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

International Tables for Crystallography (2006). Vol. D, ch. 1.10, pp. 256-257

Table 1.10.5.1 

T. Janssena*

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

Table 1.10.5.1| top | pdf |
Character tables of some point groups for quasicrystals

(a) [C_5] [[\omega =\exp (2\pi i/5)]].

[C_{5}][\varepsilon][\alpha][\alpha^{2}][\alpha^{3}][\alpha^{4}]
n11111
Order 15555
[\Gamma_{1}] 1 1 1 1 1
[\Gamma_{2}] 1 [\omega] [\omega^{2}] [\omega^{3}] [\omega^{4}]
[\Gamma_{3}] 1 [\omega^{2}] [\omega^{4}] [\omega] [\omega^{3}]
[\Gamma_{4}] 1 [\omega^{3}] [\omega] [\omega^{4}] [\omega^{2}]
[\Gamma_{5}] 1 [\omega^{4}] [\omega^{3}] [\omega^{2}] [\omega]

 Generators Vector representationPerpendicular representation
5 [\alpha = C_{5z}] [\Gamma_{1}\oplus\Gamma_{2}\oplus\Gamma_{5}] [\Gamma_{3}\oplus\Gamma_{4}]

(b) [D_5] [[\tau = (\sqrt{5}-1)/2]].

[D_{5}][\varepsilon][\alpha][\alpha^{2}] [\beta]
n1225
Order 1 5 5 2
[\Gamma_{1}] 1 1 1 1
[\Gamma_{2}] 1 1 1 [-1]
[\Gamma_{3}] 2 [\tau] [-1-\tau] 0
[\Gamma_{4}] 2 [-1-\tau] [\tau] 0

 Generators Vector representationPerpendicular representation
52 [\alpha = C_{5z}] [\Gamma_{2}\oplus \Gamma_{3}] [\Gamma_{4}]
  [\beta = C_{2x}]    
5m [\alpha = C_{5z}] [\Gamma_{1}\oplus \Gamma_{3}] [\Gamma_{4}]
  [\beta = m_{x}]    
[\bar{5}m] [\sim 52 \times {\bb Z}_{2}] [\Gamma_{1u}\oplus\Gamma_{3u}] [\Gamma_{4u}]

(c) [C_8] [[\omega =\exp (\pi i/4) = (1+i)/\sqrt{2}]].

[C_{8}][\varepsilon][\alpha][\alpha^{2}][\alpha^{3}][\alpha^{4}][\alpha^{5}][\alpha^{6}][\alpha^{7}]
[ n]11111111
Order 18482868
[\Gamma_{1}] 1 1 1 1 1 1 1 1
[\Gamma_{2}] 1 [\omega] i [\omega^{3}] [-1] [\omega^{5}] [-i] [\omega^{7}]
[\Gamma_{3}] 1 i [-1] [-i] [1] i [-1] [-i]
[\Gamma_{4}] 1 [\omega^{3}] [-i] [\omega] [-1] [\omega^{7}] i [\omega^{5}]
[\Gamma_{5}] 1 [-1] [1] [-1] [1] [-1] [1] [-1]
[\Gamma_{6}] 1 [\omega^{5}] i [\omega^{7}] [-1] [\omega] [-i] [\omega^{3}]
[\Gamma_{7}] 1 [-i] [-1] i [1] [-i] [-1] i
[\Gamma_{8}] 1 [\omega^{7}] [-i] [\omega^{5}] [-1] [\omega^{3}] i [\omega]

 Generators Vector representationPerpendicular representation
8 [\alpha = C_{8z}] [\Gamma_{1}\oplus\Gamma_{2}\oplus\Gamma_{8}] [\Gamma_{4}\oplus\Gamma_{6}]
[\bar{8}] [\alpha =S_{8z}] [\Gamma_4\oplus\Gamma_5\oplus\Gamma_6] [\Gamma_2\oplus\Gamma_8]
[8/m] [\sim 8\times {\bb Z}_2] [\Gamma_{1u}\oplus\Gamma_{2u}\oplus\Gamma_{8u}] [\Gamma_{4u}\oplus\Gamma_{6u}]

(d) [D_8]

[D_{8}][\varepsilon][\alpha] [\alpha^{2}][\alpha^{3}][\alpha^{4}][\beta][\alpha \beta]
[ n]1222144
Order 1 8 4 8 2 2 2
[\Gamma_{1}] 1 1 1 1 1 1 1
[\Gamma_{2}] 1 1 1 1 1 [-1] [-1]
[\Gamma_{3}] 1 [-1] 1 [-1 ] 1 1 [-1]
[\Gamma_{4}] 1 [-1 ] 1 [-1] 1 [-1] 1
[\Gamma_{5}] 2 [\sqrt{2}] 0 [-\sqrt{2}] [-2] 0 0
[\Gamma_{6}] 2 0 [-2 ] 0 2 0 0
[\Gamma_{7}] 2 [-\sqrt{2}] 0 [\sqrt{2}] [-2] 0 0

 GeneratorsVector representationPerpendicular representation
822 [\alpha = C_{8z}] [\Gamma_{2}\oplus\Gamma_{5}] [\Gamma_{7}]
  [\beta = C_{2x}]    
[8mm] [\alpha = C_{8z}] [\Gamma_{1}\oplus\Gamma_{5}] [\Gamma_{7}]
  [\beta = m_{x}]    
[\bar{8}2m] [\alpha = S_{8z}] [\Gamma_{3}\oplus\Gamma_{7}] [\Gamma_{5}]
  [\beta = C_{2x}]    
[8/mmm] [\sim 822\times{\bb Z}_{2}] [\Gamma_{2u}\oplus\Gamma_{5u}] [\Gamma_{7u}]

(e) [C_{10}] [[\omega=exp(2\pi i/5) ]].

[C_{10}][ \varepsilon][\alpha^2][\alpha^4][\alpha^6][\alpha^8]
n11111
Order 15555
[\Gamma_1] 1 1 1 1 1
[\Gamma_2] 1 [\omega] [\omega^2] [\omega^3] [\omega^4]
[\Gamma_3] 1 [\omega^2] [\omega^4] [\omega] [\omega^3]
[\Gamma_4] 1 [\omega^3] [\omega] [\omega^4] [\omega^2]
[\Gamma_5] 1 [\omega^4] [\omega^3] [\omega^2] [\omega]
[\Gamma_6] 1 1 1 1 1
[\Gamma_7] 1 [\omega] [\omega^2] [\omega^3] [\omega^4]
[\Gamma_8] 1 [\omega^2] [\omega^4] [\omega] [\omega^3]
[\Gamma_9] 1 [\omega^3] [\omega] [\omega^4] [\omega^2]
[\Gamma_{10}] 1 [\omega^4] [\omega^3] [\omega^2] [\omega]

[C_{10}][\alpha^5][\alpha^7][\alpha^9][\alpha][\alpha^3]
n11111
Order 210101010
[\Gamma_1] 1 1 1 1 1
[\Gamma_2] 1 [\omega] [\omega^2] [\omega^3] [\omega^4]
[\Gamma_3] 1 [\omega^2] [\omega^4] [\omega] [\omega^3]
[\Gamma_4] 1 [\omega^3] [\omega] [\omega^4] [\omega^2]
[\Gamma_5] 1 [\omega^4] [\omega^3] [\omega^2] [\omega]
[\Gamma_6] [-1] [-1] [-1] [-1] [-1]
[\Gamma_7] [-1] [-\omega] [-\omega^2] [-\omega^3] [-\omega^4]
[\Gamma_8] [-1] [-\omega^2] [-\omega^4] [-\omega] [-\omega^3]
[\Gamma_9] [-1] [-\omega^3] [-\omega] [-\omega^4] [-\omega^2]
[\Gamma_{10}] [-1] [-\omega^4] [-\omega^3] [-\omega^2] [-\omega]

 Generators Vector representationPerpendicular representation
10 [\alpha = C_{10z}] [\Gamma_{1}\oplus\Gamma_{7}\oplus\Gamma_{10}] [\Gamma_{8}\oplus\Gamma_{9}]
[{\bar{5}}] [\alpha = S_{5z}] [\Gamma_6\oplus\Gamma_8\oplus\Gamma_9] [\Gamma_7\oplus\Gamma_{10}]
[\overline{10}] [\alpha =S_{10z}] [\Gamma_2\oplus\Gamma_4\oplus\Gamma_6] [\Gamma_3\oplus\Gamma_5]
[10/m ] [\sim 10\times {\bb Z}_2] [\Gamma_{1u}\oplus\Gamma_{7u}\oplus\Gamma_{10u}] [\Gamma_{8u}\oplus\Gamma_{9u}]

(f) [D_{10}] [[\tau = (\sqrt{5}-1)/2]].

[D_{10}] [\varepsilon] [\alpha] [\alpha^{2}] [\alpha^{3}]
n1222
Order 1 10 5 10
[\Gamma_{1}] 1 1 1 1
[\Gamma_{2}] 1 1 1 1
[\Gamma_{3}] 1 [-1 ] 1 [-1]
[\Gamma_{4}] 1 [-1] 1 [-1]
[\Gamma_{5}] 2 [1+\tau] [\tau] [-\tau]
[\Gamma_{6}] 2 [\tau] [-1-\tau] [-1-\tau]
[\Gamma_{7}] 2 [-\tau] [-1-\tau] [1+\tau]
[\Gamma_{8}] 2 [-1-\tau] [\tau] [\tau]

[D_{10}][\alpha^{4}] [\alpha^{5}] [\beta] [\alpha \beta]
n2155
Order 5 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
[\Gamma_{5}] [-1-\tau] [-2 ] 0 0
[\Gamma_{6}] [\tau] 2 0 0
[\Gamma_{7}] [\tau ] [-2 ] 0 0
[\Gamma_{8}] [-1-\tau] 2 0 0

 GeneratorsVector representationPerpendicular representation
[10\,22] [\alpha = C_{10z}] [\Gamma_{2}\oplus\Gamma_{5}] [\Gamma_7]
  [\beta = C_{2x}]    
[10\,mm] [\alpha = C_{10z}] [\Gamma_{1}\oplus\Gamma_{5}] [\Gamma_7]
  [\beta = m_{x}]    
[\overline{10}\,2m] [\alpha = S_{10z}] [\Gamma_{4}\oplus\Gamma_{8}] [\Gamma_6]
  [\beta = C_{2x}]    
[\bar{5}m] [\alpha = S_{5z}] [\Gamma_{3}\oplus\Gamma_{7}] [\Gamma_5]
  [\beta = C_{2x}]    
[10/mmm] [\sim 10\,22\times{\bb Z}_{2}] [\Gamma_{2u}\oplus\Gamma_{5u}] [\Gamma_{7u}]

(g) [C_{12}] [[\omega = \exp (\pi i/6)]].

[C_{12}][\varepsilon] [\alpha][\alpha^{2}][\alpha^{3}][\alpha^4][\alpha^{5}]
n111111
Order11264312
[\Gamma_{1}] 1 1 1 1 1 1
[\Gamma_{2}] 1 [\omega] [\omega^2] i [\omega^4] [\omega^5]
[\Gamma_{3}] 1 [\omega^2] [\omega^4] [-1] [-\omega^2] [-\omega^4]
[\Gamma_4] 1 i [-1] [-i] 1 i
[\Gamma_{5}] 1 [\omega^4] [-\omega^2] [1] [\omega^4] [-\omega^2]
[\Gamma_{6}] 1 [\omega^5] [-\omega^4] i [-\omega^2] [\omega]
[\Gamma_{7}] 1 [-1] 1 [-1] 1 [-1]
[\Gamma_{8}] 1 [-\omega] [\omega^2] [-i] [\omega^4] [-\omega^5]
[\Gamma_{9}] 1 [-\omega^2] [\omega^4] [1] [-\omega^2] [\omega^4]
[\Gamma_{10}] 1 [-i] [-1] i 1 [-i]
[\Gamma_{11}] 1 [-\omega^4] [-\omega^2] [-1] [\omega^4] [\omega^2]
[\Gamma_{12}] 1 [-\omega^5] [-\omega^4] [-i] [-\omega^2] [-\omega]

[C_{12}][\alpha^{6}][\alpha^{7}][\alpha^{8}][\alpha^{9}][\alpha^{10}][\alpha^{11}]
n111111
Order21234612
[\Gamma_{1}] 1 1 1 1 1 1
[\Gamma_{2}] [-1] [-\omega] [-\omega^2] [-i] [-\omega^4] [-\omega^5]
[\Gamma_{3}] 1 [\omega^2] [\omega^4] [-1] [-\omega^2] [-\omega^4]
[\Gamma_4] [-1] [-i] 1 i [-1] [-i]
[\Gamma_{5}] 1 [\omega^4] [-\omega^2] [1] [\omega^4] [-\omega^2]
[\Gamma_{6}] [-1] [-\omega^5] [\omega^4] [-i] [\omega^2] [-\omega]
[\Gamma_{7}] 1 [-1] 1 [-1] 1 [-1]
[\Gamma_{8}] [-1] [\omega] [-\omega^2] i [-\omega^4] [\omega^5]
[\Gamma_{9}] 1 [-\omega^2] [\omega^4] [1] [-\omega^2] [\omega^4]
[\Gamma_{10}] [-1] i 1 [-i] [-1] i
[\Gamma_{11}] 1 [-\omega^4] [-\omega^2] [-1] [\omega^4] [\omega^2]
[\Gamma_{12}] [-1] [\omega^5] [\omega^4] i [\omega^2] [\omega]

 Generators Vector representationPerpendicular representation
12 [\alpha=C_{12z}] [\Gamma_{1}\oplus\Gamma_{2}\oplus\Gamma_{12}] [\Gamma_{6}\oplus\Gamma_{8}]
[\overline{12}] [\alpha=S_{12z}] [\Gamma_6\oplus\Gamma_7\oplus\Gamma_8] [\Gamma_2\oplus\Gamma_{12}]
[12/m ] [\sim 12\times {\bb Z}_2] [\Gamma_{1u}\oplus\Gamma_{2u}\oplus\Gamma_{12u}] [\Gamma_{6u}\oplus\Gamma_{8u}]

(h) [D_{12}]

[D_{12}][\varepsilon] [\alpha] [\alpha^{2}] [\alpha^{3}]
n1222
Order 1 12 6 4
[\Gamma_{1}] 1 1 1 1
[\Gamma_{2}] 1 1 1 1
[\Gamma_{3}] 1 [-1] 1 [-1 ]
[\Gamma_{4}] 1 [-1] 1 [-1]
[\Gamma_{5}] 2 [\sqrt{3}] 1 0
[\Gamma_{6}] 2 1 [-1 ] [-2]
[\Gamma_{7}] 2 0 [-2] 0
[\Gamma_{8}] 2 [-1] [-1 ] 2
[\Gamma_{9}] 2 [-\sqrt{3}] 1 0

[D_{12}][\alpha^{4}] [\alpha^{5}] [\alpha^{6}] [\beta] [\alpha \beta]
n22166
Order 3 12 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}] [-1] [-\sqrt{3}] [-2 ] 0 0
[\Gamma_{6}] [-1] 1 2 0 0
[\Gamma_{7}] 2 0 [-2 ] 0 0
[\Gamma_{8}] [-1 ] [-1 ] 2 0 0
[\Gamma_{9}] [-1 ] [\sqrt{3}] [-2] 0 0

 GeneratorsVector representationPerpendicular representation
[12 \,22] [\alpha = C_{12z}] [\Gamma_{2}\oplus\Gamma_{5}] [\Gamma_9]
  [\beta = C_{2x}]    
[12\,mm] [\alpha = C_{12z}] [\Gamma_{1}\oplus\Gamma_{5}] [\Gamma_{9}]
  [\beta = m_{x}]    
[\overline{12}\,2m] [\alpha = S_{12z}] [\Gamma_{4}\oplus\Gamma_{9}] [\Gamma_5]
  [\beta = C_{2x}]    
[12/mmm] [\sim 12 \,22\times{\bb Z}_{2}] [\Gamma_{2u}\oplus\Gamma_{5u}] [\Gamma_{9u}]

(i) I [[\tau = (\sqrt 5 -1)/2]].

I[\varepsilon] [\alpha] [\alpha^{2}] [\beta] [\alpha \beta]
n112122015
Order 1 5 5 3 2
[\Gamma_{1}] 1 1 1 1 1
[\Gamma_{2}] 3 [1+\tau] [-\tau] 0 [-1]
[\Gamma_{3}] 3 [-\tau] [1+\tau] 0 [-1]
[\Gamma_{4}] 4 [-1] [-1] 1 0
[\Gamma_{5}] 5 0 0 [-1] 1

 GeneratorsVector representationPerpendicular representation
532 [\alpha =C_{5}] [\Gamma_2] [\Gamma_3]
  [\beta =C_{3d}]    
[\bar{5}\bar{3}m] [\sim 532 \times{\bb Z}_{2}] [\Gamma_{2u}] [\Gamma_{3u}]