amd]
142 [I4_1/acd]
Trigonal 3 38 3P 143 P3
144 [P3_1]
145 [P3_2]
39 3R 146 R3
[\bar3] 40 [\bar3P] 147 [P\bar3]
41 [\bar3R] 148 [R\bar3]
32 42 312P 149 P312
151 [P3_112]
153 [P3_212]
43 321P 150 P321
152 [P3_121]
154 [P3_221]
44 32R 155 R32
3m 45 3m1P 156 [P3m1]
158 [P3c1]
46 31mP 157 [P31m]
159 [P31c]
47 3mR 160 [R3m]
161 [R3c]
[\bar3m] 48 [\bar31mP] 162 [P\bar31m]
163 [P\bar31c]
49 [\bar3m1P] 164 [P\bar3m1]
165 [P\bar3c1]
50 [\bar3mR] 166 [R\bar3m]
167 [R\bar3c]
Hexagonal 6 51 6P 168 P6
169 [P6_1]
170 [P6_5]
171 [P6_2]
172 [P6_4]
173 [P6_3]
[\bar6] 52 [\bar6P] 174 [P\bar6]
6/m 53 6/mP 175 [P6/m]
176 [P6_3/m]
622 54 622P 177 [P622]
178 [P6_122]
179 [P6_522]
180 [P6_222]
181 [P6_422]
182 [P6_322]
6mm 55 6mmP 183 P6mm
184 [P6cc]
185 [P6_3cm]
186 [P6_3mc]
[\bar6m] 56 [\bar6m2P] 187 [P\bar6m2]
188 [P\bar6c2]
57 [\bar62mP] 189 [P\bar62m]
190 [P\bar62c]
6/mmm 58 6/mmmP 191 [P6/mmm]
192 [P6/mmc]
193 [P6_3/mcm]
194 [P6_3/mmc]
Cubic 23 59 23P 195 P23
198 [P2_13]
60 23F 196 F23
61 23I 197 I23
199 [I2_13]
[m\bar3] 62 [m\bar3P] 200 [Pm\bar3]
201 [Pn\bar3]
205 [Pa\bar3]
63 [m\bar3F] 202 [Fm\bar3]
203 [Fd\bar3]
64 [m\bar3I] 204 [Im\bar3]
206 [Ia\bar3]
432 65 432P 207 P432
208 [P4_232]
213 [P4_132]
212 [P4_332]
66 432F 209 [F432]
210 [F4_132]
67 432I 211 I432
214 [I4_132]
[\bar43m] 68 [\bar43mP] 215 [P\bar43m]
218 [P\bar43n]
69 [\bar43mF] 216 [F\bar43m]
219 [F\bar43c]
70 [\bar43mI] 217 [I\bar43m]
220 [I\bar43d]
[m\bar3m] 71 [m\bar3mP] 221 [Pm\bar3m]
222 [Pn\bar3n]
223 [Pm\bar3n]
224 [Pn\bar3m]
72 [m\bar3mF] 225 [Fm\bar3m]
226 [Fm\bar3c]
227 [Fd\bar3m]
228 [Fd\bar3c]
73 [m\bar3mI] 229 [Im\bar3m]
230 [Ia\bar3d]

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 [\bar1P]        
2C          
222F          
[\bar4P] [\bar4I]        
3R [\bar3P] [\bar3R] 32R    
[\bar6P]          
23F          
2 2P mP mC [2/mC]    
222C 222I mm2F mmmF    
4I [4/mI] 422I [\bar4m2I] [\bar42mI]  
3P 312P 321P [3m1P] 31mP 3mR
[\bar31mP] [\bar3m1P] [\bar3mR]      
[6/mP] [\bar6m2P] [\bar62mP]      
23P 23I [m\bar3F] [m\bar3I] 432F 432I
[\bar43mP] [\bar43mF] [\bar43mI] [m\bar3mI]    
3 [mm2C] [mm2I]        
3P 312P 321P      
4P          
[m\bar3P] [432P]        
4 2/mP          
222P 2mmC [(=mm2A)] mmmI    
4P [4/mP] 4mmI [\bar42mP] [\bar4m2P] [4/mmmI]
6P 622P 6mmP [6/mmmP]    
432P [m\bar3mP] [m\bar3mF]      
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[link] 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, [m{\bar 3}], 432, and [{\bar 4}3m]. This point is discussed in detail in Volume A of International Tables[link], Section 8.3.4[link] . The symbols of five space groups [[C2me] (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[link]) 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[link]); 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[link]), but these are clumsy and not commonly used; those for C2 and Cm are [C1^{ 2}_{ 2_{1}}\!1] and [C1^{m}_{ a}1], respectively. In the more symmetric crystal systems, even symmorphic space groups with primitive cells contain screw axes and/or glide planes; [P422] ([P42^{ 2}_{2_{1}}]) contains many diad screw axes and P4/mmm ([P4/m2/m^{ 2/m}_{ 2_{1}/g}]) 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[link] .

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.








































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