Table 1.2.6.8

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

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

T. Janssen^a ^*

^a Institute for Theoretical Physics, University of Nijmegen, 6524 ED Nijmegen, The Netherlands
Correspondence e-mail: ted@sci.kun.nl

Table 1.2.6.8 | top | pdf |
Projective spin representations of the 32 crystallographic point groups

Point group	Relations giving $[\lambda_{i}]$	Double group	Extra representations
1		$[1^{d}]$	No
$[\bar{1}]$	$[{A}^{2}= {E}]$		No
2, m	$[{A}^{2}=-{E}]$	$[2^{d}]$	No
	$[{A}^{2}={B}^{2}=-{E}]$ , $[({AB})^{2}={E}]$
222,	$[{A}^{2}={B}^{2}=({AB})^{2}=-{E}]$	$[222^{d}]$	Yes
	$[{A}^{2}={B}^{2}=({AB})^{2}=-{E}]$
	$[{C}^{2}={E},{AC}={CA}]$ ,
4, $[\bar{4}]$	$[{A}^{4}=-{E}]$	4^d	No
	$[{A}^{4}={B}^{2}=-{E}]$ ,
422, , $[\bar{4}2m]$	$[{A}^{4}={B}^{2}=({AB})^{2}=-{E}]$	422^d	Yes
	As above, plus $[{C}^{2}={E}]$ , ,
3	$[{A}^{3}=-{E}]$	3^d	No
$[\bar{3}]$	$[{A}^{6}={E}]$
32,	$[{A}^{3}={B}^{2}=({AB})^{2}=-{E}]$	32^d	No
$[\bar{3}m]$	$[{A}^{6}={E}]$ , $[{B}^{2}=({AB})^{2}=-{E}]$
6, $[\bar{6}]$	$[{A}^{6}=-{E}]$	6^d	No
	$[{A}^{6}={B}^{2}=-{E}]$ ,
622, , $[\bar{6}2m]$	$[{A}^{6}={B}^{2}=({AB})^{2}=-{E}]$	622^d	Yes
	As above, plus $[{C}^{2}={E}]$ , ,
23	$[{A}^{3}={B}^{2}=({AB})^{3}=-{E}]$	23^d	Yes
	As above, plus $[{C}^{2}={E}]$ , ,
432, $[\bar{4}3m]$	$[{A}^{4}={B}^{3}=({AB})^{2}=-{E}]$	432^d	Yes
$[m\bar{3}m]$	As above, plus $[{C}^{2}={E}]$ , ,