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. 63-64

Table 1.2.6.10 

T. Janssena*

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

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

(a) [D_{2}]

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

(b) [D_{4}]

[{A}^{4}=-{E},{B}^{2}=({AB})^{2}={E}]
ElementsEA2[A]A3BA2BABA3B
[ \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}]
ElementsE[A^{2}][A^{4}]BA2BA4BA3[A][A^{5}]ABA3BA5B
[ \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 }]
ElementsEA[A^{2}]A3BABA2BA3B
[ \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}]
ElementsEAA2A3A4A5BABA2BA3BA4BA5B
[ \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}]
ElementsE[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]