Character Table for Point Group C120

Character table for the C₁₂₀ point group

C120 E 2 C120 2 C60 2 C40 2 C30 2 C24 2 C20 2 C120^7 2 C15 2 C40^3 2 C12 2 C120^11 2 C10 2 C120^13 2 C60^7 2 C8 2 C15^2 2 C120^17 2 C20^3 2 C120^19 2 C6 2 C40^7 2 C60^11 2 C120^23 2 C5 2 C24^5 2 C60^13 2 C40^9 2 C30^7 2 C120^29 2 C4 2 C120^31 2 C15^4 2 C40^11 2 C60^17 2 C24^7 2 C10^3 2 C120^37 2 C60^19 2 C40^13 2 C3 2 C120^41 2 C20^7 2 C120^43 2 C30^11 2 C8^3 2 C60^23 2 C120^47 2 C5^2 2 C120^49 2 C12^5 2 C40^17 2 C30^13 2 C120^53 2 C20^9 2 C24^11 2 C15^7 2 C40^19 2 C60^29 2 C120^59 C2 <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> A 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 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 ........T ..........T ............T B 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 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 1.99726 1.98904 1.97538 1.95630 1.93185 1.90211 1.86716 1.82709 1.78201 1.73205 1.67734 1.61803 1.55429 1.48629 1.41421 1.33826 1.25864 1.17557 1.08928 1.00000 0.90798 0.81347 0.71674 0.61803 0.51764 0.41582 0.31287 0.20906 0.10467 0.00000 -0.10467 -0.20906 -0.31287 -0.41582 -0.51764 -0.61803 -0.71674 -0.81347 -0.90798 -1.00000 -1.08928 -1.17557 -1.25864 -1.33826 -1.41421 -1.48629 -1.55429 -1.61803 -1.67734 -1.73205 -1.78201 -1.82709 -1.86716 -1.90211 -1.93185 -1.95630 -1.97538 -1.98904 -1.99726 -2.00000 TT. TT. ..TT. ....TT. ......TT. ........TT. ..........TT. E2 * 2.00000 1.98904 1.95630 1.90211 1.82709 1.73205 1.61803 1.48629 1.33826 1.17557 1.00000 0.81347 0.61803 0.41582 0.20906 0.00000 -0.20906 -0.41582 -0.61803 -0.81347 -1.00000 -1.17557 -1.33826 -1.48629 -1.61803 -1.73205 -1.82709 -1.90211 -1.95630 -1.98904 -2.00000 -1.98904 -1.95630 -1.90211 -1.82709 -1.73205 -1.61803 -1.48629 -1.33826 -1.17557 -1.00000 -0.81347 -0.61803 -0.41582 -0.20906 0.00000 0.20906 0.41582 0.61803 0.81347 1.00000 1.17557 1.33826 1.48629 1.61803 1.73205 1.82709 1.90211 1.95630 1.98904 2.00000 ... ... TT... ..TT... ....TT... ......TT... ........TT... E3 * 2.00000 1.97538 1.90211 1.78201 1.61803 1.41421 1.17557 0.90798 0.61803 0.31287 0.00000 -0.31287 -0.61803 -0.90798 -1.17557 -1.41421 -1.61803 -1.78201 -1.90211 -1.97538 -2.00000 -1.97538 -1.90211 -1.78201 -1.61803 -1.41421 -1.17557 -0.90798 -0.61803 -0.31287 0.00000 0.31287 0.61803 0.90798 1.17557 1.41421 1.61803 1.78201 1.90211 1.97538 2.00000 1.97538 1.90211 1.78201 1.61803 1.41421 1.17557 0.90798 0.61803 0.31287 0.00000 -0.31287 -0.61803 -0.90798 -1.17557 -1.41421 -1.61803 -1.78201 -1.90211 -1.97538 -2.00000 ... ... ..... TT..... ..TT..... ....TT..... ......TT..... E4 * 2.00000 1.95630 1.82709 1.61803 1.33826 1.00000 0.61803 0.20906 -0.20906 -0.61803 -1.00000 -1.33826 -1.61803 -1.82709 -1.95630 -2.00000 -1.95630 -1.82709 -1.61803 -1.33826 -1.00000 -0.61803 -0.20906 0.20906 0.61803 1.00000 1.33826 1.61803 1.82709 1.95630 2.00000 1.95630 1.82709 1.61803 1.33826 1.00000 0.61803 0.20906 -0.20906 -0.61803 -1.00000 -1.33826 -1.61803 -1.82709 -1.95630 -2.00000 -1.95630 -1.82709 -1.61803 -1.33826 -1.00000 -0.61803 -0.20906 0.20906 0.61803 1.00000 1.33826 1.61803 1.82709 1.95630 2.00000 ... ... ..... ....... TT....... ..TT....... ....TT....... E5 * 2.00000 1.93185 1.73205 1.41421 1.00000 0.51764 0.00000 -0.51764 -1.00000 -1.41421 -1.73205 -1.93185 -2.00000 -1.93185 -1.73205 -1.41421 -1.00000 -0.51764 0.00000 0.51764 1.00000 1.41421 1.73205 1.93185 2.00000 1.93185 1.73205 1.41421 1.00000 0.51764 0.00000 -0.51764 -1.00000 -1.41421 -1.73205 -1.93185 -2.00000 -1.93185 -1.73205 -1.41421 -1.00000 -0.51764 0.00000 0.51764 1.00000 1.41421 1.73205 1.93185 2.00000 1.93185 1.73205 1.41421 1.00000 0.51764 0.00000 -0.51764 -1.00000 -1.41421 -1.73205 -1.93185 -2.00000 ... ... ..... ....... ......... TT......... ..TT......... E6 * 2.00000 1.90211 1.61803 1.17557 0.61803 0.00000 -0.61803 -1.17557 -1.61803 -1.90211 -2.00000 -1.90211 -1.61803 -1.17557 -0.61803 0.00000 0.61803 1.17557 1.61803 1.90211 2.00000 1.90211 1.61803 1.17557 0.61803 0.00000 -0.61803 -1.17557 -1.61803 -1.90211 -2.00000 -1.90211 -1.61803 -1.17557 -0.61803 0.00000 0.61803 1.17557 1.61803 1.90211 2.00000 1.90211 1.61803 1.17557 0.61803 0.00000 -0.61803 -1.17557 -1.61803 -1.90211 -2.00000 -1.90211 -1.61803 -1.17557 -0.61803 0.00000 0.61803 1.17557 1.61803 1.90211 2.00000 ... ... ..... ....... ......... ........... TT........... E7 * 2.00000 1.86716 1.48629 0.90798 0.20906 -0.51764 -1.17557 -1.67734 -1.95630 -1.97538 -1.73205 -1.25864 -0.61803 0.10467 0.81347 1.41421 1.82709 1.99726 1.90211 1.55429 1.00000 0.31287 -0.41582 -1.08928 -1.61803 -1.93185 -1.98904 -1.78201 -1.33826 -0.71674 0.00000 0.71674 1.33826 1.78201 1.98904 1.93185 1.61803 1.08928 0.41582 -0.31287 -1.00000 -1.55429 -1.90211 -1.99726 -1.82709 -1.41421 -0.81347 -0.10467 0.61803 1.25864 1.73205 1.97538 1.95630 1.67734 1.17557 0.51764 -0.20906 -0.90798 -1.48629 -1.86716 -2.00000 ... ... ..... ....... ......... ........... ............. E8 * 2.00000 1.82709 1.33826 0.61803 -0.20906 -1.00000 -1.61803 -1.95630 -1.95630 -1.61803 -1.00000 -0.20906 0.61803 1.33826 1.82709 2.00000 1.82709 1.33826 0.61803 -0.20906 -1.00000 -1.61803 -1.95630 -1.95630 -1.61803 -1.00000 -0.20906 0.61803 1.33826 1.82709 2.00000 1.82709 1.33826 0.61803 -0.20906 -1.00000 -1.61803 -1.95630 -1.95630 -1.61803 -1.00000 -0.20906 0.61803 1.33826 1.82709 2.00000 1.82709 1.33826 0.61803 -0.20906 -1.00000 -1.61803 -1.95630 -1.95630 -1.61803 -1.00000 -0.20906 0.61803 1.33826 1.82709 2.00000 ... ... ..... ....... ......... ........... ............. E9 * 2.00000 1.78201 1.17557 0.31287 -0.61803 -1.41421 -1.90211 -1.97538 -1.61803 -0.90798 0.00000 0.90798 1.61803 1.97538 1.90211 1.41421 0.61803 -0.31287 -1.17557 -1.78201 -2.00000 -1.78201 -1.17557 -0.31287 0.61803 1.41421 1.90211 1.97538 1.61803 0.90798 0.00000 -0.90798 -1.61803 -1.97538 -1.90211 -1.41421 -0.61803 0.31287 1.17557 1.78201 2.00000 1.78201 1.17557 0.31287 -0.61803 -1.41421 -1.90211 -1.97538 -1.61803 -0.90798 0.00000 0.90798 1.61803 1.97538 1.90211 1.41421 0.61803 -0.31287 -1.17557 -1.78201 -2.00000 ... ... ..... ....... ......... ........... ............. E10 * 2.00000 1.73205 1.00000 0.00000 -1.00000 -1.73205 -2.00000 -1.73205 -1.00000 0.00000 1.00000 1.73205 2.00000 1.73205 1.00000 0.00000 -1.00000 -1.73205 -2.00000 -1.73205 -1.00000 0.00000 1.00000 1.73205 2.00000 1.73205 1.00000 0.00000 -1.00000 -1.73205 -2.00000 -1.73205 -1.00000 0.00000 1.00000 1.73205 2.00000 1.73205 1.00000 0.00000 -1.00000 -1.73205 -2.00000 -1.73205 -1.00000 0.00000 1.00000 1.73205 2.00000 1.73205 1.00000 0.00000 -1.00000 -1.73205 -2.00000 -1.73205 -1.00000 0.00000 1.00000 1.73205 2.00000 ... ... ..... ....... ......... ........... ............. E11 * 2.00000 1.67734 0.81347 -0.31287 -1.33826 -1.93185 -1.90211 -1.25864 -0.20906 0.90798 1.73205 1.99726 1.61803 0.71674 -0.41582 -1.41421 -1.95630 -1.86716 -1.17557 -0.10467 1.00000 1.78201 1.98904 1.55429 0.61803 -0.51764 -1.48629 -1.97538 -1.82709 -1.08928 0.00000 1.08928 1.82709 1.97538 1.48629 0.51764 -0.61803 -1.55429 -1.98904 -1.78201 -1.00000 0.10467 1.17557 1.86716 1.95630 1.41421 0.41582 -0.71674 -1.61803 -1.99726 -1.73205 -0.90798 0.20906 1.25864 1.90211 1.93185 1.33826 0.31287 -0.81347 -1.67734 -2.00000 ... ... ..... ....... ......... ........... ............. E12 * 2.00000 1.61803 0.61803 -0.61803 -1.61803 -2.00000 -1.61803 -0.61803 0.61803 1.61803 2.00000 1.61803 0.61803 -0.61803 -1.61803 -2.00000 -1.61803 -0.61803 0.61803 1.61803 2.00000 1.61803 0.61803 -0.61803 -1.61803 -2.00000 -1.61803 -0.61803 0.61803 1.61803 2.00000 1.61803 0.61803 -0.61803 -1.61803 -2.00000 -1.61803 -0.61803 0.61803 1.61803 2.00000 1.61803 0.61803 -0.61803 -1.61803 -2.00000 -1.61803 -0.61803 0.61803 1.61803 2.00000 1.61803 0.61803 -0.61803 -1.61803 -2.00000 -1.61803 -0.61803 0.61803 1.61803 2.00000 ... ... ..... ....... ......... ........... ............. E13 * 2.00000 1.55429 0.41582 -0.90798 -1.82709 -1.93185 -1.17557 0.10467 1.33826 1.97538 1.73205 0.71674 -0.61803 -1.67734 -1.98904 -1.41421 -0.20906 1.08928 1.90211 1.86716 1.00000 -0.31287 -1.48629 -1.99726 -1.61803 -0.51764 0.81347 1.78201 1.95630 1.25864 0.00000 -1.25864 -1.95630 -1.78201 -0.81347 0.51764 1.61803 1.99726 1.48629 0.31287 -1.00000 -1.86716 -1.90211 -1.08928 0.20906 1.41421 1.98904 1.67734 0.61803 -0.71674 -1.73205 -1.97538 -1.33826 -0.10467 1.17557 1.93185 1.82709 0.90798 -0.41582 -1.55429 -2.00000 ... ... ..... ....... ......... ........... ............. E14 * 2.00000 1.48629 0.20906 -1.17557 -1.95630 -1.73205 -0.61803 0.81347 1.82709 1.90211 1.00000 -0.41582 -1.61803 -1.98904 -1.33826 0.00000 1.33826 1.98904 1.61803 0.41582 -1.00000 -1.90211 -1.82709 -0.81347 0.61803 1.73205 1.95630 1.17557 -0.20906 -1.48629 -2.00000 -1.48629 -0.20906 1.17557 1.95630 1.73205 0.61803 -0.81347 -1.82709 -1.90211 -1.00000 0.41582 1.61803 1.98904 1.33826 0.00000 -1.33826 -1.98904 -1.61803 -0.41582 1.00000 1.90211 1.82709 0.81347 -0.61803 -1.73205 -1.95630 -1.17557 0.20906 1.48629 2.00000 ... ... ..... ....... ......... ........... ............. E15 * 2.00000 1.41421 0.00000 -1.41421 -2.00000 -1.41421 0.00000 1.41421 2.00000 1.41421 0.00000 -1.41421 -2.00000 -1.41421 0.00000 1.41421 2.00000 1.41421 0.00000 -1.41421 -2.00000 -1.41421 0.00000 1.41421 2.00000 1.41421 0.00000 -1.41421 -2.00000 -1.41421 0.00000 1.41421 2.00000 1.41421 0.00000 -1.41421 -2.00000 -1.41421 0.00000 1.41421 2.00000 1.41421 0.00000 -1.41421 -2.00000 -1.41421 0.00000 1.41421 2.00000 1.41421 0.00000 -1.41421 -2.00000 -1.41421 0.00000 1.41421 2.00000 1.41421 0.00000 -1.41421 -2.00000 ... ... ..... ....... ......... ........... ............. E16 * 2.00000 1.33826 -0.20906 -1.61803 -1.95630 -1.00000 0.61803 1.82709 1.82709 0.61803 -1.00000 -1.95630 -1.61803 -0.20906 1.33826 2.00000 1.33826 -0.20906 -1.61803 -1.95630 -1.00000 0.61803 1.82709 1.82709 0.61803 -1.00000 -1.95630 -1.61803 -0.20906 1.33826 2.00000 1.33826 -0.20906 -1.61803 -1.95630 -1.00000 0.61803 1.82709 1.82709 0.61803 -1.00000 -1.95630 -1.61803 -0.20906 1.33826 2.00000 1.33826 -0.20906 -1.61803 -1.95630 -1.00000 0.61803 1.82709 1.82709 0.61803 -1.00000 -1.95630 -1.61803 -0.20906 1.33826 2.00000 ... ... ..... ....... ......... ........... ............. E17 * 2.00000 1.25864 -0.41582 -1.78201 -1.82709 -0.51764 1.17557 1.99726 1.33826 -0.31287 -1.73205 -1.86716 -0.61803 1.08928 1.98904 1.41421 -0.20906 -1.67734 -1.90211 -0.71674 1.00000 1.97538 1.48629 -0.10467 -1.61803 -1.93185 -0.81347 0.90798 1.95630 1.55429 0.00000 -1.55429 -1.95630 -0.90798 0.81347 1.93185 1.61803 0.10467 -1.48629 -1.97538 -1.00000 0.71674 1.90211 1.67734 0.20906 -1.41421 -1.98904 -1.08928 0.61803 1.86716 1.73205 0.31287 -1.33826 -1.99726 -1.17557 0.51764 1.82709 1.78201 0.41582 -1.25864 -2.00000 ... ... ..... ....... ......... ........... ............. E18 * 2.00000 1.17557 -0.61803 -1.90211 -1.61803 0.00000 1.61803 1.90211 0.61803 -1.17557 -2.00000 -1.17557 0.61803 1.90211 1.61803 0.00000 -1.61803 -1.90211 -0.61803 1.17557 2.00000 1.17557 -0.61803 -1.90211 -1.61803 0.00000 1.61803 1.90211 0.61803 -1.17557 -2.00000 -1.17557 0.61803 1.90211 1.61803 0.00000 -1.61803 -1.90211 -0.61803 1.17557 2.00000 1.17557 -0.61803 -1.90211 -1.61803 0.00000 1.61803 1.90211 0.61803 -1.17557 -2.00000 -1.17557 0.61803 1.90211 1.61803 0.00000 -1.61803 -1.90211 -0.61803 1.17557 2.00000 ... ... ..... ....... ......... ........... ............. E19 * 2.00000 1.08928 -0.81347 -1.97538 -1.33826 0.51764 1.90211 1.55429 -0.20906 -1.78201 -1.73205 -0.10467 1.61803 1.86716 0.41582 -1.41421 -1.95630 -0.71674 1.17557 1.99726 1.00000 -0.90798 -1.98904 -1.25864 0.61803 1.93185 1.48629 -0.31287 -1.82709 -1.67734 0.00000 1.67734 1.82709 0.31287 -1.48629 -1.93185 -0.61803 1.25864 1.98904 0.90798 -1.00000 -1.99726 -1.17557 0.71674 1.95630 1.41421 -0.41582 -1.86716 -1.61803 0.10467 1.73205 1.78201 0.20906 -1.55429 -1.90211 -0.51764 1.33826 1.97538 0.81347 -1.08928 -2.00000 ... ... ..... ....... ......... ........... ............. E20 * 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 -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 ... ... ..... ....... ......... ........... ............. E21 * 2.00000 0.90798 -1.17557 -1.97538 -0.61803 1.41421 1.90211 0.31287 -1.61803 -1.78201 0.00000 1.78201 1.61803 -0.31287 -1.90211 -1.41421 0.61803 1.97538 1.17557 -0.90798 -2.00000 -0.90798 1.17557 1.97538 0.61803 -1.41421 -1.90211 -0.31287 1.61803 1.78201 0.00000 -1.78201 -1.61803 0.31287 1.90211 1.41421 -0.61803 -1.97538 -1.17557 0.90798 2.00000 0.90798 -1.17557 -1.97538 -0.61803 1.41421 1.90211 0.31287 -1.61803 -1.78201 0.00000 1.78201 1.61803 -0.31287 -1.90211 -1.41421 0.61803 1.97538 1.17557 -0.90798 -2.00000 ... ... ..... ....... ......... ........... ............. E22 * 2.00000 0.81347 -1.33826 -1.90211 -0.20906 1.73205 1.61803 -0.41582 -1.95630 -1.17557 1.00000 1.98904 0.61803 -1.48629 -1.82709 0.00000 1.82709 1.48629 -0.61803 -1.98904 -1.00000 1.17557 1.95630 0.41582 -1.61803 -1.73205 0.20906 1.90211 1.33826 -0.81347 -2.00000 -0.81347 1.33826 1.90211 0.20906 -1.73205 -1.61803 0.41582 1.95630 1.17557 -1.00000 -1.98904 -0.61803 1.48629 1.82709 0.00000 -1.82709 -1.48629 0.61803 1.98904 1.00000 -1.17557 -1.95630 -0.41582 1.61803 1.73205 -0.20906 -1.90211 -1.33826 0.81347 2.00000 ... ... ..... ....... ......... ........... ............. E23 * 2.00000 0.71674 -1.48629 -1.78201 0.20906 1.93185 1.17557 -1.08928 -1.95630 -0.31287 1.73205 1.55429 -0.61803 -1.99726 -0.81347 1.41421 1.82709 -0.10467 -1.90211 -1.25864 1.00000 1.97538 0.41582 -1.67734 -1.61803 0.51764 1.98904 0.90798 -1.33826 -1.86716 0.00000 1.86716 1.33826 -0.90798 -1.98904 -0.51764 1.61803 1.67734 -0.41582 -1.97538 -1.00000 1.25864 1.90211 0.10467 -1.82709 -1.41421 0.81347 1.99726 0.61803 -1.55429 -1.73205 0.31287 1.95630 1.08928 -1.17557 -1.93185 -0.20906 1.78201 1.48629 -0.71674 -2.00000 ... ... ..... ....... ......... ........... ............. E24 * 2.00000 0.61803 -1.61803 -1.61803 0.61803 2.00000 0.61803 -1.61803 -1.61803 0.61803 2.00000 0.61803 -1.61803 -1.61803 0.61803 2.00000 0.61803 -1.61803 -1.61803 0.61803 2.00000 0.61803 -1.61803 -1.61803 0.61803 2.00000 0.61803 -1.61803 -1.61803 0.61803 2.00000 0.61803 -1.61803 -1.61803 0.61803 2.00000 0.61803 -1.61803 -1.61803 0.61803 2.00000 0.61803 -1.61803 -1.61803 0.61803 2.00000 0.61803 -1.61803 -1.61803 0.61803 2.00000 0.61803 -1.61803 -1.61803 0.61803 2.00000 0.61803 -1.61803 -1.61803 0.61803 2.00000 ... ... ..... ....... ......... ........... ............. 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E53 * 2.00000 -1.86716 1.48629 -0.90798 0.20906 0.51764 -1.17557 1.67734 -1.95630 1.97538 -1.73205 1.25864 -0.61803 -0.10467 0.81347 -1.41421 1.82709 -1.99726 1.90211 -1.55429 1.00000 -0.31287 -0.41582 1.08928 -1.61803 1.93185 -1.98904 1.78201 -1.33826 0.71674 0.00000 -0.71674 1.33826 -1.78201 1.98904 -1.93185 1.61803 -1.08928 0.41582 0.31287 -1.00000 1.55429 -1.90211 1.99726 -1.82709 1.41421 -0.81347 0.10467 0.61803 -1.25864 1.73205 -1.97538 1.95630 -1.67734 1.17557 -0.51764 -0.20906 0.90798 -1.48629 1.86716 -2.00000 ... ... ..... ....... ......... ........... ............. E54 * 2.00000 -1.90211 1.61803 -1.17557 0.61803 0.00000 -0.61803 1.17557 -1.61803 1.90211 -2.00000 1.90211 -1.61803 1.17557 -0.61803 0.00000 0.61803 -1.17557 1.61803 -1.90211 2.00000 -1.90211 1.61803 -1.17557 0.61803 0.00000 -0.61803 1.17557 -1.61803 1.90211 -2.00000 1.90211 -1.61803 1.17557 -0.61803 0.00000 0.61803 -1.17557 1.61803 -1.90211 2.00000 -1.90211 1.61803 -1.17557 0.61803 0.00000 -0.61803 1.17557 -1.61803 1.90211 -2.00000 1.90211 -1.61803 1.17557 -0.61803 0.00000 0.61803 -1.17557 1.61803 -1.90211 2.00000 ... ... ..... ....... ......... ........... ............. E55 * 2.00000 -1.93185 1.73205 -1.41421 1.00000 -0.51764 0.00000 0.51764 -1.00000 1.41421 -1.73205 1.93185 -2.00000 1.93185 -1.73205 1.41421 -1.00000 0.51764 0.00000 -0.51764 1.00000 -1.41421 1.73205 -1.93185 2.00000 -1.93185 1.73205 -1.41421 1.00000 -0.51764 0.00000 0.51764 -1.00000 1.41421 -1.73205 1.93185 -2.00000 1.93185 -1.73205 1.41421 -1.00000 0.51764 0.00000 -0.51764 1.00000 -1.41421 1.73205 -1.93185 2.00000 -1.93185 1.73205 -1.41421 1.00000 -0.51764 0.00000 0.51764 -1.00000 1.41421 -1.73205 1.93185 -2.00000 ... ... ..... ....... ......... ........... ............. E56 * 2.00000 -1.95630 1.82709 -1.61803 1.33826 -1.00000 0.61803 -0.20906 -0.20906 0.61803 -1.00000 1.33826 -1.61803 1.82709 -1.95630 2.00000 -1.95630 1.82709 -1.61803 1.33826 -1.00000 0.61803 -0.20906 -0.20906 0.61803 -1.00000 1.33826 -1.61803 1.82709 -1.95630 2.00000 -1.95630 1.82709 -1.61803 1.33826 -1.00000 0.61803 -0.20906 -0.20906 0.61803 -1.00000 1.33826 -1.61803 1.82709 -1.95630 2.00000 -1.95630 1.82709 -1.61803 1.33826 -1.00000 0.61803 -0.20906 -0.20906 0.61803 -1.00000 1.33826 -1.61803 1.82709 -1.95630 2.00000 ... ... ..... ....... ......... ........... ............. E57 * 2.00000 -1.97538 1.90211 -1.78201 1.61803 -1.41421 1.17557 -0.90798 0.61803 -0.31287 0.00000 0.31287 -0.61803 0.90798 -1.17557 1.41421 -1.61803 1.78201 -1.90211 1.97538 -2.00000 1.97538 -1.90211 1.78201 -1.61803 1.41421 -1.17557 0.90798 -0.61803 0.31287 0.00000 -0.31287 0.61803 -0.90798 1.17557 -1.41421 1.61803 -1.78201 1.90211 -1.97538 2.00000 -1.97538 1.90211 -1.78201 1.61803 -1.41421 1.17557 -0.90798 0.61803 -0.31287 0.00000 0.31287 -0.61803 0.90798 -1.17557 1.41421 -1.61803 1.78201 -1.90211 1.97538 -2.00000 ... ... ..... ....... ......... ........... ............. E58 * 2.00000 -1.98904 1.95630 -1.90211 1.82709 -1.73205 1.61803 -1.48629 1.33826 -1.17557 1.00000 -0.81347 0.61803 -0.41582 0.20906 0.00000 -0.20906 0.41582 -0.61803 0.81347 -1.00000 1.17557 -1.33826 1.48629 -1.61803 1.73205 -1.82709 1.90211 -1.95630 1.98904 -2.00000 1.98904 -1.95630 1.90211 -1.82709 1.73205 -1.61803 1.48629 -1.33826 1.17557 -1.00000 0.81347 -0.61803 0.41582 -0.20906 0.00000 0.20906 -0.41582 0.61803 -0.81347 1.00000 -1.17557 1.33826 -1.48629 1.61803 -1.73205 1.82709 -1.90211 1.95630 -1.98904 2.00000 ... ... ..... ....... ......... ........... ............. E59 * 2.00000 -1.99726 1.98904 -1.97538 1.95630 -1.93185 1.90211 -1.86716 1.82709 -1.78201 1.73205 -1.67734 1.61803 -1.55429 1.48629 -1.41421 1.33826 -1.25864 1.17557 -1.08928 1.00000 -0.90798 0.81347 -0.71674 0.61803 -0.51764 0.41582 -0.31287 0.20906 -0.10467 0.00000 0.10467 -0.20906 0.31287 -0.41582 0.51764 -0.61803 0.71674 -0.81347 0.90798 -1.00000 1.08928 -1.17557 1.25864 -1.33826 1.41421 -1.48629 1.55429 -1.61803 1.67734 -1.73205 1.78201 -1.82709 1.86716 -1.90211 1.93185 -1.95630 1.97538 -1.98904 1.99726 -2.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.997259069509 = 2*cos(2*π/120) = 2*cos(π/60) = (√2−√6−√10+√30+2*(1+√3)*(√5+√5))/8 1.989043790737 = 2*cos(4*π/120) = 2*cos(π/30) = (√15+√3+√10−2*√5)/4 1.975376681190 = 2*cos(6*π/120) = 2*cos(π/20) = (√2+√10+2*(√5−√5))/4 1.956295201468 = 2*cos(8*π/120) = 2*cos(π/15) = (√5−1+√30+6*√5)/4 1.931851652578 = 2*cos(10*π/120) = 2*cos(π/12) = √2+√3 1.902113032590 = 2*cos(12*π/120) = 2*cos(π/10) = (√10+2*√5)/2 1.867160852994 = 2*cos(14*π/120) = 2*cos(7*π/60) = (√2+√6+√10+√30−2*(1−√3)*(√5−√5))/8 1.827090915285 = 2*cos(16*π/120) = 2*cos(2*π/15) = (1+√5+√30−6*√5)/4 1.782013048377 = 2*cos(18*π/120) = 2*cos(3*π/20) = (√10−√2+2*√5+√5)/4 1.732050807569 = 2*cos(20*π/120) = 2*cos(π/6) = √3 1.677341135891 = 2*cos(22*π/120) = 2*cos(11*π/60) = (√6−√2+√10−√30+2*(1+√3)*(√5+√5))/8 1.618033988750 = 2*cos(24*π/120) = 2*cos(π/5) = (√5+1)/2 1.554291922914 = 2*cos(26*π/120) = 2*cos(13*π/60) = (√6−√2−√10+√30+2*(1+√3)*(√5−√5))/8 1.486289650955 = 2*cos(28*π/120) = 2*cos(7*π/30) = (√15−√3+√10+2*√5)/4 1.414213562373 = 2*cos(30*π/120) = 2*cos(π/4) = √2 1.338261212718 = 2*cos(32*π/120) = 2*cos(4*π/15) = (1−√5+√30+6*√5)/4 1.258640782100 = 2*cos(34*π/120) = 2*cos(17*π/60) = (√2+√6+√10+√30+2*(1−√3)*(√5−√5))/8 1.175570504585 = 2*cos(36*π/120) = 2*cos(3*π/10) = (√10−2*√5)/2 1.089278070030 = 2*cos(38*π/120) = 2*cos(19*π/60) = (√10−√2−√6+√30−2*(1−√3)*(√5+√5))/8 0.907980999479 = 2*cos(42*π/120) = 2*cos(7*π/20) = (√2−√10+2*√5+√5)/4 0.813473286152 = 2*cos(44*π/120) = 2*cos(11*π/30) = (√15+√3−√10−2*√5)/4 0.716735899091 = 2*cos(46*π/120) = 2*cos(23*π/60) = (√2−√6+√10−√30+2*(1+√3)*(√5−√5))/8 0.618033988750 = 2*cos(48*π/120) = 2*cos(2*π/5) = (√5−1)/2 0.517638090205 = 2*cos(50*π/120) = 2*cos(5*π/12) = √2−√3 0.415823381636 = 2*cos(52*π/120) = 2*cos(13*π/30) = (√3−√15+√10+2*√5)/4 0.312868930080 = 2*cos(54*π/120) = 2*cos(9*π/20) = (√2+√10−2*(√5−√5))/4 0.209056926535 = 2*cos(56*π/120) = 2*cos(7*π/15) = (√30−6*√5−√5−1)/4 0.104671912486 = 2*cos(58*π/120) = 2*cos(29*π/60) = (√10−√2−√6+√30+2*(1−√3)*(√5+√5))/8 Symmetry of Rotations and Cartesian products A R+p+d+f+g+h+i+j+k+l+m R_z, z, z², z³, z⁴, z⁵, z⁶ E1 R+p+d+f+g+h+i+j+k+l+m {R_x, R_y}, {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 C₁₂₀ point group is 120, and the order of the principal axis (C₁₂₀) is 120. The group has 61 irreducible representations. β The C₁₂₀ point group is isomorphic to S₁₂₀. γ The C₁₂₀ point group is generated by one single symmetry element, C₁₂₀. Therefore, it is a cyclic group. δ The lowest nonvanishing multipole moment in C₁₂₀ is 2 (dipole moment). ε This is an Abelian point group (the commutative law holds between all symmetry operations). The C₁₂₀ group is Abelian because it contains only one symmetry element, all the powers of which necessarily commute (sufficient condition). In Abelian groups, all symmetry operations form a class of their own, and all irreducible representations are one-dimensional. ζ Because the group is Abelian and the maximum order of rotation is >2, some irreducible representations have complex characters. These 118 cases have been combined into 59 two-dimensional representations that are no longer irreducible but have real-valued characters. Accordingly, 59 pairs of left and right rotations have been combined into one two-membered pseudo-class each. η The 59 reducible “E” representations almost behave like true irreducible representations. Their norm, however, is twice the group order. Therefore, they have been marked with an asterisk in the table. This is essential when trying to decompose a reducible representation into “irreducible” ones using the familiar projection formula. θ The point group is chiral, as it does not contain any mirroring operation. ι 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. κ The point group corresponds to a constructible polygon, as the order of the principal axis is a product of any number of different Fermat primes (3,5,17,257,65537) times an arbitrary power of two. Therefore, all characters have an algebraic degree which is a power of two and can be expressed as radicals involving only square roots and integer numbers. λ The fact that the regular pentagon is constructible is known since antiquity; Eukleides already discovered a construction for it. The double cosine of 2π/5 is equal to the reciprocal of the Golden Ratio of (1+√5)/2 = 1.61803. Regular polygons of order 30,60,120 etc. are derived from the pentagon by overlaying it with a triangle and successive halving of angles.

This Character Table for the C₁₂₀ point group was created by Gernot Katzer.

For other groups and some explanations, see the Main Page.

	C₁₁₈
	C₁₁₉
	C₁₂₀	C_120v C_120h D₁₂₀ D_120h D_120d S₁₂₀
	C₁₂₁
	C₁₂₂

Character table for the C120 point group

Character table for the C₁₂₀ point group