D33h E 2 C33 2 C33^2 2 C11 2 C33^4 2 C33^5 2 C11^2 2 C33^7 2 C33^8 2 C11^3 2 C33^10 2 C3 2 C11^4 2 C33^13 2 C33^14 2 C11^5 2 C33^16 33 C2' sh 2 S33 2 S11 2 S33^5 2 S33^7 2 S11^3 2 S3 2 S33^13 2 S11^5 2 S33^17 2 S33^19 2 S11^7 2 S33^23 2 S33^25 2 S11^9 2 S33^29 2 S33^31 33 sv <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> A1' 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 ... ... ....T ....... ........T ........... ............T A1" 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 ... ... ..... ....... ......... ........... ............. A2' 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 -1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 -1.00000 ..T ... ..... ....... ......... ........... ............. A2" 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 1.00000 ... ..T ..... ......T ......... ..........T ............. E1' 2.00000 1.96386 1.85674 1.68251 1.44747 1.16011 0.83083 0.47152 0.09516 -0.28463 -0.65414 -1.00000 -1.30972 -1.57211 -1.77767 -1.91899 -1.99094 0.00000 2.00000 1.96386 1.68251 1.16011 0.47152 -0.28463 -1.00000 -1.57211 -1.91899 -1.99094 -1.77767 -1.30972 -0.65414 0.09516 0.83083 1.44747 1.85674 0.00000 ... TT. ..... ....TT. ......... ........TT. ............. E1" 2.00000 1.96386 1.85674 1.68251 1.44747 1.16011 0.83083 0.47152 0.09516 -0.28463 -0.65414 -1.00000 -1.30972 -1.57211 -1.77767 -1.91899 -1.99094 0.00000 -2.00000 -1.96386 -1.68251 -1.16011 -0.47152 0.28463 1.00000 1.57211 1.91899 1.99094 1.77767 1.30972 0.65414 -0.09516 -0.83083 -1.44747 -1.85674 0.00000 TT. ... ..TT. ....... ......TT. ........... ..........TT. E2' 2.00000 1.85674 1.44747 0.83083 0.09516 -0.65414 -1.30972 -1.77767 -1.99094 -1.91899 -1.57211 -1.00000 -0.28463 0.47152 1.16011 1.68251 1.96386 0.00000 2.00000 1.85674 0.83083 -0.65414 -1.77767 -1.91899 -1.00000 0.47152 1.68251 1.96386 1.16011 -0.28463 -1.57211 -1.99094 -1.30972 0.09516 1.44747 0.00000 ... ... TT... ....... ....TT... ........... ........TT... E2" 2.00000 1.85674 1.44747 0.83083 0.09516 -0.65414 -1.30972 -1.77767 -1.99094 -1.91899 -1.57211 -1.00000 -0.28463 0.47152 1.16011 1.68251 1.96386 0.00000 -2.00000 -1.85674 -0.83083 0.65414 1.77767 1.91899 1.00000 -0.47152 -1.68251 -1.96386 -1.16011 0.28463 1.57211 1.99094 1.30972 -0.09516 -1.44747 0.00000 ... ... ..... ..TT... ......... ......TT... ............. E3' 2.00000 1.68251 0.83083 -0.28463 -1.30972 -1.91899 -1.91899 -1.30972 -0.28463 0.83083 1.68251 2.00000 1.68251 0.83083 -0.28463 -1.30972 -1.91899 0.00000 2.00000 1.68251 -0.28463 -1.91899 -1.30972 0.83083 2.00000 0.83083 -1.30972 -1.91899 -0.28463 1.68251 1.68251 -0.28463 -1.91899 -1.30972 0.83083 0.00000 ... ... ..... TT..... ......... ....TT..... ............. E3" 2.00000 1.68251 0.83083 -0.28463 -1.30972 -1.91899 -1.91899 -1.30972 -0.28463 0.83083 1.68251 2.00000 1.68251 0.83083 -0.28463 -1.30972 -1.91899 0.00000 -2.00000 -1.68251 0.28463 1.91899 1.30972 -0.83083 -2.00000 -0.83083 1.30972 1.91899 0.28463 -1.68251 -1.68251 0.28463 1.91899 1.30972 -0.83083 0.00000 ... ... ..... ....... ..TT..... ........... ......TT..... E4' 2.00000 1.44747 0.09516 -1.30972 -1.99094 -1.57211 -0.28463 1.16011 1.96386 1.68251 0.47152 -1.00000 -1.91899 -1.77767 -0.65414 0.83083 1.85674 0.00000 2.00000 1.44747 -1.30972 -1.57211 1.16011 1.68251 -1.00000 -1.77767 0.83083 1.85674 -0.65414 -1.91899 0.47152 1.96386 -0.28463 -1.99094 0.09516 0.00000 ... ... ..... ....... TT....... ........... ....TT....... E4" 2.00000 1.44747 0.09516 -1.30972 -1.99094 -1.57211 -0.28463 1.16011 1.96386 1.68251 0.47152 -1.00000 -1.91899 -1.77767 -0.65414 0.83083 1.85674 0.00000 -2.00000 -1.44747 1.30972 1.57211 -1.16011 -1.68251 1.00000 1.77767 -0.83083 -1.85674 0.65414 1.91899 -0.47152 -1.96386 0.28463 1.99094 -0.09516 0.00000 ... ... ..... ....... ......... ..TT....... ............. E5' 2.00000 1.16011 -0.65414 -1.91899 -1.57211 0.09516 1.68251 1.85674 0.47152 -1.30972 -1.99094 -1.00000 0.83083 1.96386 1.44747 -0.28463 -1.77767 0.00000 2.00000 1.16011 -1.91899 0.09516 1.85674 -1.30972 -1.00000 1.96386 -0.28463 -1.77767 1.44747 0.83083 -1.99094 0.47152 1.68251 -1.57211 -0.65414 0.00000 ... ... ..... ....... ......... TT......... ............. E5" 2.00000 1.16011 -0.65414 -1.91899 -1.57211 0.09516 1.68251 1.85674 0.47152 -1.30972 -1.99094 -1.00000 0.83083 1.96386 1.44747 -0.28463 -1.77767 0.00000 -2.00000 -1.16011 1.91899 -0.09516 -1.85674 1.30972 1.00000 -1.96386 0.28463 1.77767 -1.44747 -0.83083 1.99094 -0.47152 -1.68251 1.57211 0.65414 0.00000 ... ... ..... ....... ......... ........... ..TT......... E6' 2.00000 0.83083 -1.30972 -1.91899 -0.28463 1.68251 1.68251 -0.28463 -1.91899 -1.30972 0.83083 2.00000 0.83083 -1.30972 -1.91899 -0.28463 1.68251 0.00000 2.00000 0.83083 -1.91899 1.68251 -0.28463 -1.30972 2.00000 -1.30972 -0.28463 1.68251 -1.91899 0.83083 0.83083 -1.91899 1.68251 -0.28463 -1.30972 0.00000 ... ... ..... ....... ......... ........... TT........... E6" 2.00000 0.83083 -1.30972 -1.91899 -0.28463 1.68251 1.68251 -0.28463 -1.91899 -1.30972 0.83083 2.00000 0.83083 -1.30972 -1.91899 -0.28463 1.68251 0.00000 -2.00000 -0.83083 1.91899 -1.68251 0.28463 1.30972 -2.00000 1.30972 0.28463 -1.68251 1.91899 -0.83083 -0.83083 1.91899 -1.68251 0.28463 1.30972 0.00000 ... ... ..... ....... ......... ........... ............. E7' 2.00000 0.47152 -1.77767 -1.30972 1.16011 1.85674 -0.28463 -1.99094 -0.65414 1.68251 1.44747 -1.00000 -1.91899 0.09516 1.96386 0.83083 -1.57211 0.00000 2.00000 0.47152 -1.30972 1.85674 -1.99094 1.68251 -1.00000 0.09516 0.83083 -1.57211 1.96386 -1.91899 1.44747 -0.65414 -0.28463 1.16011 -1.77767 0.00000 ... ... ..... ....... ......... ........... ............. E7" 2.00000 0.47152 -1.77767 -1.30972 1.16011 1.85674 -0.28463 -1.99094 -0.65414 1.68251 1.44747 -1.00000 -1.91899 0.09516 1.96386 0.83083 -1.57211 0.00000 -2.00000 -0.47152 1.30972 -1.85674 1.99094 -1.68251 1.00000 -0.09516 -0.83083 1.57211 -1.96386 1.91899 -1.44747 0.65414 0.28463 -1.16011 1.77767 0.00000 ... ... ..... ....... ......... ........... ............. E8' 2.00000 0.09516 -1.99094 -0.28463 1.96386 0.47152 -1.91899 -0.65414 1.85674 0.83083 -1.77767 -1.00000 1.68251 1.16011 -1.57211 -1.30972 1.44747 0.00000 2.00000 0.09516 -0.28463 0.47152 -0.65414 0.83083 -1.00000 1.16011 -1.30972 1.44747 -1.57211 1.68251 -1.77767 1.85674 -1.91899 1.96386 -1.99094 0.00000 ... ... ..... ....... ......... ........... ............. E8" 2.00000 0.09516 -1.99094 -0.28463 1.96386 0.47152 -1.91899 -0.65414 1.85674 0.83083 -1.77767 -1.00000 1.68251 1.16011 -1.57211 -1.30972 1.44747 0.00000 -2.00000 -0.09516 0.28463 -0.47152 0.65414 -0.83083 1.00000 -1.16011 1.30972 -1.44747 1.57211 -1.68251 1.77767 -1.85674 1.91899 -1.96386 1.99094 0.00000 ... ... ..... ....... ......... ........... ............. E9' 2.00000 -0.28463 -1.91899 0.83083 1.68251 -1.30972 -1.30972 1.68251 0.83083 -1.91899 -0.28463 2.00000 -0.28463 -1.91899 0.83083 1.68251 -1.30972 0.00000 2.00000 -0.28463 0.83083 -1.30972 1.68251 -1.91899 2.00000 -1.91899 1.68251 -1.30972 0.83083 -0.28463 -0.28463 0.83083 -1.30972 1.68251 -1.91899 0.00000 ... ... ..... ....... ......... ........... ............. E9" 2.00000 -0.28463 -1.91899 0.83083 1.68251 -1.30972 -1.30972 1.68251 0.83083 -1.91899 -0.28463 2.00000 -0.28463 -1.91899 0.83083 1.68251 -1.30972 0.00000 -2.00000 0.28463 -0.83083 1.30972 -1.68251 1.91899 -2.00000 1.91899 -1.68251 1.30972 -0.83083 0.28463 0.28463 -0.83083 1.30972 -1.68251 1.91899 0.00000 ... ... ..... ....... ......... ........... ............. E10' 2.00000 -0.65414 -1.57211 1.68251 0.47152 -1.99094 0.83083 1.44747 -1.77767 -0.28463 1.96386 -1.00000 -1.30972 1.85674 0.09516 -1.91899 1.16011 0.00000 2.00000 -0.65414 1.68251 -1.99094 1.44747 -0.28463 -1.00000 1.85674 -1.91899 1.16011 0.09516 -1.30972 1.96386 -1.77767 0.83083 0.47152 -1.57211 0.00000 ... ... ..... ....... ......... ........... ............. E10" 2.00000 -0.65414 -1.57211 1.68251 0.47152 -1.99094 0.83083 1.44747 -1.77767 -0.28463 1.96386 -1.00000 -1.30972 1.85674 0.09516 -1.91899 1.16011 0.00000 -2.00000 0.65414 -1.68251 1.99094 -1.44747 0.28463 1.00000 -1.85674 1.91899 -1.16011 -0.09516 1.30972 -1.96386 1.77767 -0.83083 -0.47152 1.57211 0.00000 ... ... ..... ....... ......... ........... ............. E11' 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 0.00000 2.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 0.00000 ... ... ..... ....... ......... ........... ............. E11" 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 0.00000 -2.00000 1.00000 -2.00000 1.00000 1.00000 -2.00000 1.00000 1.00000 -2.00000 1.00000 1.00000 -2.00000 1.00000 1.00000 -2.00000 1.00000 1.00000 0.00000 ... ... ..... ....... ......... ........... ............. E12' 2.00000 -1.30972 -0.28463 1.68251 -1.91899 0.83083 0.83083 -1.91899 1.68251 -0.28463 -1.30972 2.00000 -1.30972 -0.28463 1.68251 -1.91899 0.83083 0.00000 2.00000 -1.30972 1.68251 0.83083 -1.91899 -0.28463 2.00000 -0.28463 -1.91899 0.83083 1.68251 -1.30972 -1.30972 1.68251 0.83083 -1.91899 -0.28463 0.00000 ... ... ..... ....... ......... ........... ............. E12" 2.00000 -1.30972 -0.28463 1.68251 -1.91899 0.83083 0.83083 -1.91899 1.68251 -0.28463 -1.30972 2.00000 -1.30972 -0.28463 1.68251 -1.91899 0.83083 0.00000 -2.00000 1.30972 -1.68251 -0.83083 1.91899 0.28463 -2.00000 0.28463 1.91899 -0.83083 -1.68251 1.30972 1.30972 -1.68251 -0.83083 1.91899 0.28463 0.00000 ... ... ..... ....... ......... ........... ............. E13' 2.00000 -1.57211 0.47152 0.83083 -1.77767 1.96386 -1.30972 0.09516 1.16011 -1.91899 1.85674 -1.00000 -0.28463 1.44747 -1.99094 1.68251 -0.65414 0.00000 2.00000 -1.57211 0.83083 1.96386 0.09516 -1.91899 -1.00000 1.44747 1.68251 -0.65414 -1.99094 -0.28463 1.85674 1.16011 -1.30972 -1.77767 0.47152 0.00000 ... ... ..... ....... ......... ........... ............. E13" 2.00000 -1.57211 0.47152 0.83083 -1.77767 1.96386 -1.30972 0.09516 1.16011 -1.91899 1.85674 -1.00000 -0.28463 1.44747 -1.99094 1.68251 -0.65414 0.00000 -2.00000 1.57211 -0.83083 -1.96386 -0.09516 1.91899 1.00000 -1.44747 -1.68251 0.65414 1.99094 0.28463 -1.85674 -1.16011 1.30972 1.77767 -0.47152 0.00000 ... ... ..... ....... ......... ........... ............. E14' 2.00000 -1.77767 1.16011 -0.28463 -0.65414 1.44747 -1.91899 1.96386 -1.57211 0.83083 0.09516 -1.00000 1.68251 -1.99094 1.85674 -1.30972 0.47152 0.00000 2.00000 -1.77767 -0.28463 1.44747 1.96386 0.83083 -1.00000 -1.99094 -1.30972 0.47152 1.85674 1.68251 0.09516 -1.57211 -1.91899 -0.65414 1.16011 0.00000 ... ... ..... ....... ......... ........... ............. E14" 2.00000 -1.77767 1.16011 -0.28463 -0.65414 1.44747 -1.91899 1.96386 -1.57211 0.83083 0.09516 -1.00000 1.68251 -1.99094 1.85674 -1.30972 0.47152 0.00000 -2.00000 1.77767 0.28463 -1.44747 -1.96386 -0.83083 1.00000 1.99094 1.30972 -0.47152 -1.85674 -1.68251 -0.09516 1.57211 1.91899 0.65414 -1.16011 0.00000 ... ... ..... ....... ......... ........... ............. E15' 2.00000 -1.91899 1.68251 -1.30972 0.83083 -0.28463 -0.28463 0.83083 -1.30972 1.68251 -1.91899 2.00000 -1.91899 1.68251 -1.30972 0.83083 -0.28463 0.00000 2.00000 -1.91899 -1.30972 -0.28463 0.83083 1.68251 2.00000 1.68251 0.83083 -0.28463 -1.30972 -1.91899 -1.91899 -1.30972 -0.28463 0.83083 1.68251 0.00000 ... ... ..... ....... ......... ........... ............. E15" 2.00000 -1.91899 1.68251 -1.30972 0.83083 -0.28463 -0.28463 0.83083 -1.30972 1.68251 -1.91899 2.00000 -1.91899 1.68251 -1.30972 0.83083 -0.28463 0.00000 -2.00000 1.91899 1.30972 0.28463 -0.83083 -1.68251 -2.00000 -1.68251 -0.83083 0.28463 1.30972 1.91899 1.91899 1.30972 0.28463 -0.83083 -1.68251 0.00000 ... ... ..... ....... ......... ........... ............. E16' 2.00000 -1.99094 1.96386 -1.91899 1.85674 -1.77767 1.68251 -1.57211 1.44747 -1.30972 1.16011 -1.00000 0.83083 -0.65414 0.47152 -0.28463 0.09516 0.00000 2.00000 -1.99094 -1.91899 -1.77767 -1.57211 -1.30972 -1.00000 -0.65414 -0.28463 0.09516 0.47152 0.83083 1.16011 1.44747 1.68251 1.85674 1.96386 0.00000 ... ... ..... ....... ......... ........... ............. E16" 2.00000 -1.99094 1.96386 -1.91899 1.85674 -1.77767 1.68251 -1.57211 1.44747 -1.30972 1.16011 -1.00000 0.83083 -0.65414 0.47152 -0.28463 0.09516 0.00000 -2.00000 1.99094 1.91899 1.77767 1.57211 1.30972 1.00000 0.65414 0.28463 -0.09516 -0.47152 -0.83083 -1.16011 -1.44747 -1.68251 -1.85674 -1.96386 0.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.990943845146 = 2*cos(2*π/66) = 2*cos(π/33) 1.963857394525 = 2*cos(4*π/66) = 2*cos(2*π/33) 1.918985947229 = 2*cos(6*π/66) = 2*cos(π/11) 1.856735866032 = 2*cos(8*π/66) = 2*cos(4*π/33) 1.777670897310 = 2*cos(10*π/66) = 2*cos(5*π/33) 1.682507065662 = 2*cos(12*π/66) = 2*cos(2*π/11) 1.572106189486 = 2*cos(14*π/66) = 2*cos(7*π/33) 1.447468076210 = 2*cos(16*π/66) = 2*cos(8*π/33) 1.309721467891 = 2*cos(18*π/66) = 2*cos(3*π/11) 1.160113819142 = 2*cos(20*π/66) = 2*cos(10*π/33) 0.830830026004 = 2*cos(24*π/66) = 2*cos(4*π/11) 0.654135926635 = 2*cos(26*π/66) = 2*cos(13*π/33) 0.471517871019 = 2*cos(28*π/66) = 2*cos(14*π/33) 0.284629676547 = 2*cos(30*π/66) = 2*cos(5*π/11) 0.095163831647 = 2*cos(32*π/66) = 2*cos(16*π/33) Symmetry of Rotations and Cartesian products A1' d+g+i+k+m z2, z4, z6 A2' R Rz A2" p+f+h+j+l z, z3, z5 E1' p+f+h+j+l {x, y}, {xz2, yz2}, {xz4, yz4} E1" R+d+g+i+k+m {Rx, Ry}, {xz, yz}, {xz3, yz3}, {xz5, yz5} E2' d+g+i+k+m {x2−y2, xy}, {z2(x2−y2), xyz2}, {z4(x2−y2), xyz4} E2" f+h+j+l {z(x2−y2), xyz}, {z3(x2−y2), xyz3} E3' f+h+j+l {x(x2−3y2), y(3x2−y2)}, {xz2(x2−3y2), yz2(3x2−y2)} E3" g+i+k+m {xz(x2−3y2), yz(3x2−y2)}, {xz3(x2−3y2), yz3(3x2−y2)} E4' g+i+k+m {(x2−y2)2−4x2y2, xy(x2−y2)}, {z2((x2−y2)2−4x2y2), xyz2(x2−y2)} E4" h+j+l {z((x2−y2)2−4x2y2), xyz(x2−y2)} E5' h+j+l {x(x2−(5+2√5)y2)(x2−(5−2√5)y2), y((5+2√5)x2−y2)((5−2√5)x2−y2)} E5" i+k+m {xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), yz((5+2√5)x2−y2)((5−2√5)x2−y2)} E6' i+k+m {x2(x2−3y2)2−y2(3x2−y2)2, xy(x2−3y2)(3x2−y2)} E6" j+l E7' j+l E7" k+m E8' k+m E8" l E9' l E9" m E10' m Notes: α The order of the D33h point group is 132, and the order of the principal axis (S33) is 66. The group has 36 irreducible representations. β The D33h point group is isomorphic to D33d, C66v and D66. γ The D33h point group is generated by two symmetry elements, S33 and either a perpendicular C2′ or a vertical σv. Also, the group may be generated from two σv planes, or a σv and a C2′ (some pairs will yield smaller groups, though; choosing a minimum angle is safe). The canonical choice, however, is to use redundant generators: C33, C2′ and σh. δ The group contains one set of C2′ symmetry axes perpendicular to the principal (z) axis. The x axis (but not the y axis) is a member of that set. Similarly, the single set of symmetry planes denoted σd contains the xz plane but not the yz plane. ε The lowest nonvanishing multipole moment in D33h is 4 (quadrupole moment). ζ This point group is non-Abelian (some symmetry operations are not commutative). Therefore, the character table contains multi-membered classes and degenerate irreducible representations. η Some of the characters in the table are irrational because the order of the principal axis is neither 1,2,3,4 nor 6. These irrational values can be expressed as cosine values, or as solutions of algebraic equations with a leading coefficient of 1. All characters are algebraic integers of a degree much less than half the order of the principal axis. For this group, however, none of the irrational characters can be expressed by a closed algebraic form using real numbers only.
D31h | ||
D32h | ||
C33 C33v C33h D33 | D33h | D33d |
D34h | ||
D35h |
This Character Table for the D33h point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.