Table 11.2.2.1

Fischer, W.; Koch, E.

doi:10.1107/97809553602060000523

International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by Th. Hahn

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International Tables for Crystallography (2006). Vol. A. ch. 11.2, p. 815

Table 11.2.2.1

W. Fischer^a and E. Koch^a ^*

^a Institut für Mineralogie, Petrologie und Kristallographie, Philipps-Universität, D-35032 Marburg, Germany
Correspondence e-mail: kochelke@mailer.uni-marburg.de

Table 11.2.2.1 | top | pdf |
Matrices for point-group symmetry operations and orientation of corresponding symmetry elements, referred to a cubic, tetragonal, orthorhombic, monoclinic, triclinic or rhombohedral coordinate system (cf. Table 2.1.2.1 )

Symbol of symmetry operation and orientation of symmetry element	Transformed coordinates $[\tilde{x},\tilde{y},\tilde{z}]$	Matrix W	Symbol of symmetry operation and orientation of symmetry element	Transformed coordinates $[\tilde{x},\tilde{y},\tilde{z}]$	Matrix W	Symbol of symmetry operation and orientation of symmetry element	Transformed coordinates $[\tilde{x},\tilde{y},\tilde{z}]$	Matrix W	Symbol of symmetry operation and orientation of symmetry element	Transformed coordinates $[\tilde{x},\tilde{y},\tilde{z}]$	Matrix W
1		$[\pmatrix{1 &0 &0\cr 0 &1 &0\cr 0 &0 &1\cr}]$	$[\matrix{2 {\hbox to 12pt{}}0,0,z\cr \cr {\hbox to 13pt{}}[001]\cr}]$	$[\bar{x},\bar{y},z]$	$[\pmatrix{\bar{1} &0 &0\cr 0 &\bar{1} &0\cr 0 &0 &1\cr}]$	$[\matrix{2 {\hbox to 12pt{}}0,y,0\cr \cr {\hbox to 13pt{}}[010]\cr}]$	$[\bar{x},y,\bar{z}]$	$[\pmatrix{\bar{1} &0 &0\cr 0 &1 &0\cr 0 &0 &\bar{1}\cr}]$	$[\matrix{2 {\hbox to 12pt{}}x,0,0\cr \cr {\hbox to 12pt{}}[100]\cr}]$	$[x,\bar{y},\bar{z}]$	$[\pmatrix{1 &0 &0\cr 0 &\bar{1} &0\cr 0 &0 &\bar{1}\cr}]$
$[\matrix{3^{+} &x,x,x\cr \cr &[111]\cr}]$		$[\pmatrix{0 &0 &1\cr 1 &0 &0\cr 0 &1 &0\cr}]$	$[\matrix{3^{+} {\hbox to 6pt{}}x,\bar{x},\bar{x}\cr \cr{\hbox to 14pt{}}[1\bar{1}\bar{1}]\cr}]$	$[\bar{z},\bar{x},y]$	$[\pmatrix{0 &0 &\bar{1}\cr \bar{1} &0 &0\cr 0 &1 &0\cr}]$	$[\matrix{3^{+} {\hbox to 6pt{}}\bar{x},x,\bar{x}\cr \cr{\hbox to 14pt{}}[\bar{1}1\bar{1}]\cr}]$	$[z,\bar{x},\bar{y}]$	$[\pmatrix{0 &0 &1\cr \bar{1} &0 &0\cr 0 &\bar{1} &0\cr}]$	$[\matrix{3^{+} {\hbox to 6pt{}}\bar{x},\bar{x},x\cr \cr{\hbox to 13pt{}}[\bar{1}\bar{1}1]\cr}]$	$[\bar{z},x,\bar{y}]$	$[\pmatrix{0 &0 &\bar{1}\cr 1 &0 &0\cr 0 &\bar{1} &0\cr}]$
$[\matrix{3^{-} &x,x,x\cr \cr &[111]\cr}]$		$[\pmatrix{0 &1 &0\cr 0 &0 &1\cr 1 &0 &0\cr}]$	$[\matrix{3^{-} {\hbox to 6pt{}}x,\bar{x},\bar{x}\cr \cr {\hbox to 14pt{}}[1\bar{1}\bar{1}]\cr}]$	$[\bar{y},z,\bar{x}]$	$[\pmatrix{0 &\bar{1} &0\cr 0 &0 &1\cr \bar{1} &0 &0\cr}]$	$[\matrix{3^{-} {\hbox to 6pt{}}\bar{x},x,\bar{x}\cr \cr {\hbox to 14pt{}}[\bar{1}1\bar{1}]\cr}]$	$[\bar{y},\bar{z},x]$	$[\pmatrix{0 &\bar{1} &0\cr 0 &0 &\bar{1}\cr 1 &0 &0\cr}]$	$[\matrix{3^{-} {\hbox to 6pt{}}\bar{x},\bar{x},x\cr \cr{\hbox to 13pt{}}[\bar{1}\bar{1}1]\cr}]$	$[y,\bar{z},\bar{x}]$	$[\pmatrix{0 &1 &0\cr 0 &0 &\bar{1}\cr \bar{1} &0 &0\cr}]$
			$[\matrix{2 {\hbox to 12pt{}}x,x,0\cr \cr{\hbox to 13pt{}}[110]\cr}]$	$[y,x,\bar{z}]$	$[\pmatrix{0 &1 &0\cr 1 &0 &0\cr 0 &0 &\bar{1}\cr}]$	$[\matrix{2 {\hbox to 12pt{}}x,0,x\cr \cr{\hbox to 13.5pt{}}[101]\cr}]$	$[z,\bar{y},x]$	$[\pmatrix{0 &0 &1\cr 0 &\bar{1} &0\cr 1 &0 &0\cr}]$	$[\matrix{2 {\hbox to 11pt{}}0,y,y\cr \cr{\hbox to 14pt{}}[011]\cr}]$	$[\bar{x},z,y]$	$[\pmatrix{\bar{1} &0 &0\cr 0 &0 &1\cr0 &1 &0\cr}]$
			$[\matrix{2 {\hbox to 12pt{}}x,\bar{x},0\cr \cr{\hbox to 13pt{}}[1\bar{1}0]\cr}]$	$[\bar{y},\bar{x},\bar{z}]$	$[\pmatrix{0 &\bar{1} &0\cr \bar{1} &0 &0\cr 0 &0 &\bar{1}\cr}]$	$[\matrix{2 {\hbox to 12pt{}}\bar{x},0,x\cr \cr{\hbox to 13.5pt{}}[\bar{1}01]\cr}]$	$[\bar{z},\bar{y},\bar{x}]$	$[\pmatrix{0 &0 &\bar{1}\cr 0 &\bar{1} &0\cr \bar{1} &0 &0\cr}]$	$[\matrix{2 {\hbox to 11pt{}}0,y,\bar{y}\cr \cr{\hbox to 13.5pt{}}[01\bar{1}]\cr}]$	$[\bar{x},\bar{z},\bar{y}]$	$[\pmatrix{\bar{1} &0 &0\cr 0 &0 &\bar{1}\cr 0 &\bar{1} &0\cr}]$
			$[\matrix{4^{+} {\hbox to 5pt{}}0,0,z\cr \cr{\hbox to 14.5pt{}}[001]\cr}]$	$[\bar{y},x,z]$	$[\pmatrix{0 &\bar{1} &0\cr 1 &0 &0\cr 0 &0 &1\cr}]$	$[\matrix{4^{+} {\hbox to 6pt{}}0,y,0\cr \cr{\hbox to 13pt{}}[010]\cr}]$	$[z,y,\bar{x}]$	$[\pmatrix{0 &0 &1\cr 0 &1 &0\cr\bar{1} &0 &0\cr}]$	$[\matrix{4^{+} {\hbox to 6pt{}}x,0,0\cr \cr{\hbox to 12pt{}}[100]\cr}]$	$[x,\bar{z},y]$	$[\pmatrix{1 &0 &0\cr 0 &0 &\bar{1}\cr 0 &1 &0\cr}]$
			$[\matrix{4^{-} {\hbox to 5pt{}}0,0,z\cr \cr{\hbox to 14.5pt{}}[001]\cr}]$	$[y,\bar{x},z]$	$[\pmatrix{0 &1 &0\cr \bar{1} &0 &0\cr 0 &0 &1\cr}]$	$[\matrix{4^{-} {\hbox to 6pt{}}0,y,0\cr \cr{\hbox to 13pt{}}[010]\cr}]$	$[\bar{z},y,x]$	$[\pmatrix{0 &0 &\bar{1}\cr 0 &1 &0\cr 1 &0 &0\cr}]$	$[\matrix{4^{-} {\hbox to 6pt{}}x,0,0\cr \cr{\hbox to 12pt{}}[100]\cr}]$	$[x,z,\bar{y}]$	$[\pmatrix{1 &0 &0\cr 0 &0 &1\cr 0 &\bar{1} &0\cr}]$
$[\matrix{\bar{1} &0,0,0\cr}]$	$[\bar{x},\bar{y},\bar{z}]$	$[\pmatrix{\bar{1} &0 &0\cr 0 &\bar{1} &0\cr 0 &0 &\bar{1}\cr}]$	$[\matrix{m {\hbox to 10pt{}}x,y,0\cr \cr{\hbox to 13pt{}}[001]\cr}]$	$[x,y,\bar{z}]$	$[\pmatrix{1 &0 &0\cr 0 &1 &0\cr 0 &0 &\bar{1}\cr}]$	$[\matrix{m {\hbox to 10pt{}}x,0,z\cr \cr{\hbox to 14pt{}}[010]\cr}]$	$[x,\bar{y},z]$	$[\pmatrix{1 &0 &0\cr 0 &\bar{1} &0\cr 0 &0 &1\cr}]$	$[\matrix{m {\hbox to 9pt{}}0,y,z\cr \cr{\hbox to 14pt{}}[100]\cr}]$	$[\bar{x},y,z]$	$[\pmatrix{\bar{1} &0 &0\cr 0 &1 &0\cr 0 &0 &1\cr}]$
$[\matrix{\bar{3}^{+} &x,x,x\cr \cr&[111]\cr}]$	$[\bar{z},\bar{x},\bar{y}]$	$[\pmatrix{0 &0 &\bar{1}\cr \bar{1} &0 &0\cr 0 &\bar{1} &0\cr}]$	$[\matrix{\bar{3}^{+} {\hbox to 6pt{}}x,\bar{x},\bar{x}\cr \cr{\hbox to 14pt{}}[1\bar{1}\bar{1}]\cr}]$	$[z,x,\bar{y}]$	$[\pmatrix{0 &0 &1\cr 1 &0 &0\cr 0 &\bar{1} &0\cr}]$	$[\matrix{\bar{3}^{+} {\hbox to 6pt{}}\bar{x},x,\bar{x}\cr \cr{\hbox to 14pt{}}[\bar{1}1\bar{1}]\cr}]$	$[\bar{z},x,y]$	$[\pmatrix{0 &0 &\bar{1}\cr 1 &0 &0\cr 0 &1 &0\cr}]$	$[\matrix{\bar{3}^{+} {\hbox to 6pt{}}\bar{x},\bar{x},x\cr \cr{\hbox to 13pt{}}[\bar{1}\bar{1}1]\cr}]$	$[z,\bar{x},y]$	$[\pmatrix{0 &0 &1\cr \bar{1} &0 &0\cr 0 &1 &0\cr}]$
$[\matrix{\bar{3}^{-} &x,x,x\cr \cr&[111]\cr}]$	$[\bar{y},\bar{z},\bar{x}]$	$[\pmatrix{0 &\bar{1} &0\cr 0 &0 &\bar{1}\cr \bar{1} &0 &0\cr}]$	$[\matrix{\bar{3}^{-} {\hbox to 6pt{}}x,\bar{x},\bar{x}\cr \cr{\hbox to 14pt{}}[1\bar{1}\bar{1}]\cr}]$	$[y,\bar{z},x]$	$[\pmatrix{0 &1 &0\cr 0 &0 &\bar{1}\cr 1 &0 &0\cr}]$	$[\matrix{\bar{3}^{-} {\hbox to 6pt{}}\bar{x},x,\bar{x}\cr \cr{\hbox to 14pt{}}[\bar{1}1\bar{1}]\cr}]$	$[y,z,\bar{x}]$	$[\pmatrix{0 &1 &0\cr 0 &0 &1\cr \bar{1} &0 &0\cr}]$	$[\matrix{\bar{3}^{-} {\hbox to 6pt{}}\bar{x},\bar{x},x\cr \cr{\hbox to 13pt{}}[\bar{1}\bar{1}1]\cr}]$	$[\bar{y},z,x]$	$[\pmatrix{0 &\bar{1} &0\cr 0 &0 &1\cr 1 &0 &0\cr}]$
			$[\matrix{m {\hbox to 10pt{}}x,\bar{x},z\cr \cr{\hbox to 14pt{}}[110]\cr}]$	$[\bar{y},\bar{x},z]$	$[\pmatrix{0 &\bar{1} &0\cr \bar{1} &0 &0\cr 0 &0 &1\cr}]$	$[\matrix{m {\hbox to 10pt{}}\bar{x},y,x\cr \cr{\hbox to 14pt{}}[101]\cr}]$	$[\bar{z},y,\bar{x}]$	$[\pmatrix{0 &0 &\bar{1}\cr 0 &1 &0\cr \bar{1} &0 &0\cr}]$	$[\matrix{m {\hbox to 10pt{}}x,y,\bar{y}\cr \cr{\hbox to 13pt{}}[011]\cr}]$	$[x,\bar{z},\bar{y}]$	$[\pmatrix{1 &0 &0\cr 0 &0 &\bar{1}\cr 0 &\bar{1} &0\cr}]$
			$[\matrix{m {\hbox to 10pt{}}x,x,z\cr \cr{\hbox to 14pt{}}[1\bar{1}0]\cr}]$		$[\pmatrix{0 &1 &0\cr 1 &0 &0\cr 0 &0 &1\cr}]$	$[\matrix{m {\hbox to 10pt{}}x,y,x\cr \cr{\hbox to 14pt{}}[\bar{1}01]\cr}]$		$[\pmatrix{0 &0 &1\cr 0 &1 &0\cr 1 &0 &0\cr}]$	$[\matrix{m {\hbox to 10pt{}}x,y,y\cr \cr{\hbox to 13pt{}}[01\bar{1}]\cr}]$		$[\pmatrix{1 &0 &0\cr 0 &0 &1\cr 0 &1 &0\cr}]$
			$[\matrix{\bar{4}^{+} {\hbox to 5pt{}}0,0,z\cr \cr{\hbox to 14.5pt{}}[001]\cr}]$	$[y,\bar{x},\bar{z}]$	$[\pmatrix{0 &1 &0\cr \bar{1} &0 &0\cr 0 &0 &\bar{1}\cr}]$	$[\matrix{\bar{4}^{+} {\hbox to 6pt{}}0,y,0\cr \cr{\hbox to 13.5pt{}}[010]\cr}]$	$[\bar{z},\bar{y},x]$	$[\pmatrix{0 &0 &\bar{1}\cr 0 &\bar{1} &0\cr 1 &0 &0\cr}]$	$[\matrix{\bar{4}^{+} {\hbox to 6pt{}}x,0,0\cr \cr{\hbox to 12pt{}}[100]\cr}]$	$[\bar{x},z,\bar{y}]$	$[\pmatrix{\bar{1} &0 &0\cr 0 &0 &1\cr 0 &\bar{1} &0\cr}]$
			$[\matrix{\bar{4}^{-} {\hbox to 5pt{}}0,0,z\cr \cr{\hbox to 14.5pt{}}[001]\cr}]$	$[\bar{y},x,\bar{z}]$	$[\pmatrix{0 &\bar{1} &0\cr 1 &0 &0\cr 0 &0 &\bar{1}\cr}]$	$[\matrix{\bar{4}^{-} {\hbox to 6pt{}}0,y,0\cr \cr{\hbox to 13pt{}}[010]\cr}]$	$[z,\bar{y},\bar{x}]$	$[\pmatrix{0 &0 &1\cr 0 &\bar{1} &0\cr \bar{1} &0 &0\cr}]$	$[\matrix{\bar{4}^{-} {\hbox to 6pt{}}x,0,0\cr \cr{\hbox to 12pt{}}[100]\cr}]$	$[\bar{x},\bar{z},y]$	$[\pmatrix{\bar{1} &0 &0\cr 0 &0 &\bar{1}\cr 0 &1 &0\cr}]$