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

© International Union of Crystallography 2006
International Tables for Crystallography (2006). Vol. D. ch. 1.2, pp. 56-62

Section 1.2.6. Tables

T. Janssena*

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

1.2.6. Tables

| top | pdf |

In the following, a short description of the tables is given in order to facilitate consultation without reading the introductory theoretical Sections 1.2.2[link] to 1.2.5[link][link][link].

Table 1.2.6.1[link] . Finite point groups in three dimensions. The point groups are grouped by isomorphism class. There are four infinite families and six other isomorphism classes. (Notation: [C_{n}] for the cyclic group of order n, [D_{n}] for the dihedral group of order 2n, T, O and I the tetrahedral, octahedral and icosahedral groups, respectively). Point groups of the first class are subgroups of SO(3), those of the second class contain −E, and those of the third class are not subgroups of SO(3), but do not contain −E either. The families [C_n] and [D_n] are also isomorphism classes of two-dimensional finite point groups.

Table 1.2.6.1 | top | pdf |
Finite point groups in three dimensions

Isomorphism class First class with determinants [\gt\,0] Second class with −E Third class without −E Order
[ C_{n}] n   [\bar{n}] (n even, [\,\gt\,]2) [n]
      m ([n=2])  
[D_{n}] n 22 (n even)   [nmm] (n even) [2n]
  n 2 (n odd, [\gt\,1])   [\bar{n}2m] (n even)  
      [nm] (n odd)  
[C_{n}\times C_{2}]   [\bar{n}] (n odd)   [2n]
    [n/m] (n even)    
[D_{n}\times C_{2}]   [n/mmm] (n even, [\geq 4])   [4n]
    [mmm] ([n=2])    
    [\bar{n}m] (n odd, [\gt\,0])    
T 23     [12]
O 432   [\bar{4}3m] [24]
I 532     [60]
[T\times C_{2}]   [m\bar{3}]   [24]
[O\times C_{2}]   [m\bar{3}m]   [48]
[I\times C_{2}]   [\bar{5}\bar{3}m]   [120]

Table 1.2.6.2[link] . Among the infinite number of finite three-dimensional point groups, 32 are crystallographic.

Table 1.2.6.2 | top | pdf |
Crystallographic point groups in three dimensions

Isomorphism class First class Second class with −E Third class without −E Order
[ C_{1}] 1     1
[C_{2}] 2 [\bar{1}] m 2
[C_{3}] 3     3
[C_{4}] 4   4 4
[D_{2}] 222 [2/m] [2mm] 4
[C_{6}] 6 [\bar{3}] [\bar{6}] 6
[D_{3}] 32   [3m] 6
[C_{4}\times C_{2}]   [4/m]   8
[D_{4}] 422   [4mm], [\bar{4}2m] 8
[D_{2}\times C_{2}]   [mmm]   8
[D_{6}] 622 [\bar{3}m] [6mm], [\bar{6}2m] 12
T 23     12
[C_{6}\times C_{2}]   [6/m]   12
[D_{4}\times C_{2}]   [4/mmm]   16
O 432   [\bar{4}3m] 24
[D_{6}\times C_{2}]   [6/mmm ]   24
[T\times C_{2}]   [m\bar{3}]   24
[O\times C_{2}]   [m\bar{3}m]   48

Table 1.2.6.3[link] . Character table for the cyclic groups [C_{n}]. The generator is denoted by [\alpha]. The number of elements in the conjugacy classes ([n_{i}]) is one for each class. The order is the smallest nonnegative power p for which [A^{p}=E]. The n irreducible representations are denoted by [\Gamma_{i}].

Table 1.2.6.3 | top | pdf |
Irreducible representations for cyclic groups [C_{n}]

[\omega =\exp (2\pi i/n)], s.c.m = smallest common multiple.

  [\varepsilon] [\alpha] [\alpha^{2}] [\alpha^{3}] [\ldots] [\alpha^{n-1}]
[n_{i}] 1 1 1 1 [\ldots] 1
Order 1 n s.c.m.(n, 2) s.c.m.(n, 3) [\ldots] n
[ \Gamma_{1}] 1 1 1 1 [\ldots] 1
[\Gamma_{2}] 1 [\omega] [\omega^{2}] [\omega^{3}] [\ldots] [\omega^{-1}]
[\vdots] 1 [\vdots] [\vdots] [\vdots] [\ddots] [\vdots]
[\Gamma_{n}] 1 [\omega^{-1}] [\omega^{-2}] [\omega^{-3}] [\ldots] [\omega]

Table 1.2.6.4[link] . Character tables for the dihedral groups [D_{n}] of order [2n]. [n_{i}] is the number of elements in the conjugacy class [C_{i}]. The irreducible representations are denoted by [\Gamma_{i}].

Table 1.2.6.4 | top | pdf |
Irreducible representations for dihedral groups [D_{n}]

(a) n odd. [m=1,\ldots, (n-1)/2;\,j=1,\ldots, (n-1)/2], s.c.m = smallest common multiple.

  [\varepsilon] [\alpha^{j}] [\ldots] [\beta]
[n_i] 1 1 2 n
Order 1 s.c.m.([n,j]) [\ldots] 2
[ \Gamma_{1}] 1 1 [\ldots] 1
[\Gamma_{2}] 1 1 [\ldots] −1
[\Gamma_{2+m}] 2 [2\cos(2\pi mj/n) ] [\ldots] 0

(b) n even. [m=1,\ldots, (n/2- 1);\,j=1,\ldots, (n/2- 1)], s.c.m = smallest common multiple.

  [\varepsilon] [\alpha^{n/2}] [\alpha^{j}] [\ldots] [\beta] [\alpha \beta]
[n_{i}] 1 1 2 [\ldots] [n/2] [n/2]
Order 1 2 s.c.m.[(n,j)] [\ldots] 2 2
[ \Gamma_{1}] 1 1 1 [\ldots] 1 1
[\Gamma_{2}] 1 1 1 [\ldots] −1 −1
[\Gamma_{3}] 1 [(-1)^{n/2}] [(-1)^{j}] [\ldots] 1 −1
[\Gamma_{4}] 1 [(-1)^{n/2}] [(-1)^{j}] [\ldots] −1 1
[\Gamma_{4+m}] 2 [(-1)^m]2 2cos([2 \pi mj/n]) [\ldots] 0 0

Table 1.2.6.5[link] . The character tables for the 32 three-dimensional crystallographic point groups. The groups are grouped by isomorphism class (there are 18 isomorphism classes).

Table 1.2.6.5 | top | pdf |
Irreducible representations and character tables for the 32 crystallographic point groups in three dimensions

(a) [C_{1}]

[C_{1}] [\varepsilon]
n 1
Order 1
[ \Gamma_{1}] 1

[1] [\Gamma_{1}: A = \chi_1] [x,y,z] [x^2,y^2,z^2,yz,xz,xy]
[C_1]      

(b) [C_{2}]

[C_{2}] [\varepsilon] [\alpha]
n 1 1
Order 1 2
[ \Gamma_{1}] 1 1
[\Gamma_{2}] 1 −1

[{2}] [\alpha =C_{2z}] [\Gamma_{1}: A =\chi_1] z [x^{2},y^{2},z^{2},xy]
[C_2]   [\Gamma_{2}: B =\chi_3] [x,y] [yz,xz]
         
[{m}] [\alpha =\sigma_{z}] [\Gamma_{1}: A' =\chi_1] [x,y] [x^{2},y^{2},z^{2},xy]
[C_s]   [\Gamma_{2}: A'' =\chi_3] z [yz,xz]
         
[\bar{1}] [\alpha =I] [\Gamma_{1}: A_{g} =\chi_1^+]   [x^2,y^2,z^2,yz,xz,xy]
[C_i]   [\Gamma_{2}: A_{u} =\chi_{1}^{-}] [x,y,z]  

(c) [C_{3}] [[\omega = \exp(2\pi i/3)]].

[C_{3}] [\varepsilon] [\alpha] [\alpha^{2}]
n 1 1 1
Order 1 3 3
[\Gamma_{1}] 1 1 1
[\Gamma_{2}] 1 [\omega] [\omega^{2}]
[\Gamma_{3}] 1 [\omega^{2}] [\omega]

Matrices of the real two-dimensional representation:

  [\varepsilon] [\alpha] [\alpha ^2]
[ \Gamma_2\oplus \Gamma_3] [\pmatrix{ 1&0\cr 0&1 }] [\pmatrix{ 0&-1\cr 1& -1 }] [\pmatrix{ -1&1 \cr -1&0 }]

[3] [\alpha =C_{3z}] [\Gamma_{1}: A =\chi_1] z [x^{2}+y^{2},z^{2}]
[C_3]   [\Gamma_{2}\oplus\Gamma_{3}: E=\chi_{1c}+\chi_{1c}^{*}] [x,y] [x^{2}-y^{2},xz,yz,xy]

(d) [C_{4}]

[C_{4}] [\varepsilon] [\alpha] [\alpha^{2}] [\alpha^{3}]
n 1 [1] [1] [1]
Order 1 [4] [2] [4]
[ \Gamma_{1}] 1 [1] [1] [1]
[\Gamma_{2}] 1 [i] [-1] [-i]
[\Gamma_{3}] 1 [-1] [1] [-1]
[\Gamma_{4}] 1 [-i] [-1] [i]

Matrices of the real two-dimensional representation:

  [\varepsilon] [\alpha] [\alpha ^2] [\alpha^3]
[ \Gamma_2\oplus\Gamma_4] [\pmatrix{ 1&0 \cr 0&1 }] [\pmatrix{ 0&-1 \cr 1&0 }] [\pmatrix{-1&0 \cr 0&-1 }] [\pmatrix{0&1 \cr -1&0 }]

[4] [\alpha =C_{4z}] [\Gamma_{1}: A =\chi_1] z [x^{2}+y^{2},z^{2}]
[C_4]   [\Gamma_{3}: B =\chi_3]   [x^{2}-y^{2},xy]
    [\Gamma_{2}\oplus\Gamma_{4}: E =\chi_{1c}+\chi_{1c}^{*}] [x,y] [yz,xz]
         
[\bar{4}] [\alpha =S_{4}] [\Gamma_{1}: A =\chi_1]   [x^{2}+y^{2},z^{2}]
[S_4]   [\Gamma_{3}: B =\chi_3] z [x^{2}-y^{2},xy]
    [\Gamma_{2}\oplus\Gamma_{4}: E=\chi_{1c}+\chi_{1c}^{*}] [x,y] [yz,xz]

(e) [C_{6}] [[\omega=\exp(\pi i/3)]].

[C_{6}] [\varepsilon] [\alpha] [\alpha^{2}] [\alpha^{3}] [\alpha^{4}] [\alpha^{5}]
n 1 [1] [1] [1] [1] [1]
Order 1 [6] [3] [2] [3] [6]
[ \Gamma_{1}] 1 [1] [1] [1] [1] [1]
[\Gamma_{2}] 1 [\omega] [\omega^{2}] [-1] [-\omega] [-\omega^{2}]
[\Gamma_{3}] 1 [\omega^{2}] [-\omega] [1] [\omega^{2}] [-\omega]
[\Gamma_{4}] 1 [-1] [1] [-1] [1] [-1]
[\Gamma_{5}] 1 [-\omega] [\omega^{2}] [1] [-\omega] [\omega^{2}]
[\Gamma_{6}] 1 [-\omega^{2}] [-\omega] [-1] [\omega^{2}] [\omega]

Matrices of the real representations:

  [\Gamma_2\oplus\Gamma_6] [\Gamma_3\oplus\Gamma_5]
[\varepsilon] [\pmatrix{ 1&0 \cr 0&1 }] [\pmatrix{ 1&0 \cr 0&1 }]
[\alpha] [\pmatrix{ 1&-1 \cr 1&0 }] [\pmatrix{ 0&-1 \cr 1&-1 }]
[\alpha ^2] [\pmatrix{ 0&-1 \cr 1&-1 } ] [\pmatrix{ -1&1 \cr -1 & 0 }]
[\alpha^3] [\pmatrix{ -1&0 \cr 0&-1 }] [\pmatrix{ 1&0 \cr 0&1 }]
[\alpha^4] [\pmatrix{ -1&1 \cr -1&0 }] [\pmatrix{ 0&-1 \cr 1&-1 }]
[\alpha^5] [\pmatrix{ 0&1 \cr -1&1 } ] [\pmatrix{ -1&1 \cr -1 & 0 }]

[6] [\alpha =C_{6z}] [\Gamma_{1}: A =\chi_1] z [x^{2}+y^{2},z^{2}]
[C_6]   [\Gamma_{4}: B =\chi_3]    
    [\Gamma_{2}\oplus\Gamma_{6}]: [E_{1} = \chi_{1c}+\chi_{1c}^{*}] [x,y] [xz,yz]
    [\Gamma_{3}\oplus\Gamma_{5}]: [E_{2}= \chi_{2c}+\chi_{2c}^{*}]   [x^{2}-y^{2},xy]
         
[\bar{3}] [\alpha =S_{3z}] [\Gamma_{1}: A_{g} =\chi_1^+]   [x^{2}+y^{2},z^{2}]
[S_6]   [\Gamma_{4}: A_{u} =\chi_1^-] z  
    [\Gamma_{2}\oplus\Gamma_{6}]: [E_{u} = \chi_{1c}^-+\chi_{1c}^{-*}] [x,y]  
    [\Gamma_{3}\oplus\Gamma_{5}]: [E_{g}= \chi_{1c}^++\chi_{1c}^{+*}] [x^{2}-y^{2},xy, xz,yz]  
         
[\bar{6}] [\alpha =S_{6z}] [\Gamma_{1}: A' =\chi_1]   [x^{2}+y^{2},z^{2}]
[C_{3h}]   [\Gamma_{4}: A'' =\chi_3] z  
    [\Gamma_{2}\oplus\Gamma_{6}]: [E'= \chi_{2c}+\chi_{2c}^{*}]   [xz,yz]
    [\Gamma_{3}\oplus\Gamma_{5}]: [E''= \chi_{1c}+\chi_{1c}^{*}] [x,y] [x^{2}-y^{2},xy]

(f) [D_{2}]

[D_{2}] [\varepsilon] [\alpha] [\beta] [\alpha \beta]
n 1 [1] [1] [1]
Order 1 [2] [2] [2]
[ \Gamma_{1}] 1 [1] [1] [1]
[\Gamma_{2}] 1 [1] [-1] [-1]
[\Gamma_{3}] 1 [-1] [1] [-1]
[\Gamma_{4}] 1 [-1] [-1] [1]

[222] [\alpha =C_{2x}] [\Gamma_{1}: A_1 =\chi_1]   [x^{2},y^{2},z^{2}]
[D_2] [\beta =C_{2y}] [\Gamma_{2}: B_{3} =\chi_3] x [yz]
  [\alpha\beta =C_{2z}] [\Gamma_{3}: B_{2} =\chi_4] y [xz]
    [\Gamma_{4}: B_{1} =\chi_2] z [xz]
         
[mm2] [\alpha =C_{2z}] [\Gamma_{1}: A_{1} =\chi_1] z [x^{2},y^{2},z^{2}]
[C_{2v}] [\beta =\sigma_{x}] [\Gamma_{2}: A_{2} =\chi_2]   [xy]
  [\alpha\beta =\sigma_{y}] [\Gamma_{3}: B_{2} =\chi_3] y [yz]
    [\Gamma_{4}: B_{1} =\chi_4] x [xz]
         
[2/m] [\alpha =C_{2z}] [\Gamma_{1}: A_{g} =\chi_1^+]   [x^{2},y^{2},z^{2},xy]
[C_{2h}] [\beta =\sigma_{z}] [\Gamma_{2}: A_{u} =\chi_1^-] z z
  [\alpha\beta =I] [\Gamma_{3}: B_{u} =\chi_3^-] [x,y]  
    [\Gamma_{4}: B_{g} =\chi_3^+]    

(g) [D_{3}]

[D_{3}] [\varepsilon] [\alpha] [\beta]
n [1] [2] [3]
Order [1] [3] [2]
[ \Gamma_{1}] [1] [1] [1]
[\Gamma_{2}] [1] [1] [-1]
[\Gamma_{3}] [2] [-1] [0]

Matrices of the two-dimensional representation:

  [\varepsilon] [\alpha] [\beta]
[ \Gamma_3] [\pmatrix{1&0 \cr 0&1 }] [\pmatrix{0&-1 \cr 1&-1 }] [\pmatrix{-1&1 \cr 0&1 }]

[32] [\alpha =C_{3z}] [\Gamma_{1}: A_{1} =\chi_1]   [x^{2}+y^{2},z^{2}]
[D_3] [\beta =C_{2x}] [\Gamma_{2}: A_{2} =\chi_2] z  
    [\Gamma_{3}: E ={\chi}_1] [x,y] [xz,yz,xy,x^{2}-y^{2}]
         
[3m] [\alpha =C_{3z}] [\Gamma_{1}: A_{1} =\chi_1] z [x^{2}+y^{2},z^{2}]
[C_{3v}] [\beta =\sigma_{v}] [\Gamma_{2}: A_{2} =\chi_2]    
    [\Gamma_{3}: E ={\chi}_1] [x,y] [xz,yz,xy,x^{2}-y^{2}]

(h) [D_{4}]

[D_{4}] [\varepsilon] [\alpha] [\alpha^{2}] [\beta] [\alpha \beta]
n 1 [2] [1] [2] [2]
Order 1 [4] [2] [2] [2]
[ \Gamma_{1}] 1 [1] [1] [1] [1]
[\Gamma_{2}] 1 [1] [1] [-1] [-1]
[\Gamma_{3}] 1 [-1] [1] [1] [-1]
[\Gamma_{4}] 1 [-1] [1] [-1] [1]
[\Gamma_{5}] 2 [0] [-2] [0] [0]

Matrices of the two-dimensional representation:

  [\Gamma_5]
[\varepsilon] [\pmatrix{1&0 \cr 0&1 }]
[\alpha] [\pmatrix{0&-1 \cr 1&0 }]
[\alpha^2] [\pmatrix{-1&0 \cr 0&-1 }]
[\beta ] [\pmatrix{ -1&0 \cr 0&1 }]
[\alpha\beta] [\pmatrix{0&-1 \cr -1&0 }]

422 [\alpha =C_{4z}] [\Gamma_{1}: A_{1} =\chi_1]   [x^{2}+y^{2},z^{2}]
[D_4] [\beta =C_{2x}] [\Gamma_{2}: A_{2} =\chi_2] z  
    [\Gamma_{3}: B_{1} =\chi_3]   [x^{2}-y^{2}]
    [\Gamma_{4}: B_{2} =\chi_4]   [xy]
    [\Gamma_{5}: E ={\chi}_1] [x,y] [xz,yz]
         
[4mm] [\alpha =C_{4z}] [\Gamma_{1}: A_{1} =\chi_1] z [x^{2}+y^{2},z^{2}]
[C_{4v}] [\beta =\sigma_{v}] [\Gamma_{2}: A_{2} =\chi_2]    
    [\Gamma_{3}: B_{1} =\chi_3]   [x^{2}-y^{2}]
    [\Gamma_{4}: B_{2} =\chi_4]   [xy]
    [\Gamma_{5}: E ={\chi}_1] [x,y] [xz,yz]
         
[\bar{4}2m] [\alpha =S_{4z}] [\Gamma_{1}: A_{1} =\chi_1]   [x^{2}+y^{2},z^{2}]
[D_{2d}] [\beta =C_{2v}] [\Gamma_{2}: A_{2} =\chi_2]    
  [\alpha\beta =\sigma_{d}] [\Gamma_{3}: B_{1} =\chi_3]   [x^{2}-y^{2}]
    [\Gamma_{4}: B_{2} =\chi_4] z [xy]
    [\Gamma_{5}: E ={\chi}_1] [x,y] [xz,yz]

(i) [D_{6}]

[D_{6}] [\varepsilon] [\alpha] [\alpha^{2}] [\alpha^{3}] [\beta] [\alpha \beta]
n 1 [2] [2] [1] [3] [3]
Order 1 [6] [3] [2] [2] [2]
[ \Gamma_{1}] 1 [1] [1] [1] [1] [1]
[\Gamma_{2}] 1 [1] [1] [1] [-1] [-1]
[\Gamma_{3}] 1 [-1] [1] [-1] [1] [-1]
[\Gamma_{4}] 1 [-1] [1] [-1] [-1] [1]
[\Gamma_{5}] 2 [1] [-1] [-2] [0] [0]
[\Gamma_{6}] 2 [-1] [-1] [2] [0] [0]

Matrices of the two-dimensional representations:

  [\Gamma_5] [\Gamma_6]
[\varepsilon] [\pmatrix{1&0 \cr 0&1 }] [\pmatrix{1&0 \cr 0&1 }]
[\alpha] [\pmatrix{1&-1 \cr 1&0 }] [\pmatrix{0&-1 \cr 1&-1}]
[\alpha^2] [\pmatrix{0&-1 \cr 1&-1 }] [\pmatrix{-1&1 \cr -1&0 }]
[\alpha^3] [\pmatrix{-1&0 \cr 0&-1 }] [\pmatrix{ 1&0 \cr 0&1 }]
[\beta] [\pmatrix{-1&1 \cr 0&1 }] [\pmatrix{ -1&1 \cr 0&1 }]
[\alpha\beta] [\pmatrix{ -1&0 \cr -1&1 }] [\pmatrix{ 0&-1 \cr -1&0 }]

[622] [\alpha =C_{6z}] [\Gamma_{1}: A_{1} =\chi_1]   [x^{2}+y^{2},z^{2}]
[D_6] [\beta =C_{2x}] [\Gamma_{2}: A_{2} =\chi_2] z  
    [\Gamma_{3}: B_{1} =\chi_3]   [x^{2}-y^{2}]
    [\Gamma_{4}: B_{2} =\chi_4]   [xy]
    [\Gamma_{5}: E_{1} ={\chi}_1] [x,y] [xz,yz]
    [\Gamma_{6}: E_{2} ={\chi}_2]    
         
[6mm] [\alpha =C_{6z}] [\Gamma_{1}: A_{1} =\chi_1] z [x^{2}+y^{2},z^{2}]
[C_{6v}] [\beta =\sigma_{v}] [\Gamma_{2}: A_{2} =\chi_2]    
    [\Gamma_{3}: B_{1} =\chi_3]   [x^{2}-y^{2}]
    [\Gamma_{4}: B_{2} =\chi_4]   [xy]
    [\Gamma_{5}: E_{1} ={\chi}_1] [x,y] [xz,yz]
    [\Gamma_{6}: E_{2} ={\chi}_2]    
         
[\bar{6}2m] [\alpha =S_{6z}] [\Gamma_{1}: A'_{1} =\chi_1]   [x^{2}+y^{2},z^{2}]
[D_{3h}] [\beta =C_{2v}] [\Gamma_{2}: A'_{2} =\chi_2]    
  [\alpha\beta =\sigma_{d}] [\Gamma_{3}: A''_{1} =\chi_3]   [x^{2}-y^{2}]
    [\Gamma_{4}: A''_{2} =\chi_4] z [xy]
    [\Gamma_{5}: E' ={\chi}_2]   [xz,yz]
    [\Gamma_{6}: E'' ={\chi}_1] [x,y]  
         
[\bar{3}m] [\alpha =S_{3z}] [\Gamma_{1}: A_{1g} =\chi_1^+]   [x^{2}+y^{2},z^{2}]
[D_{3v}] [\beta =\sigma_{d}] [\Gamma_{2}: A_{2g} =\chi_2^+]    
    [\Gamma_{3}: A_{1u} =\chi_1^-] z  
    [\Gamma_{4}: A_{2u} =\chi_2^-]    
    [\Gamma_{5}: E_{u} ={\chi}_1^-] [x,y]  
    [\Gamma_{6}: E_{g} ={\chi}_1^+]   [xz.yz,xy,x^{2}-y^{2}]

(j) T [[\omega=\exp(2\pi i/3)]].

T [\varepsilon] [\alpha] [\alpha^{2}] [\beta]
n 1 4 4 [3]
Order 1 3 3 [2]
[ \Gamma_{1}] 1 1 1 [1]
[\Gamma_{2}] 1 [\omega] [\omega^{2}] [1]
[\Gamma_{3}] 1 [\omega^{2}] [\omega] [1]
[\Gamma_{4}] 3 0 0 [-1]

Real representations of dimension [d\,\gt\,1]:

  [\Gamma_2\oplus\Gamma_3] [\Gamma_4]
[\varepsilon] [\pmatrix{ 1&0 \cr 0&1 }] [\pmatrix{1&0&0 \cr 0&1&0 \cr 0&0&1 }]
[\alpha] [\pmatrix{1&-1 \cr 0&-1 }] [\pmatrix{0&1&0 \cr 0&0&1 \cr 1&0&0 }]
[\alpha^2] [\pmatrix{1&-1 \cr 0&-1 }] [\pmatrix{0&0&1 \cr 1&0&0 \cr 0&1&0 }]
[\beta] [\pmatrix{1&0 \cr 0&1 }] [\pmatrix{-1&0&0 \cr 0&-1&0 \cr 0&0&1 }]

[23] [\alpha =C_{3d}] [\Gamma_{1}: A =\chi_1]   [x^{2}+y^{2}+z^{2}]
T [\beta =C_{2z}] [\Gamma_{2}\oplus\Gamma_{3}: E=\chi_{3c}+\chi_{3c}^*]   [x^{2}-y^{2}, x^{2}-z^{2}]
    [\Gamma_{4}: T ={\chi}_1] [x,y,z] [xy,xz,yz]

(k) O

O [\varepsilon] [\beta] [\alpha^{2}] [\alpha] [\alpha \beta]
n 1 [8] [3] [6] [6]
Order 1 [3] [2] [4] [2]
[ \Gamma_{1}] 1 [1] [1] [1] [1]
[\Gamma_{2}] 1 [1] [1] [-1] [-1]
[\Gamma_{3}] 2 [-1] [2] [0] [0]
[\Gamma_{4}] 3 [0] [-1] [1] [-1]
[\Gamma_{5}] 3 [0] [-1] [-1] [1]

Higher-dimensional representations:

  [\Gamma_3] [\Gamma_4] [\Gamma_5]
[\varepsilon] [\pmatrix{ 1&0 \cr 0&1 }] [\pmatrix{1&0&0 \cr 0&1&0 \cr 0&0&1 }] [\pmatrix{1&0&0 \cr 0&1&0 \cr 0&0&1 }]
[\beta] [\pmatrix{0&-1 \cr 1&-1 }] [\pmatrix{0&0&1 \cr 1&0&0 \cr 0&1&0 }] [\pmatrix{0&0&1 \cr 1&0&0 \cr 0&1&0 }]
[\alpha^2] [\pmatrix{ 1&0 \cr 0&1 }] [\pmatrix{ -1&0&0 \cr 0&-1&0 \cr 0&0&1 }] [\pmatrix{-1&0&0 \cr 0&-1&0 \cr 0&0&1 }]
[\alpha] [\pmatrix{ 0&1 \cr 1&0 }] [\pmatrix{ 0&-1&0 \cr 1&0&0 \cr 0&0&1 }] [\pmatrix{0&1&0 \cr -1&0&0 \cr 0&0&-1 }]
[\alpha\beta] [\pmatrix{ -1&0 \cr -1&1 }] [\pmatrix{ -1 &0&0 \cr 0&0&1 \cr 0&1&0 } ] [\pmatrix{ 1 &0&0 \cr 0&0&-1 \cr 0&-1&0 }]

[432] [\alpha =C_{4z}] [\Gamma_{1}: A_{1} =\chi_1]   [x^{2}+y^{2}+z^{2}]
O [\beta =C_{3d}] [\Gamma_{2}: A_{2} =\chi_2]    
  [\alpha\beta =C_{2}] [\Gamma_{3}: E ={\chi}_3]   [x^{2}-y^{2}, y^{2}-z^{2}]
    [\Gamma_{4}: T_{1} ={\chi}_1] [x,y,z]  
    [\Gamma_{5}: T_{2} ={\chi}_2]   [xy,xz,yz]
         
[\bar{4}3m] [\alpha =S_{4z}] [\Gamma_{1}: A_{1}= \chi_1]   [x^{2}+y^{2}+z^{2}]
[T_d] [\beta =C_{3d}] [\Gamma_{2}: A_{2} =\chi_2]    
  [\alpha\beta =\sigma_{d}] [\Gamma_{3}: E ={\chi}_3]   [x^{2}-y^{2}, y^{2}-z^{2}]
    [\Gamma_{4}: T_{1} ={\chi}_1]    
    [\Gamma_{5}: T_{2} ={\chi}_2] [x,y,z] [xy,yz,xz]

Other point groups which are of second class and contain [-E]. See Table 1.2.6.6[link](a).

Group Isomorphism class Rotation subgroup
[4/m] [C_{4}\times {\bb Z}_2] 4
[6/m] [C_{6}\times {\bb Z}_2] 6
[mmm] [D_{2}\times {\bb Z}_{2}] 222
[4/mmm] [D_{4}\times {\bb Z}_{2}] 422
[6/mmm ] [D_{6}\times {\bb Z}_{2}] 622
[m\bar{3}] [T\times {\bb Z}_{2}] 23
[m\bar{3}m] [O\times{\bb Z}_{2}] 432

For each isomorphism class, the character table is given, including the symbol for the isomorphism class, the number n of elements per conjugacy class and the order of the elements in each such class. The conjugation classes are specified by representative elements expressed in terms of the generators [\alpha, \beta, \ldots]. The irreps are denoted by [\Gamma_i], where i takes as many values as there are conjugation classes. In each isomorphism class for each point group, given by its international symbol and its Schoenflies symbol, identification is made between the generators of the abstract group ([\alpha, \beta ]) and the generating orthogonal transformations. Notation: [C_{nx}] is a rotation of [2\pi /n] along the x axis, [\sigma_{x}] is a reflection from a plane perpendicular to the x axis, [S_{nz}] is a rotation over [2\pi /n] along the z axis multiplied by [-E] and [\sigma_{v}] is a reflection from a plane through the unique axis.

The notation for the irreducible representations can be given as [\Gamma_{i}], but other systems have been used as well. Indicated below are the relations between [\Gamma_{i}] and a system that uses a characterization according to the dimension of the representation and (for groups of the second kind) the sign of the representative of [-E]. This nomenclature is often used by spectroscopists.[\matrix{ A, A_{1}, A_{2}, A', A''\hfill & \hbox{one-dimensional}\hfill\cr B, B_{1}, B_{2}, B_{3}\hfill & \hbox{one-dimensional}\hfill\cr E\hfill & \hbox{two-dimensional}\hfill\cr T, T_{1}, T_{2}\hfill& \hbox{three-dimensional}\hfill\cr A_{g}, B_{g}\,\,etc.\hfill & gerade\hfill\cr A_{u}, B_{u}\,\,etc.\hfill & ungerade\hfill}]The other notation for which the relation with the present notation is indicated is that of Kopský, and is used in the accompanying software.

The three functions x, y and z transform according to the vector representation of the point group, which is generally reducible. The reduction into irreducible components of this three-dimensional vector representation is indicated.

The six bilinear functions [x^{2}], [xy], [xz], [y^{2}], [yz], [z^{2}] transform according to the symmetrized product of the vector representation. The basis functions of the irreducible components are indicated. Because the basis functions are real, one should consider the physically irreducible representations.

Table 1.2.6.6[link] . The point groups of the second class containing [-E] are obtained from those of the first class by taking the direct product with the group generated by [\bar{1}]. From the point groups, one obtains nonmagnetic point groups by the direct product with the group generated by the time reversal [1']. The relation between the characters of a point group and its direct products with groups generated by [\bar{1}], [1'] and [\{\bar{1},1'\}] are given in Tables 1.2.6.6[link](a), (b) and (c), respectively.

Table 1.2.6.6 | top | pdf |
Direct products with [\{E,\bar{1}\}] and [\{E,1'\}]

(a) With [\{E,\bar{1}\}].

[K\times {\bb Z}_{2}] [R\in K] [\bar{R}]
[ \Gamma_{g}] [\chi (R)] [\chi (R)]
[\Gamma_{u}] [\chi (R)] [-\chi (R)]

[4/m] [C_{4}\times {\bb Z}_{2}] cf. 4
[6/m] [C_{6}\times {\bb Z}_{2}] cf. 6
[mmm] [D_{2}\times {\bb Z}_{2}] cf. 222
[4/mmm] [D_{4}\times{\bb Z}_{2}] cf. 422
[6/mmm] [D_{6}\times{\bb Z}_{2}] cf. 622
[m\bar{3}] [T\times{\bb Z}_{2}] cf. 23
[m\bar{3}m] [O\times{\bb Z}_{2}] cf. 432

(b) With [\{E,1'\}].

[K\times {\bb Z}_{2}] [R\in K] [R']
[ \Gamma_{+}] [\chi (R)] [\chi (R)]
[\Gamma_{-}] [\chi (R)] [-\chi (R)]

[1'] [ C_{1}\times {\bb Z}_{2}] cf. 1
[21'] [C_{2}\times {\bb Z}_{2}] cf. 2
[m1'] [C_{2}\times {\bb Z}_{2}] cf. m
[2221'] [D_{2}\times {\bb Z}_{2}] cf. 222
[2mm1'] [D_{2}\times {\bb Z}_{2}] cf. [2mm]
[41'] [C_{4}\times {\bb Z}_{2}] cf. 4
[\bar{4}1'] [C_{4}\times {\bb Z}_{2}] cf. [\bar{4}]
[4mm1'] [D_{4}\times{\bb Z}_{2}] cf. [4mm]
[4221'] [D_{4}\times{\bb Z}_{2}] cf. 422
[\bar{4}2m1'] [D_{4}\times{\bb Z}_{2}] cf. [\bar{4}2m]
[31'] [C_{3}\times {\bb Z}_{2}] cf. 3
[321'] [D_{3}\times{\bb Z}_{2}] cf. 32
[\bar{3}1'] [C_{6}\times{\bb Z}_{2}] cf. [\bar{3}]
[3m1'] [D_{3}\times{\bb Z}_{2}] cf. [3m]
[6mm1'] [D_{6}\times{\bb Z}_{2}] cf . [6mm]
[61'] [C_{6}\times{\bb Z}_{2}] cf. 6
[\bar{6}1'] [C_{6}\times {\bb Z}_{2}] cf. [\bar{6}]
[6221'] [D_{6}\times{\bb Z}_{2}] cf. 622
[\bar{6}2m1'] [D_{6}\times{\bb Z}_{2}] cf. [\bar{6}2m]
[231'] [T\times{\bb Z}_{2}] cf. 23
[4321'] [O\times{\bb Z}_{2}] cf. 432
[\bar{4}3m1'] [O\times{\bb Z}_{2}] cf. [\bar{4}3m]

(c) With [\{E,\bar{1}\}] and [\{E,1'\}].

[K\times {\bb Z}_{2} \times {\bb Z}_{2}] [R\in K] [\bar{R}] [R'] [\bar{R}']
[ \Gamma_{g+}] [\chi (R)] [\chi (R)] [\chi (R)] [\chi (R)]
[\Gamma_{u+}] [\chi (R)] [-\chi (R)] [\chi (R)] [-\chi (R)]
[\Gamma_{g-}] [\chi (R)] [\chi (R)] [-\chi (R)] [-\chi (R)]
[\Gamma_{u-}] [\chi (R)] [-\chi (R)] [-\chi (R)] [\chi (R)]

[\bar{1}'] [C_{1}\times {\bb Z}_{2}\times {\bb Z}_{2}] cf. 1
[21'/m] [C_{2}\times {\bb Z}_{2}\times {\bb Z}_{2}] cf. 2
[4/m1'] [C_{4}\times {\bb Z}_{2}\times {\bb Z}_{2}] cf. 4
[6/m1'] [C_{6}\times {\bb Z}_{2}\times {\bb Z}_{2}] cf. 6
[mmm1'] [D_{2}\times {\bb Z}_{2}\times {\bb Z}_{2}] cf. 222
[4/mmm1'] [D_{4}\times{\bb Z}_{2}\times {\bb Z}_{2}] cf. 422
[\bar{3}m1'] [D_{6}\times{\bb Z}_{2}\times {\bb Z}_{2}] cf. [3m]
[6/mmm1'] [D_{6}\times{\bb Z}_{2}\times {\bb Z}_{2}] cf. 622
[m31'] [T\times{\bb Z}_{2}\times {\bb Z}_{2}] cf. 23
[m(\bar{3})m1'] [O\times{\bb Z}_{2}\times {\bb Z}_{2}] cf. 432

Table 1.2.6.7[link] . The representations of a point group are also representations of their double groups. In addition, there are extra representations which give projective representations of the point groups. For several cases, these are associated with an ordinary representation. As extra representations, those irreducible representations of the double point groups that give rise to projective representations of the point groups with a factor system that is not associated with the trivial one are given. These do not correspond to ordinary representations of the single group.

Table 1.2.6.7 | top | pdf |
Extra representations of double groups

222d E −E [\pm] A [\pm] B [\pm] AB        
[\Gamma_{5}'] 2 −2 [0] [0] [0]        
422d E −E [\pm] A 2 [A] −A [\pm] B [\pm] AB    
[\Gamma_{6}'] 2 −2 [0] [\sqrt{2}] [-\sqrt{2}] [0] [0]    
[\Gamma_{7}'] 2 −2 [0] [-\sqrt{2}] [\sqrt{2}] [0] [0]    
622d E −E [A^{2}] −A2 [\pm] B [\pm] A 3 [A^{5}] −A5 [\pm ] A 3 B
[\Gamma_{8}'] 2 −2 [1] −1 [0] [0] [\sqrt{3}] [-\sqrt{3}] [0]
[\Gamma_{9}'] 2 −2 [1] −1 [0] [0] [-\sqrt{3}] [\sqrt{3}] [0]
[\Gamma_{7}'] 2 −2 −2 [2] [0] [0] [0] [0] [0]
23d E −E [A] −A [A^{2}] −A2 [\pm] B    
[\Gamma_{5}'] 2 −2 [1] −1 [1] −1 [0]    
[\Gamma_{6}'] 2 −2 [\omega] [\omega^{4}] [\omega^{2}] [\omega^{5}] [0]    
[\Gamma_{7}'] 2 −2 [\omega^{5}] [\omega^{2}] [\omega^{4}] [\omega] [0]    
432d E −E [B] −B [\pm] A 2 [A] −A [\pm] AB  
[\Gamma_{6}'] 2 −2 [1] −1 [0] [\sqrt{2}] [-\sqrt{2}] [0]  
[\Gamma_{7}'] 2 −2 [1] −1 [0] [-\sqrt{2}] [\sqrt{2}] [0]  
[\Gamma_{8}'] 4 −4 −1 [1] [0] [0] [0] [0]  

Table 1.2.6.8[link]. If one chooses for each element of a point group one of the two corresponding [SU(2)] elements, the latter form a projective representation of the point group. If one selects for the rotation [R\in K \subset SO(3)] the element [u(R) = E\cos (\varphi /2) + i({\boldsigma}\cdot{\bf n})\sin (\varphi /2),]where [\varphi] is the rotation angle and [{\bf n}] the rotation axis, and for [R\in K\subset O(3)\backslash SO(3)] the element [u(R) = E\cos (\psi /2) + i({\boldsigma}\cdot{\bf n})\sin (\psi /2),]where [\psi] and [{\bf n}] are the rotation angle and axis of the rotation [-R], the matrices [u(R)] form a projective representation: [u(R)u(R') = \omega_{s}(R,R')u(RR').]The factor system [\omega_{s}] is the spin factor system. It is determined via the generators and defining relations [W_{i}(A_{1},\ldots, A_{p}) = E]of the point group K. Then [W_{i}(u(A_{1}),\ldots, u(A_{p})) = \lambda_{i}E, ]and the factors [\lambda_{i}] fix uniquely the class of the factor system [\omega_{s}]. These factors are given in the table.

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 [{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}]    

Because [\bar{1}] is represented by the unit matrix in spin space, the double groups of two isomorphic point groups obtained from each other by replacing the elements [R\in O(3)\backslash SO(3)] by [-R] are the same.

The projective representations with factor system [\omega_{s}] may sometimes be associated with one with a trivial factor system. If this is the case, there are actually no extra representations of the double group. If there are extra representations, these are irreducible representations of the double group: see Table 1.2.6.7[link].

Table 1.2.6.9[link] . For the 32 three-dimensional crystallographic point groups, the character of the vector representation [\Gamma] and the number of times the identity representation occurs in a number of tensor products of this vector representation are given. This is identical to the number of free parameters in a tensor of the corresponding type. For the direct products [{K}\times C_{2}], the character is equal to that of K on the rotation subgroup, and its opposite [[\chi (-R)= -\chi (R)]] for the coset [-{K}].

Table 1.2.6.9 | top | pdf |
Number of free parameters of some tensors

Group Isomorphism class Character of the vector representation Multiplicity identity representation in
[\Gamma^{\otimes 2}] [\Gamma_{s}^{\otimes 2}] [\Gamma^{\otimes 3}] [\Gamma \otimes \Gamma_{s}^{\otimes 2}] [\left(\Gamma_{s}^{\otimes 2} \right)_{s} ^{\otimes 2}]
1 [C_{1}] 3 9 6 27 18 21
[\bar{1}] [C_{2}] 3, −3 9 6 0 0 21
2 [C_{2}] 3, −1 5 4 13 8 13
m [C_{2}] 3, 1 5 4 14 10 13
[2/m] [C_{2}\times C_2]   5 4 0 0 13
222 [D_2] 3, −1, −1, −1 3 3 6 3 9
[2mm] [ D_2] 3, 1, 1, −1 3 3 7 5 9
[mmm] [D_{2}\times C_2]   3 3 0 0 9
3 [C_3] 3, 0, 0 3 2 9 6 9
[\bar{3}] [C_3\times C_2]   3 2 0 0 9
32 [D_3] 3, 0, −1 2 2 4 2 6
[3m] [D_3] 3, 0, 1 2 2 5 4 6
[\bar{3}m] [D_3\times C_2]   2 2 0 0 6
6 [C_6] 3, 2, 0, −1, 0, 2 3 2 7 4 5
[\bar{6}] [C_6] 3, 2, 0, 1, 0, −2 3 2 2 2 5
[6/m] [C_6\times C_2]   3 2 0 0 5
622 [D_6] 3, 2, 0, −1, −1, −1 2 2 3 1 5
[6mm] [D_6] 3, 2, 0, −1, 1, 1 2 2 4 3 5
[\bar {6}2m] [D_6] 3, −2, 0, 1, −1, 1 2 2 1 1 5
[6/mmm] [D_6\times C_2]   2 2 0 0 5
4 [C_4] 3, 1, −1, 1 3 2 7 4 7
[\bar{4}] [C_4] 3, −1, −1, −1 3 2 6 4 7
[4/m] [C_4\times C_2]   3 2 0 0 7
422 [D_4] 3, 1, −1, −1, −1 2 2 3 1 6
[4mm] [D_4] 3, 1, −1, 1, 1 2 2 4 3 6
[\bar {4}2m] [D_4] 3, −1, −1, −1, 1 2 2 3 2 6
[4/mmm] [D_4\times C_2]   2 2 0 0 6
23 T 3, 0, 0, −1 1 1 2 1 3
[m3] [T\times C_2]   1 1 0 0 3
432 O 3, 0, −1, 1, −1 1 1 1 0 3
[\bar{4}3m] O 3, 0, −1, −1, 1 1 1 1 1 3
[m3m] [O\times C_2]   1 1 0 0 3

Table 1.2.6.10[link] . The irreducible projective representations of the 32 three-dimensional crystallographic point groups that have a factor system that is not associated to a trivial one. In three (and two) dimensions all factor systems are of order two.

Table 1.2.6.10 | top | pdf |
Irreducible projective representations of the 32 crystallographic point groups

(a) [D_{2}]

[{A}^{2}={B}^{2}={E}, ({AB})^{2}=-{E}]
Elements E A B AB
[ \Gamma_{5}'] 2 0 0 0

(b) [D_{4}]

[{A}^{4}=-{E},{B}^{2}=({AB})^{2}={E}]
Elements E A 2 [A] A 3 B A 2 B AB A 3 B
[ \Gamma_{6}'] 2 0 [i\sqrt{2}] [i\sqrt{2}] 0 0 0 0
[\Gamma_{7}'] 2 0 [-i\sqrt{2}] [-i\sqrt{2}] 0 0 0 0

(c) [D_{6}]

[{A}^{6}={B}^{2}={E}, ({AB})^{2}=-{E}]
Elements E [A^{2}] [A^{4}] B A 2 B A 4 B A 3 [A] [A^{5}] AB A 3 B A 5 B
[ \Gamma_{7}'] 2 [2] [2] 0 0 0 0 [0] [0] 0 0 0
[\Gamma_{8}'] 2 [-1] [-1] 0 0 0 0 [i\sqrt{3}] [-i\sqrt{3}] 0 0 0
[\Gamma_{9}'] 2 [-1] [-1] 0 0 0 0 [-i\sqrt{3}] [i\sqrt{3}] 0 0 0

(d) T [[\omega =\exp (2\pi i/3)]].

[{A}^{3}={E}, {B}^{2}=({AB})^{3}=-{E}]
Elements [E] [A] [BAB] [BA] [AB] [A^2]
[ \Gamma_{5}'] [2] [-1] [1] [1] [1] [-1]
[\Gamma_{6}'] [2] [\omega^{5}] [\omega^{2}] [\omega^{2}] [\omega^{2}] [\omega^{5}]
[\Gamma_{7}'] [2] [\omega] [\omega^{4}] [\omega^{4}] [\omega^{4}] [\omega]
Elements [ABA] [A^2B] [BA^2] [B] [ABA^2] [A^2BA]
[ \Gamma_{5}'] [-1] [-1] [-1] [0] [0] [0]
[\Gamma_{6}'] [\omega^{5}] [\omega^{5}] [\omega^{5}] [0] [0] [0]
[\Gamma_{7}'] [\omega] [\omega] [\omega] [0] [0] [0]

(e) O

[{A}^{4}=-{E}, {B}^{3}=({AB})^{2}={E}]
Elements [E] [B] [AB^2A] [A^2B] [BA^2] [B^2]
[ \Gamma_{6}'] [2] [-1] [1] [-1] [-1] [-1]
[\Gamma_{7}'] [2] [-1] [1] [-1] [-1] [-1]
[\Gamma_{8}'] [4] [1] [-1] [1] [1] [1]
Elements [BA^2B] [ABA^3] [A^2B^2] [A^2] [BA^2B^2] [B^2A^2B]
[ \Gamma_{6}'] [1] [1] [1] [0] [0] [0]
[\Gamma_{7}'] [1] [1] [1] [0] [0] [0]
[\Gamma_{8}'] [-1] [-1] [-1] [0] [0] [0]
Elements [A] [A^3] [A^3B] [BA^3] [B^2A] [AB^2]
[ \Gamma_{6}'] [i\sqrt{2}] [i\sqrt{2}] [-i\sqrt{2}] [-i\sqrt{2}] [-i\sqrt{2}] [-i\sqrt{2}]
[\Gamma_{7}'] [-i\sqrt{2}] [-i\sqrt{2}] [i\sqrt{2}] [i\sqrt{2}] [i\sqrt{2}] [i\sqrt{2}]
[\Gamma_{8}'] [0] [0] [0] [0] [0] [0]
Elements [A^2B^2A] [BA] [AB] [AB^2A^2] [AB^2A^2B] [B^2AB^2]
[ \Gamma_{6}'] [0] [0] [0] [0] [0] [0]
[\Gamma_{7}'] [0] [0] [0] [0] [0] [0]
[\Gamma_{8}'] [0] [0] [0] [0] [0] [0]

(f) [C_{4}\times C_{2}]

[{A}^{4}={B}^{2}={E}, {AB}=-{BA }]
Elements E A [A^{2}] A 3 B AB A 2 B A 3 B
[ \Gamma_{9}'] 2 0 [2] 0 0 0 0 0
[\Gamma_{10}'] 2 0 [-2] 0 0 0 0 0

(g) [C_{6}\times C_{2}]

[{A}^{6}={B}^{2}={E}, {AB}=-{BA}]
Elements E A A 2 A 3 A 4 A 5 B AB A 2 B A 3 B A 4 B A 5 B
[ \Gamma_{13}'] 2 0 2 0 2 0 0 0 0 0 0 0
[\Gamma_{14}'] 2 0 2[\omega^{2}] 0 2[\omega^{4}] 0 0 0 0 0 0 0
[\Gamma_{15}'] 2 0 2[\omega^{4}] 0 2[\omega^{4}] 0 0 0 0 0 0 0

(h) [D_{2}\times C_{2}]

[{A}^{2}=-{E},{B}^{2}={C}^{2}=({AB})^{2}={E},{AC}={CA},{BC}={CB}]
Elements E [A] [B] [AB] [C] [AC] [BC] [ABC]
[ \Gamma_{9}'] 2 [0] [0] [0] [2] [0] [0] [0]
[\Gamma_{10}'] 2 [0] [0] [0] [-2] [0] [0] [0]
[{A}^{2}={E},{B}^{2}={C}^{2}=({AB})^{2}={E},{AC}=-{CA},{BC}={CB}]
Elements E [A] [B] [AB] [C] [AC] [BC] [ABC]
[ \Gamma_{11}'] 2 [0] [2] [0] [0] [0] [0] [0]
[\Gamma_{12}'] 2 [0] [-2] [0] [0] [0] [0] [0]
[{A}^{2}={E},{B}^{2}={C}^{2}=({AB})^{2}={E},{AC}={CA},{BC}=-{CB}]
Elements E [A] [B] [AB] [C] [AC] [BC] [ABC]
[ \Gamma_{13}'] 2 [2i] [0] [0] [0] [0] [0] [0]
[\Gamma_{14}'] 2 [-2i] [0] [0] [0] [0] [0] [0]
[{A}^{2}=-{E},{B}^{2}={C}^{2}=({AB})^{2}={E}, {AC}=-{CA},{BC}={CB}]
Elements E [A] [B] [AB] [C] [AC] [BC] [ABC]
[ \Gamma_{15}'] 2 [0] [0] [0] [0] [0] [2] [0]
[\Gamma_{16}'] 2 [0] [0] [0] [0] [0] [-2] [0]
[{A}^{2}=-{E},{B}^{2}={C}^{2}=({AB})^{2}={E},{AC}={CA},{BC}=-{CB}]
Elements E [A] [B] [AB] [C] [AC] [BC] [ABC]
[ \Gamma_{17}'] 2 [0] [0] [0] [0] [2i] [0] [0]
[\Gamma_{18}'] 2 [0] [0] [0] [0] [-2i] [0] [0]
[{A}^{2}={E},{B}^{2}={C}^{2}=({AB})^{2}={E}, {AC}=-{CA},{BC}=-{CB}]
Elements E [A] [B] [AB] [C] [AC] [BC] [ABC]
[ \Gamma_{19}'] 2 [0] [0] [2i] [0] [0] [0] [0]
[\Gamma_{20}'] 2 [0] [0] [-2i] [0] [0] [0] [0]
[{A}^{2}=-{E},{B}^{2}={C}^{2}=({AB})^{2}={E}, {AC}=-{CA},{BC}=-{CB}]
Elements E [A] [B] [AB] [C] [AC] [BC] [ABC]
[ \Gamma_{21}'] 2 [0] [0] [0] [0] [0] [0] [2i]
[\Gamma_{22}'] 2 [0] [0] [0] [0] [0] [0] [-2i]

Table 1.2.6.11[link] . The special points in the Brillouin zones. Strata of irreducible representations of the space groups are characterized by the wavevector [{\bf k}] of such a point and a (possibly projective) irreducible representation of the point group [{K}_{{\bf k}}]. The latter is the intersection of the symmetry group of [{\bf k}] (the group of [{\bf k}] for the holohedral point group) and the point group of the space group. For each Bravais class the special points for the holohedry are given. These are given by their coordinates with respect to a basis of the reciprocal lattice of the conventional cell. These points correspond to Wyckoff positions in the corresponding dual lattice. The symbols for these Wyckoff positions and their site symmetry are given. A well known notation for the special points is that of Kovalev, as used in his book on representations of space groups. Correspondence with the notation in Kovalev (1987[link]) is given.

Table 1.2.6.11 | top | pdf |
Special points in the Brillouin zones in three dimensions

(a) Triclinic

[{\bf k}] [K_{{\bf k}}] Kovalev
a [000] [\bar{1}] [k_8]
b [00{{1}\over{2}}] [\bar{1}] [k_7]
c [0{{1}\over{2}}0] [\bar{1}] [k_{6}]
d [{{1}\over{2}}00] [\bar{1}] [k_{5}]
e [{{1}\over{2}}{{1}\over{2}}0] [\bar{1}] [k_{4}]
f [{{1}\over{2}} 0{{1}\over{2}}] [\bar{1}] [k_{3}]
g [0{{1}\over{2}}{{1}\over{2}}] [\bar{1}] [k_{2}]
h [{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}] [\bar{1}] [k_{1}]

(b) Monoclinic P

[{\bf k}] [K_{{\bf k}}] Kovalev
a [000] [2/m] [k_{7}]
b [00{{1}\over{2}}] [2/m] [k_{11}]
c [{{1}\over{2}}00] [2/m] [k_{12}]
d [0{{1}\over{2}}0] [2/m] [k_{13}]
e [0{{1}\over{2}}{{1}\over{2}}] [2/m] [k_{9}]
f [{{1}\over{2}} 0{{1}\over{2}}] [2/m] [k_{8}]
g [{{1}\over{2}}{{1}\over{2}}0] [2/m] [k_{14}]
h [{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}] [2/m] [k_{10}]
i [00\gamma] 2 [k_{3}]
j [0{{1}\over{2}}\gamma] 2 [k_{5}]
k [{{1}\over{2}} 0\gamma] 2 [k_{4}]
l [{{1}\over{2}}{{1}\over{2}}\gamma] 2 [k_{6}]
m [\alpha\beta 0] m [k_{1}]
n [\alpha\beta{{1}\over{2}}] m [k_{2}]

(c) Monoclinic A

[{\bf k}] [K_{{\bf k}}] Kovalev
a [000] [2/m ] [k_{6 }]
b [010] [2/m] [k_{8 }]
c [{{1}\over{2}} 00] [ 2/m] [k_{7 }]
d [{{1}\over{2}} 10] [ 2/m] [k_{9 }]
e [0{{1}\over{2}}{{1}\over{2}}] [\bar{1}] [k_{4 }]
f [{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}] [\bar{1}] [k_{5 }]
g [00\gamma] 2 [k_{2 }]
h [{{1}\over{2}} 0\gamma] 2 [k_{3 }]
i [\alpha\beta 0] m [k_{1 }]

(d) Orthorhombic P

[{\bf k}] [K_{{\bf k}}] Kovalev
a [000] [mmm] [k_{19}]
b [{{1}\over{2}} 00] [mmm] [k_{20}]
c [00{{1}\over{2}}] [mmm] [k_{22}]
d [{{1}\over{2}} 0{{1}\over{2}}] [mmm] [k_{24}]
e [0{{1}\over{2}}0] [mmm] [k_{21}]
f [{{1}\over{2}}{{1}\over{2}}0] [mmm] [k_{25}]
g [0{{1}\over{2}}{{1}\over{2}}] [mmm] [k_{23}]
h [{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}] [mmm] [k_{26}]
i [\alpha 00] [2mm] [k_{7 }]
j [\alpha 0{{1}\over{2}}] [2mm] [k_{12}]
k [\alpha{{1}\over{2}}0] [2mm] [k_{10}]
l [\alpha{{1}\over{2}}{{1}\over{2}}] [2mm] [k_{11}]
m [0\beta 0] [m2m] [k_{8 }]
n [0\beta{{1}\over{2}}] [m2m] [k_{15}]
o [{{1}\over{2}}\beta 0] [m2m] [k_{13}]
p [{{1}\over{2}}\beta{{1}\over{2}}] [m2m] [k_{14}]
q [00\gamma] [mm2] [k_{9 }]
r [0{{1}\over{2}}\gamma] [mm2] [k_{18}]
s [{{1}\over{2}} 0\gamma] [mm2] [k_{16}]
t [{{1}\over{2}}{{1}\over{2}}\gamma] [mm2] [k_{17}]
u [0\beta\gamma] m [k_{1 }]
v [{{1}\over{2}}\beta\gamma] [m11] [k_{2 }]
w [\alpha 0\gamma] [1m1] [k_{3 }]
x [\alpha{{1}\over{2}}\gamma] [1m1] [k_{4 }]
y [\alpha\beta 0] [11m] [k_{5 }]
z [\alpha\beta{{1}\over{2}}] [11m] [k_{6 }]

(e) Orthorhombic C

[{\bf k}] [K_{{\bf k}}] Kovalev
a [000] [mmm] [k_{14}]
b [010] [mmm] [k_{15}]
c [01{{1}\over{2}}] [mmm] [k_{17}]
d [00{{1}\over{2}}] [mmm ] [k_{16}]
e [{{1}\over{2}}{{1}\over{2}}0] [2/m] [k_{12}]
f [{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}] [2/m] [k_{13}]
g [\alpha 00] [2mm ] [k_{8}]
h [\alpha 0{{1}\over{2}}] [2mm] [k_{9}]
i [0\beta 0] [m2m] [k_{10}]
j [0\beta{{1}\over{2}}] [m2m] [k_{11}]
k [00\gamma] [mm2] [k_{6 }]
l [01\gamma] [mm2] [k_{7 }]
m [{{1}\over{2}}{{1}\over{2}}\gamma] [112] [k_{5 }]
n [0 \beta\gamma] [m11] [k_{1 }]
o [\alpha 0\gamma] [1m1] [k_{2 }]
p [\alpha\beta 0] [11m] [k_{3 }]
q [\alpha\beta{{1}\over{2}}] [11m] [k_{4 }]

(f) Orthorhombic I

[{\bf k}] [K_{{\bf k}}] Kovalev
a [000] [mmm] [k_{17}]
b [001] [mmm] [k_{18}]
c [0{{1}\over{2}}{{1}\over{2}}] [2/m11] [k_{13}]
[1{{1}\over{2}}{{1}\over{2}}] [2/m11] [k_{10}]
d [{{1}\over{2}} 0{{1}\over{2}}] [12/m1] [k_{14}]
[{{1}\over{2}} 1{{1}\over{2}}] [12/m1] [k_{11}]
e [{{1}\over{2}}{{1}\over{2}} 0] [112/m] [k_{15}]
[{{1}\over{2}}{{1}\over{2}} 1] [112/m] [k_{12}]
f [{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}] [222] [k_{16}]
g [\alpha 00] [2mm] [k_{7 }]
h [0\beta 0] [m2m] [k_{8 }]
i [00\gamma] [mm2] [k_{9 }]
j [{{1}\over{2}}{{1}\over{2}}\gamma] [112] [k_{6 }]
k [{{1}\over{2}}\beta{{1}\over{2}}] [121] [k_{5 }]
l [\alpha{{1}\over{2}}{{1}\over{2}}] [211] [k_{4 }]
m [0\beta\gamma] [m11] [k_{1 }]
n [\alpha 0\gamma] [1m1 ] [k_{2 }]
o [\alpha\beta 0] [11m] [k_{3 }]

(g) Orthorhombic F

[{\bf k}] [K_{{\bf k}}] Kovalev
a [000] [mmm] [k_{14}]
b [100] [mmm] [k_{15}]
c [010] [mmm] [k_{16}]
d [001] [ mmm ] [k_{17}]
e [\alpha00] [2mm] [k_{4 }]
f [\alpha10] [2mm] [k_{5 }]
g [0\beta0] [m2m] [k_{6 }]
h [1\beta0] [m2m] [k_{7 }]
i [00\gamma] [mm]2 [k_{8 }]
j [01\gamma] [mm2] [k_{9 }]
k   [{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}] [\bar{1}] [k_{10}]
[-{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}] [\bar{1}] [k_{11}]
[{{1}\over{2}} -{{1}\over{2}}{{1}\over{2}}] [\bar{1}] [k_{12}]
[{{1}\over{2}}{{1}\over{2}} -{{1}\over{2}}] [\bar{1}] [k_{13}]
l [0\beta\gamma] [m11] [k_{1 }]
m [\alpha 0\gamma] [1m1] [k_{2 }]
n [\alpha\beta0] [11m] [k_{3 }]

(h) Tetragonal P

[{\bf k}] [K_{{\bf k}}] Kovalev
a [000] [4/mmm] [k_{17}]
b [00{{1}\over{2}}] [4/mmm] [k_{19}]
c [{{1}\over{2}}{{1}\over{2}}0] [4/mmm] [k_{18}]
d [{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}] [4/mmm ] [k_{20}]
e [0{{1}\over{2}}{{1}\over{2}}] [mmm] [k_{16}]
f [0{{1}\over{2}}0] [mmm] [k_{15}]
g [00\gamma] [4mm] [k_{13}]
h [{{1}\over{2}}{{1}\over{2}}\gamma] [4mm] [k_{14}]
i [0{{1}\over{2}}\gamma] [mm2] [k_{12}]
j [\alpha\alpha 0] [2mm] [k_{10}]
k [\alpha\alpha{{1}\over{2}}] [2mm] [k_{11}]
l [0\beta 0 ] [m2m] [k_{8 }]
m [0\beta {{1}\over{2}}] [m2m] [k_{9 }]
n [\alpha{{1}\over{2}}0] [2mm] [k_{6 }]
o [\alpha{{1}\over{2}}{{1}\over{2}}] [2mm] [k_{7 }]
p [\alpha\beta 0] [11m] [k_{1 }]
q [\alpha\beta{{1}\over{2}}] [11m] [k_{2 }]
r [\alpha\alpha\gamma] m [k_{5 }]
s [0\beta\gamma] [m11] [k_{3 }]
t [\alpha{{1}\over{2}}\gamma] [1m1] [k_{4 }]

(i) Tetragonal I

[{\bf k}] [K_{{\bf k}}] Kovalev
a [000] [4/mmm] [k_{14}]
b [001] [4/mmm] [k_{15}]
c [{{1}\over{2}}{{1}\over{2}}0] [mmm] [k_{13}]
d [{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}] [\bar{4}m2] [k_{12}]
e [00\gamma] [4mm] [k_{10}]
f [{{1}\over{2}} 0{{1}\over{2}}] [12/m1] [k_{11}]
g [{{1}\over{2}}{{1}\over{2}}\gamma] [2mm] [k_{9 }]
h [\alpha\alpha 0] [2mm] [k_{7 }]
i [\alpha 00] [2mm] [k_{7 }]
j [\alpha (1-\alpha )0] [2mm] [k_{8 }]
k [{{1}\over{2}} \beta{{1}\over{2}}] [121] [k_{5 }]
l [\alpha\beta 0] [11m] [k_{2 }]
m [\alpha \alpha \gamma] m [k_{3 }]
[\alpha(1-\alpha)\gamma] m [k_4]
n [\alpha 0\gamma] [1m1] [k_{1 }]

(j) Trigonal R (rhombohedral axes)

[{\bf k}] [K_{{\bf k}}] Kovalev
a [000] [\bar{3}m] [k_{7}]
b [{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}] [\bar{3}m] [k_{8}]
c [\alpha\alpha\alpha] [3m] [k_{6}]
d [00{{1}\over{2}}] [2/m] [k_{4}]
e [{{1}\over{2}}{{1}\over{2}}0] [2/m] [k_{5}]
f [\alpha(-\alpha )0] [2] [k_{2}]
g [\alpha(-\alpha){{1}\over{2}}] [2] [k_{2}]
h [\alpha\beta\beta] m [k_{1}]

(k) Hexagonal P

[{\bf k}] [K_{{\bf k}}] Kovalev
a [000] [6/mmm] [k_{16}]
b [00{{1}\over{2}}] [6/mmm] [k_{17}]
c [{{1}\over{3}}{{1}\over{3}}0] [\bar{6}m2] [k_{13}]
d [{{1}\over{3}}{{1}\over{3}}{{1}\over{2}}] [\bar{6}m2] [k_{15}]
e [00\gamma] [6mm] [k_{11}]
f [{{1}\over{2}} 00] [mmm] [k_{12}]
g [{{1}\over{2}} 0{{1}\over{2}}] [mmm] [k_{14}]
h [{{1}\over{3}}{{1}\over{3}}\gamma] [3m] [k_{10}]
i [{{1}\over{2}} 0\gamma] [2mm] [k_9]
j [\alpha 00] [2mm] [k_5]
k [\alpha 0{{1}\over{2}}] [2mm] [k_7]
l [\alpha\alpha 0] [2mm] [k_6]
m [\alpha\alpha{{1}\over{2}}] [2mm] [k_8]
n [\alpha 0\gamma] m [k_3]
o [\alpha\alpha\gamma] m [k_4]
p [\alpha\beta 0] m [k_1]
q [\alpha\beta{{1}\over{2}}] m [k_2]

(l) Cubic P

[{\bf k}] [K_{{\bf k}}] Kovalev
a [000] [m\bar{3}m ] [k_{12}]
b [{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}] [m\bar{3}m ] [k_{13}]
c [{{1}\over{2}}{{1}\over{2}}0] [4/mmm] [k_{11}]
d [00{{1}\over{2}}] [4/mmm] [k_{10}]
e [00\gamma] [4mm ] [k_{8 }]
f [{{1}\over{2}}{{1}\over{2}}\gamma] [4mm ] [k_{7 }]
g [\alpha\alpha\alpha] [3m] [k_{9 }]
h [{{1}\over{2}} 0\gamma] [mm2 ] [k_{6 }]
i [\alpha\alpha 0] [2mm] [k_{4 }]
j [\alpha\alpha{{1}\over{2}}] [2mm] [k_{5 }]
k [\alpha\beta 0] [11m ] [k_{1 }]
l [\alpha\beta{{1}\over{2}}] [11m] [k_{2 }]
m [\alpha\alpha\gamma] [m ] [k_{3 }]

(m) Cubic F

[{\bf k}] [K_{{\bf k}}] Kovalev
a [000] [m\bar{3}m] [k_{11}]
b [001] [4/mmm] [k_{10 }]
c [{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}] [\bar{3}m] [k_{9}]
d [10{{1}\over{2}}] [ \bar{4}m2 ] [k_{8}]
e [\alpha 00] [4mm] [k_{6}]
f [\alpha\alpha\alpha] [3m ] [k_{5}]
g [\alpha 01] [2mm] [k_{7}]
h [\alpha\alpha 0] [2mm] [k_{4}]
i [\alpha(1-\alpha){{1}\over{2}}] [2 ] [k_{3}]
j [\alpha\beta ] [11m] [k_{1}]
k [\alpha\alpha\gamma] m [k_{2}]

(n) Cubic I

[{\bf k}] [K_{{\bf k}}] Kovalev
a [000] [m\bar{3}m] [k_{11}]
b [001] [ m\bar{3}m] [k_{10}]
c [{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}] [\bar{4}3m] [k_{10}]
d [{{1}\over{2}}{{1}\over{2}}0] [mmm] [k_{9}]
e [\alpha 00] [4mm] [k_{8}]
f [\alpha\alpha\alpha] [3m] [k_{7}]
g [\alpha{{1}\over{2}}{{1}\over{2}}] [ 2mm] [k_{6}]
h [\alpha\alpha 0] [2mm] [k_{4}]
i [\alpha(1-\alpha) 0] [2mm] [k_{9}]
j [\alpha\beta ] [11m] [k_{1}]
k [\alpha\alpha\gamma] m [k_{2}]
[\alpha (1-\alpha)\gamma] m [k_3]

Table 1.2.6.12[link] . The three-dimensional crystallographic magnetic and nonmagnetic point groups of type I (trivial magnetic, no antichronous elements), type II (nonmagnetic, containing time reversal as an element) and type III (nontrivial magnetic, without time reversal itself, but with antichronous elements).

Table 1.2.6.12 | top | pdf |
Magnetic point groups

Type I Type II Type III
1 [1']  
[\bar{1}] [\bar{1}1'] [\bar{1}']
2 [21'] [2']
m [m1'] [m']
[2/m] [21'/m] [2'/m], [2/m'], [2'/m'],
222 [2221'] [2'2']
[2mm] [2mm1'] [2'mm'], [2m'm']
[mmm] [mmm1'] [m'mm], [m'm'm], [m'm'm']
4 [41'] [4']
[\bar{4}] [\bar{4}1'] [\bar{4}']
[4/m] [41'/m] [4'/m], [4/m'], [4'/m']
422 [4221'] [4'22'], [42'2']
[4mm] [4mm1'] [4'mm'], [4m'm']
[\bar{4}2m] [\bar{4}2m1'] [\bar{4}'2'm], [\bar{4}'2m'], [\bar{4}2'm']
[4/mmm] [4/mmm1'] [4/m'mm], [4'/mm'm], [4'/m'm'm], [4/mm'm'], [4/m'm'm']
3 [31']  
[\bar{3}] [\bar{3}1'] [\bar{3}']
32 [321'] [32']
[3m] [3m1'] [3m']
[\bar{3}m] [\bar{3}m1'] [\bar{3}'m], [\bar{3}'m'], [\bar{3}m']
6 [61'] [6']
[\bar{6}] [\bar{6}1'] [\bar{6}']
[6/m] [61'/m] [6'/m], [6/m'], [6'/m']
622 [6221'] [6'22'], [62'2']
[6mm] [6mm1'] [6'mm'], [6m'm']
[\bar{6}2m] [\bar{6}2m1'] [\bar{6}'2'm], [\bar{6}'2m'], [\bar{6}2'm']
[6/mmm] [6/mmm1'] [6/m'mm], [6'/mm'm], [6'/m'm'm], [6/mm'm'], [6/m'm'm']
23 [231']  
[m\bar{3}] [m\bar{3}1'] [m' \bar {3}]
432 [4321'] [4'32']
[\bar {4}3m] [\bar {4}3m1'] [\bar {4} '3m']
[m\bar{3}m] [m\bar{3}m1'] [m' \bar{3}m], [m\bar{3}m'], [m' \bar{3}m']

References

First citation Kovalev, O. V. (1987). Representations of the crystallographic space groups. New York: Gordon and Breach.Google Scholar








































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