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

International Tables for Crystallography (2015). Vol. A, ch. 1.5, pp. 77-80

Table 1.5.1.1 

H. Wondratscheka and M. I. Aroyob

Table 1.5.1.1| top | pdf |
Selected 3 × 3 transformation matrices [{\bi P}] and [{\bi Q} = {\bi P}^{ -1}]

For inverse transformations (against the arrow) replace P by Q and vice versa.

TransformationP[{\bi Q} = {\bi P}^{-1}]Crystal system
Cell choice 1 [\rightarrow] cell choice 2: [\cases{P \rightarrow P\cr C \rightarrow A\cr}] [\pmatrix{\bar{1} &0 &1\cr 0 &1 &0\cr \bar{1} &0 &0\cr}] [\pmatrix{0 &0 &\bar{1}\cr 0 &1 &0\cr 1 &0 &\bar{1}\cr}] Monoclinic (cf. Sections 1.5.4.3 and 2.1.3.15)
Cell choice 2 [\rightarrow] cell choice 3: [\cases{P \rightarrow P\cr A \rightarrow I\cr}] Unique axis b invariant
Cell choice 3 [\rightarrow] cell choice 1: [\cases{P \rightarrow P\cr I \rightarrow C\cr}]
(Fig. 1.5.1.2[link]a)      
Cell choice 1 [\rightarrow] cell choice 2: [\cases{P \rightarrow P\cr A \rightarrow B\cr}] [\pmatrix{0 &\bar{1} &0\cr 1 &\bar{1} &0\cr 0 &0 &1\cr}] [\pmatrix{\bar{1} &1 &0\cr \bar{1} &0 &0\cr 0 &0 &1\cr}] Monoclinic (cf. Sections 1.5.4.3 and 2.1.3.15)
Cell choice 2 [\rightarrow] cell choice 3: [\cases{P \rightarrow P\cr B \rightarrow I\cr}] Unique axis c invariant
Cell choice 3 [\rightarrow] cell choice 1: [\cases{P \rightarrow P\cr I \rightarrow A\cr}]
(Fig. 1.5.1.2[link]b)      
Cell choice 1 [\rightarrow] cell choice 2: [\cases{P \rightarrow P\cr B \rightarrow C\cr}] [\pmatrix{1 &0 &0\cr 0 &0 &\bar{1}\cr 0 &1 &\bar{1}\cr}] [\pmatrix{1 &0 &0\cr 0 &\bar{1} &1\cr 0 &\bar{1} &0\cr}] Monoclinic (cf. Sections 1.5.4.3 and 2.1.3.15)
Cell choice 2 [\rightarrow] cell choice 3: [\cases{P \rightarrow P\cr C \rightarrow I\cr}] Unique axis a invariant
Cell choice 3 [\rightarrow] cell choice 1: [\cases{P \rightarrow P\cr I \rightarrow B\cr}]
(Fig. 1.5.1.2[link]c)      
Unique axis b [\rightarrow] unique axis c      
Cell choice 1: [\cases{P \rightarrow P\cr C \rightarrow A\cr}] [\pmatrix{0 &1 &0\cr 0 &0 &1\cr 1 &0 &0\cr}] [\pmatrix{0 &0 &1\cr 1 &0 &0\cr 0 &1 &0\cr}] Monoclinic (cf. Sections 1.5.4.3 and 2.1.3.15)
Cell choice 2: [\cases{P \rightarrow P\cr A \rightarrow B\cr}] Cell choice invariant
Cell choice 3: [\cases{P \rightarrow P\cr I \rightarrow I\cr}]
Unique axis b [\rightarrow] unique axis a      
Cell choice 1: [\cases{P \rightarrow P\cr C \rightarrow B\cr}] [\pmatrix{0 &0 &1\cr 1 &0 &0\cr 0 &1 &0\cr}] [\pmatrix{0 &1 &0\cr 0 &0 &1\cr 1 &0 &0\cr}] Monoclinic (cf. Sections 1.5.4.3 and 2.1.3.15)
Cell choice 2: [\cases{P \rightarrow P\cr A \rightarrow C\cr}] Cell choice invariant
Cell choice 3: [\cases{P \rightarrow P\cr I \rightarrow I\cr}]
Unique axis c [\rightarrow] unique axis a      
Cell choice 1: [\cases{P \rightarrow P\cr A \rightarrow B\cr}] [\pmatrix{0 &1 &0\cr 0 &0 &1\cr 1 &0 &0\cr}] [\pmatrix{0 &0 &1\cr 1 &0 &0\cr 0 &1 &0\cr}] Monoclinic (cf. Sections 1.5.4.3 and 2.1.3.15)
Cell choice 2: [\cases{P \rightarrow P\cr B \rightarrow C\cr}] Cell choice invariant
Cell choice 3: [\cases{P \rightarrow P\cr I \rightarrow I\cr}]
[I \rightarrow P] (Fig. 1.5.1.3[link]) [\openup2pt\pmatrix{\bar{{1 \over 2}} &{1 \over 2} &{1 \over 2}\cr {1 \over 2} &\bar{{1 \over 2}} &{1 \over 2}\cr {1 \over 2} &{1 \over 2} &\bar{{1 \over 2}}\cr}] [\pmatrix{0 &1 &1\cr 1 &0 &1\cr 1 &1 &0\cr}] Orthorhombic
Tetragonal
Cubic
[F \rightarrow P] (Fig. 1.5.1.4[link]) [\openup2pt\pmatrix{0 &{1 \over 2} &{1 \over 2}\cr {1 \over 2} &0 &{1 \over 2}\cr {1 \over 2} &{1 \over 2} &0\cr}] [\pmatrix{\bar{1} &1 &1\cr 1 &\bar{1} &1\cr 1 &1 &\bar{1}\cr}] Orthorhombic
Tetragonal
Cubic
[({\bf b}, {\bf a}, \bar{{\bf c}}) \rightarrow ({\bf a}, {\bf b}, {\bf c})] [\pmatrix{0 &1 &0\cr 1 &0 &0\cr 0 &0 &\bar{1}\cr}] [\pmatrix{0 &1 &0\cr 1 &0 &0\cr 0 &0 &\bar{1}\cr}] Unconventional orthorhombic setting
[({\bf c}, {\bf a}, {\bf b}) \rightarrow ({\bf a}, {\bf b}, {\bf c})] [\pmatrix{0 &0 &1\cr 1 &0 &0\cr 0 &1 &0\cr}] [\pmatrix{0 &1 &0\cr 0 &0 &1\cr 1 &0 &0\cr}] Unconventional orthorhombic setting
[(\bar{{\bf c}}, {\bf b}, {\bf a}) \rightarrow ({\bf a}, {\bf b}, {\bf c})] [\pmatrix{0 &0 &\bar{1}\cr 0 &1 &0\cr 1 &0 &0\cr}] [\pmatrix{0 &0 &1\cr 0 &1 &0\cr \bar{1} &0 &0\cr}] Unconventional orthorhombic setting
[({\bf b}, {\bf c}, {\bf a}) \rightarrow ({\bf a}, {\bf b}, {\bf c})] [\pmatrix{0 &1 &0\cr 0 &0 &1\cr 1 &0 &0\cr}] [\pmatrix{0 &0 &1\cr 1 &0 &0\cr 0 &1 &0\cr}] Unconventional orthorhombic setting
[({\bf a}, \bar{{\bf c}}, {\bf b}) \rightarrow ({\bf a}, {\bf b}, {\bf c})] [\pmatrix{1 &0 &0\cr 0 &0 &\bar{1}\cr 0 &1 &0\cr}] [\pmatrix{1 &0 &0\cr 0 &0 &1\cr 0 &\bar{1} &0\cr}] Unconventional orthorhombic setting
[\left.\matrix{P\rightarrow C_{1}\cr I\rightarrow F_{1}\cr}\right\}] (Fig. 1.5.1.5[link]), c axis invariant [\pmatrix{1 &1 &0\cr \bar{1} &1 &0\cr 0 &0 &1\cr}] [\openup2pt\pmatrix{{1 \over 2} &\bar{{1 \over 2}} &0\cr {1 \over 2} &{1 \over 2} &0\cr 0 &0 &1\cr}] Tetragonal (cf. Section 1.5.4.3[link])
[\left.\matrix{P\rightarrow C_{2}\cr I\rightarrow F_{2}\cr}\right\}] (Fig. 1.5.1.5[link]), c axis invariant [\pmatrix{1 &\bar{1} &0\cr 1 &1 &0\cr 0 &0 &1\cr}] [\openup2pt\pmatrix{{1 \over 2} &{1 \over 2} &0\cr \bar{1 \over 2} &{1 \over 2} &0\cr 0 &0 &1\cr}] Tetragonal (cf. Section 1.5.4.3[link])
Primitive rhombohedral cell [\rightarrow] triple hexagonal cell [R_{1}], obverse setting (Fig. 1.5.1.6[link]a,c) [\pmatrix{1 &0 &1\cr \bar{1} &1 &1\cr 0 &\bar{1} &1\cr}] [\openup2pt\pmatrix{{2 \over 3} &\bar{{1 \over 3}} &\bar{{1 \over 3}}\cr {1 \over 3} &{1 \over 3} &\bar{{2} \over 3}\cr {1 \over 3} &{1 \over 3} &{1 \over 3}\cr}] Rhombohedral space groups (cf. Section 1.5.4.3[link])
Primitive rhombohedral cell [\rightarrow] triple hexagonal cell [R_{2}], obverse setting (Fig. 1.5.1.6[link]c) [\pmatrix{0 &\bar{1} &1\cr 1 &0 &1\cr \bar{1} &1 &1\cr}] [\openup2pt\pmatrix{\bar{{1 \over 3}} &{2 \over 3} &\bar{{1 \over 3}}\cr \bar{{2} \over 3} &{1 \over 3} &{1 \over 3}\cr {1 \over 3} &{1 \over 3} &{1 \over 3}\cr}] Rhombohedral space groups (cf. Section 1.5.4.3[link])
Primitive rhombohedral cell [\rightarrow] triple hexagonal cell [R_{3}], obverse setting (Fig. 1.5.1.6[link]c) [\pmatrix{\bar{1} &1 &1\cr 0 &\bar{1} &1\cr 1 &0 &1\cr}] [\openup2pt\pmatrix{\bar{{1 \over 3}} &\bar{{1 \over 3}} &{2 \over 3}\cr {1 \over 3} &\bar{{2} \over 3} &{1 \over 3}\cr {1 \over 3} &{1 \over 3} &{1 \over 3}\cr}] Rhombohedral space groups (cf. Section 1.5.4.3[link])
Primitive rhombohedral cell [\rightarrow] triple hexagonal cell [R_{1}], reverse setting (Fig. 1.5.1.6[link]d) [\pmatrix{\bar{1} &0 &1\cr 1 &\bar{1} &1\cr 0 &1 &1\cr}] [\openup2pt\pmatrix{\bar{{2 \over 3}} &{1 \over 3} &{1 \over 3}\cr \bar{{1} \over 3} &\bar{{1} \over 3} &{2 \over 3}\cr {1 \over 3} &{1 \over 3} &{1 \over 3}\cr}] Rhombohedral space groups (cf. Section 1.5.4.3[link])
Primitive rhombohedral cell [\rightarrow] triple hexagonal cell [R_{2}], reverse setting (Fig. 1.5.1.6[link]b,d) [\pmatrix{0 &1 &1\cr \bar{1} &0 &1\cr 1 &\bar{1} &1\cr}] [\openup2pt\pmatrix{{1 \over 3} &\bar{{2 \over 3}} &{1 \over 3}\cr {2 \over 3} &\bar{{1} \over 3} &\bar{{1} \over 3}\cr {1 \over 3} &{1 \over 3} &{1 \over 3}\cr}] Rhombohedral space groups (cf. Section 1.5.4.3[link])
Primitive rhombohedral cell [\rightarrow] triple hexagonal cell [R_{3}], reverse setting (Fig. 1.5.1.6[link]d) [\pmatrix{1 &\bar{1} &1\cr 0 &1 &1\cr \bar{1} &0 &1\cr}] [\openup2pt\pmatrix{{1 \over 3} &{1 \over 3} &\bar{{2 \over 3}}\cr \bar{{1} \over 3} &{2 \over 3} &\bar{{1} \over 3}\cr {1 \over 3} &{1 \over 3} &{1 \over 3}\cr}] Rhombohedral space groups (cf. Section 1.5.4.3[link])
Hexagonal cell [P \rightarrow] orthohexagonal centred cell [C_{1}] (Fig. 1.5.1.7[link]) [\pmatrix{1 &1 &0\cr 0 &2 &0\cr 0 &0 &1\cr}] [\openup2pt\pmatrix{1 &\bar{{1 \over 2}} &0\cr 0 &{1 \over 2} &0\cr 0 &0 &1\cr}] Trigonal
Hexagonal (cf. Section 1.5.4.3[link])
Hexagonal cell [P \rightarrow] orthohexagonal centred cell [C_{2}] (Fig. 1.5.1.7[link]) [\pmatrix{1 &\bar{1} &0\cr 1 &1 &0\cr 0 &0 &1\cr}] [\openup2pt\pmatrix{{1 \over 2} &{1 \over 2} &0\cr \bar{{1} \over 2} &{1 \over 2} &0\cr 0 &0 &1\cr}] Trigonal
Hexagonal (cf. Section 1.5.4.3[link])
Hexagonal cell [P \rightarrow] orthohexagonal centred cell [C_{3}] (Fig. 1.5.1.7[link]) [\pmatrix{0 &\bar{2} &0\cr 1 &\bar{1} &0\cr 0 &0 &1\cr}] [\openup2pt\pmatrix{\bar{{1 \over 2}} &1 &0\cr \bar{1 \over 2} &0 &0\cr 0 &0 &1\cr}] Trigonal
Hexagonal (cf. Section 1.5.4.3[link])
Hexagonal cell [P \rightarrow] triple hexagonal cell [H_{1}] (Fig. 1.5.1.8[link]) [\pmatrix{1 &1 &0\cr \bar{1} &2 &0\cr 0 &0 &1\cr}] [\openup2pt\pmatrix{{2 \over 3} &\bar{{1 \over 3}} &0\cr {1 \over 3} &{1 \over 3} &0\cr 0 &0 &1\cr}] Trigonal
Hexagonal (cf. Section 1.5.4.3[link])
Hexagonal cell [P \rightarrow] triple hexagonal cell [H_{2}] (Fig. 1.5.1.8[link]) [\pmatrix{2 &\bar{1} &0\cr 1 &1 &0\cr 0 &0 &1\cr}] [\openup2pt\pmatrix{{1 \over 3} &{1 \over 3} &0\cr \bar{{1} \over 3} &{2 \over 3} &0\cr 0 &0 &1\cr}] Trigonal
Hexagonal (cf. Section 1.5.4.3[link])
Hexagonal cell [P \rightarrow] triple hexagonal cell [H_{3}] (Fig. 1.5.1.8[link]) [\pmatrix{1 &\bar{2} &0\cr 2 &\bar{1} &0\cr 0 &0 &1\cr}] [\openup2pt\pmatrix{\bar{{1 \over 3}} &{2 \over 3} &0\cr \bar{{2 \over 3}} &{1 \over 3} &0\cr 0 &0 &1\cr}] Trigonal
Hexagonal (cf. Section 1.5.4.3[link])
Hexagonal cell [P \rightarrow] triple rhombohedral cell [D_{1}] [\pmatrix{1 &0 &\bar{1}\cr 0 &1 &\bar{1}\cr 1 &1 &1\cr}] [\openup2pt\pmatrix{{2 \over 3} &\bar{{1 \over 3}} &{1 \over 3}\cr \bar{1 \over 3} &{2 \over 3} &{1 \over 3}\cr \bar{1 \over 3} &\bar{1 \over 3} &{1 \over 3}\cr}] Trigonal
Hexagonal (cf. Section 1.5.4.3[link])
Hexagonal cell [P \rightarrow] triple rhombohedral cell [D_{2}] [\pmatrix{\bar{1} &0 &1\cr 0 &\bar{1} &1\cr 1 &1 &1\cr}] [\openup2pt\pmatrix{\bar{{2 \over 3}} &{1 \over 3} &{1 \over 3}\cr {1 \over 3} &\bar{2 \over 3} &{1 \over 3}\cr {1 \over 3} &{1 \over 3} &{1 \over 3}\cr}] Trigonal
Hexagonal (cf. Section 1.5.4.3[link])
Triple hexagonal cell R, obverse setting [\rightarrow] C-centred monoclinic cell, unique axis b, cell choice 1
(Fig. 1.5.1.9[link]a)
[\openup3pt\pmatrix{{2 \over 3} &0 &0\cr {1 \over 3} &1 &0\cr \bar{{2 \over 3}} &0 &1\cr}] [\openup3pt\pmatrix{{3 \over 2} &0 &0\cr \bar{{1 \over 2}} &1 &0\cr 1 &0 &1\cr}] Rhombohedral space groups (cf. Section 1.5.4.3[link])
Triple hexagonal cell R, obverse setting [\rightarrow] C-centred monoclinic cell, unique axis b, cell choice 2
(Fig. 1.5.1.9[link]a)
[\openup2pt\pmatrix{\bar{{1 \over 3}} &\bar{1} &0\cr {1 \over 3} &\bar{1} &0\cr \bar{{2 \over 3}} &0 &1\cr}] [\openup2pt\pmatrix{\bar{{3 \over 2}} &{3 \over 2} &0\cr \bar{{1 \over 2}} &\bar{{1 \over 2}} &0\cr \bar{1} &1 &1\cr}] Rhombohedral space groups (cf. Section 1.5.4.3[link])
Triple hexagonal cell R, obverse setting [\rightarrow] C-centred monoclinic cell, unique axis b, cell choice 3
(Fig. 1.5.1.9[link]a)
[\openup3pt\pmatrix{\bar{{1 \over 3}} &1 &0\cr \bar{{2 \over 3}} &0 &0\cr \bar{{2 \over 3}} &0 &1\cr}] [\openup3pt\pmatrix{0 &\bar{{3 \over 2}} &0\cr 1 &\bar{{1 \over 2}} &0\cr 0 &\bar{1} &1\cr}] Rhombohedral space groups (cf. Section 1.5.4.3[link])
Triple hexagonal cell R, obverse setting [\rightarrow] A-centred monoclinic cell, unique axis c, cell choice 1
(Fig. 1.5.1.9[link]b)
[\openup2pt\pmatrix{0 &{2 \over 3} &0\cr 0 &{1 \over 3} &1\cr 1 &\bar{{2 \over 3}} &0\cr}] [\openup3pt\pmatrix{1 &0 &1\cr {3 \over 2} &0 &0\cr \bar{{1 \over 2}} &1 &0\cr}] Rhombohedral space groups (cf. Section 1.5.4.3[link])
Triple hexagonal cell R, obverse setting [\rightarrow] A-centred monoclinic cell, unique axis c, cell choice 2
(Fig. 1.5.1.9[link]b)
[\openup3pt\pmatrix{0 &\bar{{1 \over 3}} &\bar{1}\cr 0 &{1 \over 3} &\bar{1}\cr 1 &\bar{{2 \over 3}} &0\cr}] [\openup3pt\pmatrix{\bar{1} &1 &1\cr \bar{{3 \over 2}} &{3 \over 2} &0\cr \bar{{1 \over 2}} &\bar{{1 \over 2}} &0\cr}] Rhombohedral space groups (cf. Section 1.5.4.3[link])
Triple hexagonal cell R, obverse setting [\rightarrow] A-centred monoclinic cell, unique axis c, cell choice 3
(Fig. 1.5.1.9[link]b)
[\openup3pt\pmatrix{0 &\bar{{1 \over 3}} &1\cr 0 &\bar{{2 \over 3}} &0\cr 1 &\bar{{2 \over 3}} &0\cr}] [\openup3pt\pmatrix{0 &\bar{1} &1\cr 0 &\bar{{3 \over 2}} &0\cr 1 &\bar{{1 \over 2}} &0\cr}] Rhombohedral space groups (cf. Section 1.5.4.3[link])
Primitive rhombohedral cell [\rightarrow C]-centred monoclinic cell, unique axis b, cell choice 1
(Fig. 1.5.1.10[link]a)
[\pmatrix{0 &0 &1\cr \bar{1} &1 &1\cr \bar{1} &\bar{1} &1\cr}] [\openup2pt\pmatrix{1 &\bar{{1 \over 2}} &\bar{{1 \over 2}}\cr 0 &{1 \over 2} &\bar{{1 \over 2}}\cr 1 &0 &0\cr}] Rhombohedral space groups (cf. Section 1.5.4.3[link])
Primitive rhombohedral cell [\rightarrow C]-centred monoclinic cell, unique axis b, cell choice 2
(Fig. 1.5.1.10[link]a)
[\pmatrix{\bar{1} &\bar{1} &1\cr 0 &0 &1\cr \bar{1} &1 &1\cr}] [\openup3pt\pmatrix{\bar{{1 \over 2}} &1 &\bar{{1 \over 2}}\cr \bar{{1 \over 2}} &0 &{1 \over 2}\cr 0 &1 &0\cr}] Rhombohedral space groups (cf. Section 1.5.4.3[link])
Primitive rhombohedral cell [\rightarrow C]-centred monoclinic cell, unique axis b, cell choice 3
(Fig. 1.5.1.10[link]a)
[\pmatrix{\bar{1} &1 &1\cr \bar{1} &\bar{1} &1\cr 0 &0 &1\cr}] [\pmatrix{\bar{{1 \over 2}} &\bar{{1 \over 2}} &1\cr {1 \over 2} &\bar{{1 \over 2}} &0\cr 0 &0 &1\cr}] Rhombohedral space groups (cf. Section 1.5.4.3[link])
Primitive rhombohedral cell [\rightarrow A]-centred monoclinic cell, unique axis c, cell choice 1
(Fig.1.5.1.10b)
[\pmatrix{1 &0 &0\cr 1 &\bar{1} &1\cr 1 &\bar{1} &\bar{1}\cr}] [\openup3pt\pmatrix{1 &0 &0\cr 1 &\bar{{1 \over 2}} &\bar{{1 \over 2}}\cr 0 &{1 \over 2} &\bar{{1 \over 2}}\cr}] Rhombohedral space groups (cf. Section 1.5.4.3[link])
Primitive rhombohedral cell [\rightarrow A]-centred monoclinic cell, unique axis c, cell choice 2
(Fig. 1.5.1.10[link]b)
[\pmatrix{1 &\bar{1} &\bar{1}\cr 1 &0 &0\cr 1 &\bar{1} &1\cr}] [\openup3pt\pmatrix{0 &1 &0\cr \bar{{1 \over 2}} &1 &\bar{{1 \over 2}}\cr \bar{{1 \over 2}} &0 &{1 \over 2}\cr}] Rhombohedral space groups (cf. Section 1.5.4.3[link])
Primitive rhombohedral cell [\rightarrow A]-centred monoclinic cell, unique axis c, cell choice 3
(Fig. 1.5.1.10[link]b)
[\pmatrix{1 &\bar{1} &1\cr 1 &\bar{1} &\bar{1}\cr 1 &0 &0\cr}] [\openup3pt\pmatrix{0 &0 &1\cr \bar{{1 \over 2}} &\bar{{1 \over 2}} &1\cr {1 \over 2} &\bar{{1 \over 2}} &0\cr}] Rhombohedral space groups (cf. Section 1.5.4.3[link])