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

International Tables for Crystallography (2015). Vol. A, ch. 1.4, pp. 49-50

Section Sequence of space-group types

H. Wondratscheke Sequence of space-group types

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The sequence of space-group entries in the space-group tables follows that introduced by Schoenflies (1891[link]) and is thus established historically. Within each geometric crystal class, Schoenflies numbered the space-group types in an obscure way. As early as 1919, Niggli (1919[link]) considered this Schoenflies sequence to be unsatisfactory and suggested that another sequence might be more appropriate. Fedorov (1891[link]) used a different sequence in order to distinguish between symmorphic, hemisymmorphic and asymmorphic space groups (cf. Section[link] for a detailed discussion of symmorphic space groups).

The basis of the Schoenflies symbols and thus of the Schoenflies listing is the geometric crystal class. For the present space-group tables, a sequence might have been preferred in which, in addition, space-group types belonging to the same arithmetic crystal class were grouped together. It was decided, however, that the long-established sequence in the earlier editions of International Tables should not be changed.

In Table[link], those geometric crystal classes are listed in which the Schoenflies sequence separates space groups belonging to the same arithmetic crystal class (cf. Section[link] for the definition and discussion of arithmetic crystal classes). The space groups are rearranged in such a way that space groups of the same arithmetic crystal class are grouped together. The arithmetic crystal classes are separated by rules spanning the last three columns of the table and the geometric crystal classes are separated by rules spanning the full width of the table. In all cases not listed in Table[link], the Schoenflies sequence, as used in the space-group tables, does not break up arithmetic crystal classes. Nevertheless, some rearrangement would be desirable in other arithmetic crystal classes too. For example, the symmorphic space group should always be the first entry of each arithmetic crystal class.

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List of geometric crystal classes in which the Schoenflies sequence separates space groups belonging to the same arithmetic crystal class

Geometric crystal classSpace-group type
No.Hermann–Mauguin symbolSchoenflies symbol
2/m 10 P2/m [C_{2h}^{1}]
11 [P2_{1}/m] [C_{2h}^{2}]
13 P2/c [C_{2h}^{4}]
14 [P2_{1}/c] [C_{2h}^{5}]
12 C2/m [C_{2h}^{3}]
15 C2/c [C_{2h}^{6}]
32 149 P312 [D_{3}^{1}]
151 [P3_{1}12] [D_{3}^{3}]
153 [P3_{2}12] [D_{3}^{5}]
150 P321 [D_{3}^{2}]
152 [P3_{1}21] [D_{3}^{4}]
154 [P3_{2}21] [D_{3}^{6}]
155 R32 [D_{3}^{7}]
3m 156 P3m1 [C_{3v}^{1}]
158 P3c1 [C_{3v}^{3}]
157 P31m [C_{3v}^{2}]
159 P31c [C_{3v}^{4}]
160 R3m [C_{3v}^{5}]
161 R3c [C_{3v}^{6}]
23 195 P23 [T^{1}]
198 [P2_{1}3] [T^{4}]
196 F23 [T^{2}]
197 I23 [T^{3}]
199 [I2_{1}3] [T^{5}]
[m\bar{3}] 200 [Pm\bar{3}] [T_{h}^{1}]
201 [Pn\bar{3}] [T_{h}^{2}]
205 [Pa\bar{3}] [T_{h}^{6}]
202 [Fm\bar{3}] [T_{h}^{3}]
203 [Fd\bar{3}] [T_{h}^{4}]
204 [Im\bar{3}] [T_{h}^{5}]
206 [Ia\bar{3}] [T_{h}^{7}]
432 207 P432 [O^{1}]
208 [P4_{2}32] [O^{2}]
213 [P4_{1}32] [O^{7}]
212 [P4_{3}32] [O^{6}]
209 F432 [O^{3}]
210 [F4_{1}32] [O^{4}]
211 I432 [O^{5}]
214 [I4_{1}32] [O^{8}]
[\bar{4}3m] 215 [P\bar{4}3m] [T_{d}^{1}]
218 [P\bar{4}3n] [T_{d}^{4}]
216 [F\bar{4}3m] [T_{d}^{2}]
219 [F\bar{4}3c] [T_{d}^{5}]
217 [I\bar{4}3m] [T_{d}^{3}]
220 [I\bar{4}3d] [T_{d}^{6}]


Fedorov, E. S. (1891). The symmetry of regular systems of figures. (In Russian.) [English translation by D. & K. Harker (1971). Symmetry of crystals, pp. 50–131. American Crystallographic Association, Monograph No. 7.]
Niggli, P. (1919). Geometrische Kristallographie des Diskontinuums. Leipzig: Borntraeger. [Reprint (1973). Wiesbaden: Sändig.]
Schoenflies, A. (1891). Krystallsysteme und Krystallstruktur. Leipzig: B. G. Teubner. [Reprint (1984). Berlin: Springer-Verlag.]

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