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

International Tables for Crystallography (2016). Vol. A, ch. 1.3, p. 30

Table 1.3.3.2 

B. Souvigniera*

aRadboud University Nijmegen, Faculty of Science, Mathematics and Computing Science, Institute for Mathematics, Astrophysics and Particle Physics, Postbus 9010, 6500 GL Nijmegen, The Netherlands
Correspondence e-mail: souvi@math.ru.nl

Table 1.3.3.2| top | pdf |
Automorphism groups of three-dimensional primitive lattices

LatticeMetric tensor Bravais group
Hermann–Mauguin symbolGenerators
Triclinic [\pmatrix{ g_{11} & g_{12} & g_{13} \cr & g_{22} & g_{23} \cr & & g_{33} } ] [\bar{1}] [\bar{1}{:}\ \bar x,\bar y,\bar z]
Monoclinic [\pmatrix{ g_{11} & 0 & g_{13} \cr & g_{22} & 0 \cr & & g_{33} } ] 2/m [2_{010}{:}\ \bar x,y,\bar z]
    [m_{010}{:}\ x,\bar y,z]
Orthorhombic [\pmatrix{ g_{11} & 0 & 0 \cr & g_{22} & 0 \cr & & g_{33} } ] mmm [m_{100}{:} \ \bar x,y,z ]
    [m_{010}{:}\ x,\bar y,z]
    [m_{001}{:} \ x,y,\bar z]
Tetragonal [\pmatrix{ g_{11} & 0 & 0 \cr & g_{11} & 0 \cr & & g_{33} } ] 4/mmm [4_{001}{:}\ \bar y,x,z]
    [m_{001}{:} \ x,y,\bar z]
    [m_{100}{:} \ \bar x,y,z]
Hexagonal [\pmatrix{ g_{11} & -{{1}\over{2}} g_{11} & 0 \cr & g_{11} & 0 \cr & & g_{33} } ] 6/mmm [6_{001}{:}\ x-y,x,z]
    [m_{001}{:}\ x,y,\bar z]
    [m_{100}{:}\ \bar x+y,y,z]
Rhombohedral [\pmatrix{ g_{11} & g_{12} & g_{12} \cr & g_{11} & g_{12} \cr & & g_{11} } ] [\bar{3}m] [\bar{3}_{111}{:}\ \bar z,\bar x,\bar y]
    [m_{1\bar{1}0}{:}\ y,x,z ]
Cubic [\pmatrix{ g_{11} & 0 & 0 \cr & g_{11} & 0 \cr & & g_{11} } ] [m\bar{3}m] [m_{001}{:} \ x,y,\bar z]
    [\bar{3}_{111}{:}\ \bar z,\bar x,\bar y]
    [m_{110}{:} \ \bar y,\bar x,z]