D66 E 2 C66 2 C33 2 C22 2 C33^2 2 C66^5 2 C11 2 C66^7 2 C33^4 2 C22^3 2 C33^5 2 C6 2 C11^2 2 C66^13 2 C33^7 2 C22^5 2 C33^8 2 C66^17 2 C11^3 2 C66^19 2 C33^10 2 C22^7 2 C3 2 C66^23 2 C11^4 2 C66^25 2 C33^13 2 C22^9 2 C33^14 2 C66^29 2 C11^5 2 C66^31 2 C33^16 C2 33 C2' 33 C2" <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 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 ......... ..........T ............. B1 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 ... ... ..... ....... ......... ........... ............. B2 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 ... ... ..... ....... ......... ........... ............. E1 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 -0.09516 -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 -2.00000 0.00000 0.00000 TT. TT. ..TT. ....TT. ......TT. ........TT. ..........TT. E2 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 -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 2.00000 0.00000 0.00000 ... ... TT... ..TT... ....TT... ......TT... ........TT... E3 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 0.28463 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 -2.00000 0.00000 0.00000 ... ... ..... TT..... ..TT..... ....TT..... ......TT..... E4 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 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 2.00000 0.00000 0.00000 ... ... ..... ....... TT....... ..TT....... ....TT....... E5 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 -0.47152 -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 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... TT......... ..TT......... E6 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 -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 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... TT........... E7 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 0.65414 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 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E8 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 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 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E9 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 -0.83083 -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 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E10 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 -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 2.00000 0.00000 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 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 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E12 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 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 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E13 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 -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 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E14 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 -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 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E15 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 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 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E16 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 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 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E17 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 -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 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E18 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 -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 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E19 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 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 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E20 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 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 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E21 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 -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 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E22 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 -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 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E23 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 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 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E24 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 0.83083 -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 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E25 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 -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 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E26 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 -0.65414 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 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E27 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 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 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E28 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 0.47152 -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 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E29 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 -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 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E30 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 -0.28463 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 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E31 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 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 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E32 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 0.09516 -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 2.00000 0.00000 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+p+f+h+j+l Rz, z, z3, z5 E1 R+p+d+f+g+h+i+j+k+l+m {Rx, Ry}, {x, y}, {xz, yz}, {xz2, yz2}, {xz3, yz3}, {xz4, yz4}, {xz5, yz5} E2 d+f+g+h+i+j+k+l+m {x2−y2, xy}, {z(x2−y2), xyz}, {z2(x2−y2), xyz2}, {z3(x2−y2), xyz3}, {z4(x2−y2), xyz4} E3 f+g+h+i+j+k+l+m {x(x2−3y2), y(3x2−y2)}, {xz(x2−3y2), yz(3x2−y2)}, {xz2(x2−3y2), yz2(3x2−y2)}, {xz3(x2−3y2), yz3(3x2−y2)} E4 g+h+i+j+k+l+m {(x2−y2)2−4x2y2, xy(x2−y2)}, {z((x2−y2)2−4x2y2), xyz(x2−y2)}, {z2((x2−y2)2−4x2y2), xyz2(x2−y2)} E5 h+i+j+k+l+m {x(x2−(5+2√5)y2)(x2−(5−2√5)y2), y((5+2√5)x2−y2)((5−2√5)x2−y2)}, {xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), yz((5+2√5)x2−y2)((5−2√5)x2−y2)} E6 i+j+k+l+m {x2(x2−3y2)2−y2(3x2−y2)2, xy(x2−3y2)(3x2−y2)} E7 j+k+l+m E8 k+l+m E9 l+m E10 m Notes: α The order of the D66 point group is 132, and the order of the principal axis (C66) is 66. The group has 36 irreducible representations. β The D66 point group is isomorphic to D33d, D33h and C66v. γ The D66 point group is generated by two symmetry elements, C66 and a perpendicular C2′ (or, non-canonically, C2″). Also, the group may be generated from a C2′ plus a C2″ (some pairs will yield smaller groups, though; choosing a minimum angle is safe). δ There are two different sets of twofold symmetry axes perpendicular to the principal axis (z axis in standard orientation). By convention, the set denoted as C2′ has the x axis as a member, while the y axis is a member of the C2″ set. ε The lowest nonvanishing multipole moment in D66 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 The point group is chiral, as it does not contain any mirroring operation. θ 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.
D64 | ||
D65 | ||
C66 C66v C66h | D66 | D66h D66d S66 |
D67 | ||
D68 |
This Character Table for the D66 point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.