Tables for
Volume D
Physical properties of crystals
Edited by A. Authier

International Tables for Crystallography (2006). Vol. D, ch. 1.2, p. 61


T. Janssena*

aInstitute for Theoretical Physics, University of Nijmegen, 6524 ED Nijmegen, The Netherlands
Correspondence e-mail:

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Projective spin representations of the 32 crystallographic point groups

Point groupRelations giving [\lambda_{i}]Double groupExtra representations
1 [{A} = {E}] [1^{d}] No
[\bar{1}] [{A}^{2}= {E}]   No
2, m [{A}^{2}=-{E}] [2^{d}] No
[2/m] [{A}^{2}={B}^{2}=-{E}], [({AB})^{2}={E}]    
222, [2mm] [{A}^{2}={B}^{2}=({AB})^{2}=-{E}] [222^{d}] Yes
[mmm] [{A}^{2}={B}^{2}=({AB})^{2}=-{E}]    
  [{C}^{2}={E},{AC}={CA}], [{BC}={CB}]    
4, [\bar{4}] [{A}^{4}=-{E}] 4d No
[4/m] [{A}^{4}={B}^{2}=-{E}], [{AB}={BA}]    
422, [4mm], [\bar{4}2m] [{A}^{4}={B}^{2}=({AB})^{2}=-{E}] 422d Yes
[4/mmm] As above, plus [{C}^{2}={E}], [{AC}={CA}], [{BC}={CB}]    
3 [{A}^{3}=-{E}] 3d No
[\bar{3}] [{A}^{6}={E}]    
32, [3m] [{A}^{3}={B}^{2}=({AB})^{2}=-{E}] 32d No
[\bar{3}m] [{A}^{6}={E}], [{B}^{2}=({AB})^{2}=-{E}]    
6, [\bar{6}] [{A}^{6}=-{E}] 6d No
[6/m] [{A}^{6}={B}^{2}=-{E}], [{AB}={BA}]    
622, [6mm], [\bar{6}2m] [{A}^{6}={B}^{2}=({AB})^{2}=-{E}] 622d Yes
[6/mmm] As above, plus [{C}^{2}={E}], [{AC}={CA}], [{BC}={CB}]    
23 [{A}^{3}={B}^{2}=({AB})^{3}=-{E}] 23d Yes
[m3] As above, plus [{C}^{2}={E}], [{AC}={CA}], [{BC}={CB}]    
432, [\bar{4}3m] [{A}^{4}={B}^{3}=({AB})^{2}=-{E}] 432d Yes
[m\bar{3}m] As above, plus [{C}^{2}={E}], [{AC}={CA}], [{BC}={CB}]