Character Table for Point Group C120v

Character table for the C_120v point group

C120v 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 60 sv 60 sd <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 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 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 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 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 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 ... ..... ....... ......... ........... ............. 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 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -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 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -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 0.00000 0.00000 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 0.00000 0.00000 ... ... 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 0.00000 0.00000 ... ... ..... 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 0.00000 0.00000 ... ... ..... ....... 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 0.00000 0.00000 ... ... ..... ....... ......... 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 0.00000 0.00000 ... ... ..... ....... ......... ........... 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. 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E51 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E52 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. 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 A1 p+d+f+g+h+i+j+k+l+m z, z², z³, z⁴, z⁵, z⁶ A2 R R_z 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 C_120v point group is 240, and the order of the principal axis (C₁₂₀) is 120. The group has 63 irreducible representations. β The C_120v point group is isomorphic to D_60d and D₁₂₀. γ The C_120v point group is generated by two symmetry elements, C₁₂₀ and any σ_v (or, non-canonically, any σ_d). Also, the group may be generated from a σ_v plus a σ_d (some pairs will yield smaller groups, though; choosing a minimum angle is safe). δ There are two different sets of symmetry planes containing the principal axis (z axis in standard orientation). By convention, the set denoted as σ_v contains both the xz and the yz planes. ε The lowest nonvanishing multipole moment in C_120v is 2 (dipole 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. θ 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. ι 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 C_120v point group was created by Gernot Katzer.

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

	C_118v
	C_119v
C₁₂₀	C_120v	C_120h D₁₂₀ D_120h D_120d S₁₂₀
	C_121v
	C_122v

Character table for the C120v point group

Character table for the C_120v point group