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

International Tables for Crystallography (2006). Vol. D, ch. 1.2, pp. 58-60

Table 1.2.6.5 

T. Janssena*

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

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}]
n1[1][1][1]
Order1[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}]
n1[1][1][1][1][1]
Order1[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]
n1[1][1][1]
Order1[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]
n1[2][1][2][2]
Order1[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]
n1[2][2][1][3][3]
Order1[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]
n144[3]
Order133[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]
n1[8][3][6][6]
Order1[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).

GroupIsomorphism classRotation 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