Character Table for Point Group S240

Character table for the S₂₄₀ point group

S240 E 2 S240 2 C120 2 S80 2 C60 2 S48 2 C40 2 S240^7 2 C30 2 S80^3 2 C24 2 S240^11 2 C20 2 S240^13 2 C120^7 2 S16 2 C15 2 S240^17 2 C40^3 2 S240^19 2 C12 2 S80^7 2 C120^11 2 S240^23 2 C10 2 S48^5 2 C120^13 2 S80^9 2 C60^7 2 S240^29 2 C8 2 S240^31 2 C15^2 2 S80^11 2 C120^17 2 S48^7 2 C20^3 2 S240^37 2 C120^19 2 S80^13 2 C6 2 S240^41 2 C40^7 2 S240^43 2 C60^11 2 S16^3 2 C120^23 2 S240^47 2 C5 2 S240^49 2 C24^5 2 S80^17 2 C60^13 2 S240^53 2 C40^9 2 S48^11 2 C30^7 2 S80^19 2 C120^29 2 S240^59 2 C4 2 S240^61 2 C120^31 2 S80^21 2 C15^4 2 S48^13 2 C40^11 2 S240^67 2 C60^17 2 S80^23 2 C24^7 2 S240^71 2 C10^3 2 S240^73 2 C120^37 2 S16^5 2 C60^19 2 S240^77 2 C40^13 2 S240^79 2 C3 2 S80^27 2 C120^41 2 S240^83 2 C20^7 2 S48^17 2 C120^43 2 S80^29 2 C30^11 2 S240^89 2 C8^3 2 S240^91 2 C60^23 2 S80^31 2 C120^47 2 S48^19 2 C5^2 2 S240^97 2 C120^49 2 S80^33 2 C12^5 2 S240^101 2 C40^17 2 S240^103 2 C30^13 2 S16^7 2 C120^53 2 S240^107 2 C20^9 2 S240^109 2 C24^11 2 S80^37 2 C15^7 2 S240^113 2 C40^19 2 S48^23 2 C60^29 2 S80^39 2 C120^59 2 S240^119 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 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 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 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 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -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 1.99931 1.99726 1.99383 1.98904 1.98289 1.97538 1.96651 1.95630 1.94474 1.93185 1.91764 1.90211 1.88528 1.86716 1.84776 1.82709 1.80517 1.78201 1.75763 1.73205 1.70528 1.67734 1.64825 1.61803 1.58671 1.55429 1.52081 1.48629 1.45075 1.41421 1.37671 1.33826 1.29890 1.25864 1.21752 1.17557 1.13281 1.08928 1.04500 1.00000 0.95432 0.90798 0.86102 0.81347 0.76537 0.71674 0.66761 0.61803 0.56803 0.51764 0.46689 0.41582 0.36447 0.31287 0.26105 0.20906 0.15692 0.10467 0.05235 0.00000 -0.05235 -0.10467 -0.15692 -0.20906 -0.26105 -0.31287 -0.36447 -0.41582 -0.46689 -0.51764 -0.56803 -0.61803 -0.66761 -0.71674 -0.76537 -0.81347 -0.86102 -0.90798 -0.95432 -1.00000 -1.04500 -1.08928 -1.13281 -1.17557 -1.21752 -1.25864 -1.29890 -1.33826 -1.37671 -1.41421 -1.45075 -1.48629 -1.52081 -1.55429 -1.58671 -1.61803 -1.64825 -1.67734 -1.70528 -1.73205 -1.75763 -1.78201 -1.80517 -1.82709 -1.84776 -1.86716 -1.88528 -1.90211 -1.91764 -1.93185 -1.94474 -1.95630 -1.96651 -1.97538 -1.98289 -1.98904 -1.99383 -1.99726 -1.99931 -2.00000 ... TT. ..... ....TT. ......... ........TT. ............. E2 * 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 -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... E3 * 2.00000 1.99383 1.97538 1.94474 1.90211 1.84776 1.78201 1.70528 1.61803 1.52081 1.41421 1.29890 1.17557 1.04500 0.90798 0.76537 0.61803 0.46689 0.31287 0.15692 0.00000 -0.15692 -0.31287 -0.46689 -0.61803 -0.76537 -0.90798 -1.04500 -1.17557 -1.29890 -1.41421 -1.52081 -1.61803 -1.70528 -1.78201 -1.84776 -1.90211 -1.94474 -1.97538 -1.99383 -2.00000 -1.99383 -1.97538 -1.94474 -1.90211 -1.84776 -1.78201 -1.70528 -1.61803 -1.52081 -1.41421 -1.29890 -1.17557 -1.04500 -0.90798 -0.76537 -0.61803 -0.46689 -0.31287 -0.15692 0.00000 0.15692 0.31287 0.46689 0.61803 0.76537 0.90798 1.04500 1.17557 1.29890 1.41421 1.52081 1.61803 1.70528 1.78201 1.84776 1.90211 1.94474 1.97538 1.99383 2.00000 1.99383 1.97538 1.94474 1.90211 1.84776 1.78201 1.70528 1.61803 1.52081 1.41421 1.29890 1.17557 1.04500 0.90798 0.76537 0.61803 0.46689 0.31287 0.15692 0.00000 -0.15692 -0.31287 -0.46689 -0.61803 -0.76537 -0.90798 -1.04500 -1.17557 -1.29890 -1.41421 -1.52081 -1.61803 -1.70528 -1.78201 -1.84776 -1.90211 -1.94474 -1.97538 -1.99383 -2.00000 ... ... ..... TT..... ......... ....TT..... ............. E4 * 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 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....... E5 * 2.00000 1.98289 1.93185 1.84776 1.73205 1.58671 1.41421 1.21752 1.00000 0.76537 0.51764 0.26105 0.00000 -0.26105 -0.51764 -0.76537 -1.00000 -1.21752 -1.41421 -1.58671 -1.73205 -1.84776 -1.93185 -1.98289 -2.00000 -1.98289 -1.93185 -1.84776 -1.73205 -1.58671 -1.41421 -1.21752 -1.00000 -0.76537 -0.51764 -0.26105 0.00000 0.26105 0.51764 0.76537 1.00000 1.21752 1.41421 1.58671 1.73205 1.84776 1.93185 1.98289 2.00000 1.98289 1.93185 1.84776 1.73205 1.58671 1.41421 1.21752 1.00000 0.76537 0.51764 0.26105 0.00000 -0.26105 -0.51764 -0.76537 -1.00000 -1.21752 -1.41421 -1.58671 -1.73205 -1.84776 -1.93185 -1.98289 -2.00000 -1.98289 -1.93185 -1.84776 -1.73205 -1.58671 -1.41421 -1.21752 -1.00000 -0.76537 -0.51764 -0.26105 0.00000 0.26105 0.51764 0.76537 1.00000 1.21752 1.41421 1.58671 1.73205 1.84776 1.93185 1.98289 2.00000 1.98289 1.93185 1.84776 1.73205 1.58671 1.41421 1.21752 1.00000 0.76537 0.51764 0.26105 0.00000 -0.26105 -0.51764 -0.76537 -1.00000 -1.21752 -1.41421 -1.58671 -1.73205 -1.84776 -1.93185 -1.98289 -2.00000 ... ... ..... ....... ......... TT......... ............. E6 * 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 -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........... E7 * 2.00000 1.96651 1.86716 1.70528 1.48629 1.21752 0.90798 0.56803 0.20906 -0.15692 -0.51764 -0.86102 -1.17557 -1.45075 -1.67734 -1.84776 -1.95630 -1.99931 -1.97538 -1.88528 -1.73205 -1.52081 -1.25864 -0.95432 -0.61803 -0.26105 0.10467 0.46689 0.81347 1.13281 1.41421 1.64825 1.82709 1.94474 1.99726 1.98289 1.90211 1.75763 1.55429 1.29890 1.00000 0.66761 0.31287 -0.05235 -0.41582 -0.76537 -1.08928 -1.37671 -1.61803 -1.80517 -1.93185 -1.99383 -1.98904 -1.91764 -1.78201 -1.58671 -1.33826 -1.04500 -0.71674 -0.36447 0.00000 0.36447 0.71674 1.04500 1.33826 1.58671 1.78201 1.91764 1.98904 1.99383 1.93185 1.80517 1.61803 1.37671 1.08928 0.76537 0.41582 0.05235 -0.31287 -0.66761 -1.00000 -1.29890 -1.55429 -1.75763 -1.90211 -1.98289 -1.99726 -1.94474 -1.82709 -1.64825 -1.41421 -1.13281 -0.81347 -0.46689 -0.10467 0.26105 0.61803 0.95432 1.25864 1.52081 1.73205 1.88528 1.97538 1.99931 1.95630 1.84776 1.67734 1.45075 1.17557 0.86102 0.51764 0.15692 -0.20906 -0.56803 -0.90798 -1.21752 -1.48629 -1.70528 -1.86716 -1.96651 -2.00000 ... ... ..... ....... ......... ........... ............. E8 * 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 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 ... ... ..... ....... ......... ........... ............. E9 * 2.00000 1.94474 1.78201 1.52081 1.17557 0.76537 0.31287 -0.15692 -0.61803 -1.04500 -1.41421 -1.70528 -1.90211 -1.99383 -1.97538 -1.84776 -1.61803 -1.29890 -0.90798 -0.46689 0.00000 0.46689 0.90798 1.29890 1.61803 1.84776 1.97538 1.99383 1.90211 1.70528 1.41421 1.04500 0.61803 0.15692 -0.31287 -0.76537 -1.17557 -1.52081 -1.78201 -1.94474 -2.00000 -1.94474 -1.78201 -1.52081 -1.17557 -0.76537 -0.31287 0.15692 0.61803 1.04500 1.41421 1.70528 1.90211 1.99383 1.97538 1.84776 1.61803 1.29890 0.90798 0.46689 0.00000 -0.46689 -0.90798 -1.29890 -1.61803 -1.84776 -1.97538 -1.99383 -1.90211 -1.70528 -1.41421 -1.04500 -0.61803 -0.15692 0.31287 0.76537 1.17557 1.52081 1.78201 1.94474 2.00000 1.94474 1.78201 1.52081 1.17557 0.76537 0.31287 -0.15692 -0.61803 -1.04500 -1.41421 -1.70528 -1.90211 -1.99383 -1.97538 -1.84776 -1.61803 -1.29890 -0.90798 -0.46689 0.00000 0.46689 0.90798 1.29890 1.61803 1.84776 1.97538 1.99383 1.90211 1.70528 1.41421 1.04500 0.61803 0.15692 -0.31287 -0.76537 -1.17557 -1.52081 -1.78201 -1.94474 -2.00000 ... ... ..... ....... ......... ........... ............. E10 * 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 -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 ... ... ..... ....... ......... ........... ............. 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E109 * 2.00000 -1.91764 1.67734 -1.29890 0.81347 -0.26105 -0.31287 0.86102 -1.33826 1.70528 -1.93185 1.99931 -1.90211 1.64825 -1.25864 0.76537 -0.20906 -0.36447 0.90798 -1.37671 1.73205 -1.94474 1.99726 -1.88528 1.61803 -1.21752 0.71674 -0.15692 -0.41582 0.95432 -1.41421 1.75763 -1.95630 1.99383 -1.86716 1.58671 -1.17557 0.66761 -0.10467 -0.46689 1.00000 -1.45075 1.78201 -1.96651 1.98904 -1.84776 1.55429 -1.13281 0.61803 -0.05235 -0.51764 1.04500 -1.48629 1.80517 -1.97538 1.98289 -1.82709 1.52081 -1.08928 0.56803 0.00000 -0.56803 1.08928 -1.52081 1.82709 -1.98289 1.97538 -1.80517 1.48629 -1.04500 0.51764 0.05235 -0.61803 1.13281 -1.55429 1.84776 -1.98904 1.96651 -1.78201 1.45075 -1.00000 0.46689 0.10467 -0.66761 1.17557 -1.58671 1.86716 -1.99383 1.95630 -1.75763 1.41421 -0.95432 0.41582 0.15692 -0.71674 1.21752 -1.61803 1.88528 -1.99726 1.94474 -1.73205 1.37671 -0.90798 0.36447 0.20906 -0.76537 1.25864 -1.64825 1.90211 -1.99931 1.93185 -1.70528 1.33826 -0.86102 0.31287 0.26105 -0.81347 1.29890 -1.67734 1.91764 -2.00000 ... ... ..... ....... ......... ........... ............. E110 * 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 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 ... ... ..... ....... ......... ........... ............. E111 * 2.00000 -1.94474 1.78201 -1.52081 1.17557 -0.76537 0.31287 0.15692 -0.61803 1.04500 -1.41421 1.70528 -1.90211 1.99383 -1.97538 1.84776 -1.61803 1.29890 -0.90798 0.46689 0.00000 -0.46689 0.90798 -1.29890 1.61803 -1.84776 1.97538 -1.99383 1.90211 -1.70528 1.41421 -1.04500 0.61803 -0.15692 -0.31287 0.76537 -1.17557 1.52081 -1.78201 1.94474 -2.00000 1.94474 -1.78201 1.52081 -1.17557 0.76537 -0.31287 -0.15692 0.61803 -1.04500 1.41421 -1.70528 1.90211 -1.99383 1.97538 -1.84776 1.61803 -1.29890 0.90798 -0.46689 0.00000 0.46689 -0.90798 1.29890 -1.61803 1.84776 -1.97538 1.99383 -1.90211 1.70528 -1.41421 1.04500 -0.61803 0.15692 0.31287 -0.76537 1.17557 -1.52081 1.78201 -1.94474 2.00000 -1.94474 1.78201 -1.52081 1.17557 -0.76537 0.31287 0.15692 -0.61803 1.04500 -1.41421 1.70528 -1.90211 1.99383 -1.97538 1.84776 -1.61803 1.29890 -0.90798 0.46689 0.00000 -0.46689 0.90798 -1.29890 1.61803 -1.84776 1.97538 -1.99383 1.90211 -1.70528 1.41421 -1.04500 0.61803 -0.15692 -0.31287 0.76537 -1.17557 1.52081 -1.78201 1.94474 -2.00000 ... ... ..... ....... ......... ........... ............. E112 * 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 -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 ... ... ..... ....... ......... ........... ............. E113 * 2.00000 -1.96651 1.86716 -1.70528 1.48629 -1.21752 0.90798 -0.56803 0.20906 0.15692 -0.51764 0.86102 -1.17557 1.45075 -1.67734 1.84776 -1.95630 1.99931 -1.97538 1.88528 -1.73205 1.52081 -1.25864 0.95432 -0.61803 0.26105 0.10467 -0.46689 0.81347 -1.13281 1.41421 -1.64825 1.82709 -1.94474 1.99726 -1.98289 1.90211 -1.75763 1.55429 -1.29890 1.00000 -0.66761 0.31287 0.05235 -0.41582 0.76537 -1.08928 1.37671 -1.61803 1.80517 -1.93185 1.99383 -1.98904 1.91764 -1.78201 1.58671 -1.33826 1.04500 -0.71674 0.36447 0.00000 -0.36447 0.71674 -1.04500 1.33826 -1.58671 1.78201 -1.91764 1.98904 -1.99383 1.93185 -1.80517 1.61803 -1.37671 1.08928 -0.76537 0.41582 -0.05235 -0.31287 0.66761 -1.00000 1.29890 -1.55429 1.75763 -1.90211 1.98289 -1.99726 1.94474 -1.82709 1.64825 -1.41421 1.13281 -0.81347 0.46689 -0.10467 -0.26105 0.61803 -0.95432 1.25864 -1.52081 1.73205 -1.88528 1.97538 -1.99931 1.95630 -1.84776 1.67734 -1.45075 1.17557 -0.86102 0.51764 -0.15692 -0.20906 0.56803 -0.90798 1.21752 -1.48629 1.70528 -1.86716 1.96651 -2.00000 ... ... ..... ....... ......... ........... ............. 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E115 * 2.00000 -1.98289 1.93185 -1.84776 1.73205 -1.58671 1.41421 -1.21752 1.00000 -0.76537 0.51764 -0.26105 0.00000 0.26105 -0.51764 0.76537 -1.00000 1.21752 -1.41421 1.58671 -1.73205 1.84776 -1.93185 1.98289 -2.00000 1.98289 -1.93185 1.84776 -1.73205 1.58671 -1.41421 1.21752 -1.00000 0.76537 -0.51764 0.26105 0.00000 -0.26105 0.51764 -0.76537 1.00000 -1.21752 1.41421 -1.58671 1.73205 -1.84776 1.93185 -1.98289 2.00000 -1.98289 1.93185 -1.84776 1.73205 -1.58671 1.41421 -1.21752 1.00000 -0.76537 0.51764 -0.26105 0.00000 0.26105 -0.51764 0.76537 -1.00000 1.21752 -1.41421 1.58671 -1.73205 1.84776 -1.93185 1.98289 -2.00000 1.98289 -1.93185 1.84776 -1.73205 1.58671 -1.41421 1.21752 -1.00000 0.76537 -0.51764 0.26105 0.00000 -0.26105 0.51764 -0.76537 1.00000 -1.21752 1.41421 -1.58671 1.73205 -1.84776 1.93185 -1.98289 2.00000 -1.98289 1.93185 -1.84776 1.73205 -1.58671 1.41421 -1.21752 1.00000 -0.76537 0.51764 -0.26105 0.00000 0.26105 -0.51764 0.76537 -1.00000 1.21752 -1.41421 1.58671 -1.73205 1.84776 -1.93185 1.98289 -2.00000 ... ... ..... ....... ......... ........... ..TT......... 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E117 * 2.00000 -1.99383 1.97538 -1.94474 1.90211 -1.84776 1.78201 -1.70528 1.61803 -1.52081 1.41421 -1.29890 1.17557 -1.04500 0.90798 -0.76537 0.61803 -0.46689 0.31287 -0.15692 0.00000 0.15692 -0.31287 0.46689 -0.61803 0.76537 -0.90798 1.04500 -1.17557 1.29890 -1.41421 1.52081 -1.61803 1.70528 -1.78201 1.84776 -1.90211 1.94474 -1.97538 1.99383 -2.00000 1.99383 -1.97538 1.94474 -1.90211 1.84776 -1.78201 1.70528 -1.61803 1.52081 -1.41421 1.29890 -1.17557 1.04500 -0.90798 0.76537 -0.61803 0.46689 -0.31287 0.15692 0.00000 -0.15692 0.31287 -0.46689 0.61803 -0.76537 0.90798 -1.04500 1.17557 -1.29890 1.41421 -1.52081 1.61803 -1.70528 1.78201 -1.84776 1.90211 -1.94474 1.97538 -1.99383 2.00000 -1.99383 1.97538 -1.94474 1.90211 -1.84776 1.78201 -1.70528 1.61803 -1.52081 1.41421 -1.29890 1.17557 -1.04500 0.90798 -0.76537 0.61803 -0.46689 0.31287 -0.15692 0.00000 0.15692 -0.31287 0.46689 -0.61803 0.76537 -0.90798 1.04500 -1.17557 1.29890 -1.41421 1.52081 -1.61803 1.70528 -1.78201 1.84776 -1.90211 1.94474 -1.97538 1.99383 -2.00000 ... ... ..... ....... ..TT..... ........... ......TT..... E118 * 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 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... ............. E119 * 2.00000 -1.99931 1.99726 -1.99383 1.98904 -1.98289 1.97538 -1.96651 1.95630 -1.94474 1.93185 -1.91764 1.90211 -1.88528 1.86716 -1.84776 1.82709 -1.80517 1.78201 -1.75763 1.73205 -1.70528 1.67734 -1.64825 1.61803 -1.58671 1.55429 -1.52081 1.48629 -1.45075 1.41421 -1.37671 1.33826 -1.29890 1.25864 -1.21752 1.17557 -1.13281 1.08928 -1.04500 1.00000 -0.95432 0.90798 -0.86102 0.81347 -0.76537 0.71674 -0.66761 0.61803 -0.56803 0.51764 -0.46689 0.41582 -0.36447 0.31287 -0.26105 0.20906 -0.15692 0.10467 -0.05235 0.00000 0.05235 -0.10467 0.15692 -0.20906 0.26105 -0.31287 0.36447 -0.41582 0.46689 -0.51764 0.56803 -0.61803 0.66761 -0.71674 0.76537 -0.81347 0.86102 -0.90798 0.95432 -1.00000 1.04500 -1.08928 1.13281 -1.17557 1.21752 -1.25864 1.29890 -1.33826 1.37671 -1.41421 1.45075 -1.48629 1.52081 -1.55429 1.58671 -1.61803 1.64825 -1.67734 1.70528 -1.73205 1.75763 -1.78201 1.80517 -1.82709 1.84776 -1.86716 1.88528 -1.90211 1.91764 -1.93185 1.94474 -1.95630 1.96651 -1.97538 1.98289 -1.98904 1.99383 -1.99726 1.99931 -2.00000 TT. ... ..TT. ....... ......TT. ........... ..........TT. Irrational character values: 1.999314649951 = 2*cos(2*π/240) = 2*cos(π/120) 1.997259069509 = 2*cos(4*π/240) = 2*cos(π/60) = (√2-√6-√10+√30+2*(1+√3)*(√5+√5))/8 1.993834667466 = 2*cos(6*π/240) = 2*cos(π/40) = √8+√2+√10+2*√5-√5/2 1.989043790737 = 2*cos(8*π/240) = 2*cos(π/30) = (√15+√3+√10-2*√5)/4 1.982889722748 = 2*cos(10*π/240) = 2*cos(π/24) = √2+√2+√3 1.975376681190 = 2*cos(12*π/240) = 2*cos(π/20) = (√2+√10+2*(√5-√5))/4 1.966509815128 = 2*cos(14*π/240) = 2*cos(7*π/120) 1.956295201468 = 2*cos(16*π/240) = 2*cos(π/15) = (√5-1+√30+6*√5)/4 1.944739840795 = 2*cos(18*π/240) = 2*cos(3*π/40) = √8-√2+√10+2*√5+√5/2 1.931851652578 = 2*cos(20*π/240) = 2*cos(π/12) = √2+√3 1.917639469736 = 2*cos(22*π/240) = 2*cos(11*π/120) 1.902113032590 = 2*cos(24*π/240) = 2*cos(π/10) = (√10+2*√5)/2 1.885282982184 = 2*cos(26*π/240) = 2*cos(13*π/120) 1.867160852994 = 2*cos(28*π/240) = 2*cos(7*π/60) = (√2+√6+√10+√30-2*(1-√3)*(√5-√5))/8 1.847759065023 = 2*cos(30*π/240) = 2*cos(π/8) = √2+√2 1.827090915285 = 2*cos(32*π/240) = 2*cos(2*π/15) = (1+√5+√30-6*√5)/4 1.805170568700 = 2*cos(34*π/240) = 2*cos(17*π/120) 1.782013048377 = 2*cos(36*π/240) = 2*cos(3*π/20) = (√10-√2+2*√5+√5)/4 1.757634225324 = 2*cos(38*π/240) = 2*cos(19*π/120) 1.732050807569 = 2*cos(40*π/240) = 2*cos(π/6) = √3 1.705280328708 = 2*cos(42*π/240) = 2*cos(7*π/40) = √8+√2-√10+2*√5+√5/2 1.677341135891 = 2*cos(44*π/240) = 2*cos(11*π/60) = (√6-√2+√10-√30+2*(1+√3)*(√5+√5))/8 1.648252377244 = 2*cos(46*π/240) = 2*cos(23*π/120) 1.618033988750 = 2*cos(48*π/240) = 2*cos(π/5) = (1+√5)/2 1.586706680582 = 2*cos(50*π/240) = 2*cos(5*π/24) = √2+√2-√3 1.554291922914 = 2*cos(52*π/240) = 2*cos(13*π/60) = (√6-√2-√10+√30+2*(1+√3)*(√5-√5))/8 1.520811931200 = 2*cos(54*π/240) = 2*cos(9*π/40) = √8+√2+√10-2*√5-√5/2 1.486289650955 = 2*cos(56*π/240) = 2*cos(7*π/30) = (√15-√3+√10+2*√5)/4 1.450748742025 = 2*cos(58*π/240) = 2*cos(29*π/120) 1.414213562373 = 2*cos(60*π/240) = 2*cos(π/4) = √2 1.376709151387 = 2*cos(62*π/240) = 2*cos(31*π/120) 1.338261212718 = 2*cos(64*π/240) = 2*cos(4*π/15) = (1-√5+√30+6*√5)/4 1.298896096660 = 2*cos(66*π/240) = 2*cos(11*π/40) = √8-√2-√10+2*√5-√5/2 1.258640782100 = 2*cos(68*π/240) = 2*cos(17*π/60) = (√2+√6+√10+√30+2*(1-√3)*(√5-√5))/8 1.217522858017 = 2*cos(70*π/240) = 2*cos(7*π/24) = √2-√2-√3 1.175570504585 = 2*cos(72*π/240) = 2*cos(3*π/10) = (√10-2*√5)/2 1.132812473850 = 2*cos(74*π/240) = 2*cos(37*π/120) 1.089278070030 = 2*cos(76*π/240) = 2*cos(19*π/60) = (√10-√2-√6+√30-2*(1-√3)*(√5+√5))/8 1.044997129432 = 2*cos(78*π/240) = 2*cos(13*π/40) = √8-√2+√10-2*√5+√5/2 0.954317520519 = 2*cos(82*π/240) = 2*cos(41*π/120) 0.907980999479 = 2*cos(84*π/240) = 2*cos(7*π/20) = (√2-√10+2*√5+√5)/4 0.861022193616 = 2*cos(86*π/240) = 2*cos(43*π/120) 0.813473286151 = 2*cos(88*π/240) = 2*cos(11*π/30) = (√15+√3-√10-2*√5)/4 0.765366864730 = 2*cos(90*π/240) = 2*cos(3*π/8) = √2-√2 0.716735899090 = 2*cos(92*π/240) = 2*cos(23*π/60) = (√2-√6+√10-√30+2*(1+√3)*(√5-√5))/8 0.667613718467 = 2*cos(94*π/240) = 2*cos(47*π/120) 0.618033988750 = 2*cos(96*π/240) = 2*cos(2*π/5) = (√5-1)/2 0.568030689408 = 2*cos(98*π/240) = 2*cos(49*π/120) 0.517638090205 = 2*cos(100*π/240) = 2*cos(5*π/12) = √2-√3 0.466890727712 = 2*cos(102*π/240) = 2*cos(17*π/40) = √8+√2-√10-2*√5+√5/2 0.415823381635 = 2*cos(104*π/240) = 2*cos(13*π/30) = (√3-√15+√10+2*√5)/4 0.364471050984 = 2*cos(106*π/240) = 2*cos(53*π/120) 0.312868930080 = 2*cos(108*π/240) = 2*cos(9*π/20) = (√2+√10-2*(√5-√5))/4 0.261052384440 = 2*cos(110*π/240) = 2*cos(11*π/24) = √2-√2+√3 0.209056926535 = 2*cos(112*π/240) = 2*cos(7*π/15) = (√30-6*√5-√5-1)/4 0.156918191455 = 2*cos(114*π/240) = 2*cos(19*π/40) = √8-√2-√10-2*√5-√5/2 0.104671912486 = 2*cos(116*π/240) = 2*cos(29*π/60) = (√10-√2-√6+√30+2*(1-√3)*(√5+√5))/8 0.052353896616 = 2*cos(118*π/240) = 2*cos(59*π/120) Symmetry of Rotations and Cartesian products A R+d+g+i R_z, z², z⁴, z⁶ B p+f+h z, z³, z⁵ E1 p+f+h {x, y}, {xz², yz²}, {xz⁴, yz⁴} E2 d+g+i {x²–y², xy}, {z²(x²–y²), xyz²}, {z⁴(x²–y²), xyz⁴} E3 f+h {x(x²–3y²), y(3x²–y²)}, {xz²(x²–3y²), yz²(3x²–y²)} E4 g+i {(x²–y²)²–2x²y², xy(x²–y²)}, {z²((x²–y²)²–4x²y²), xyz²(x²–y²)} E5 h {x(x²–(5+2√5)y²)(x²–(5–2√5)y²), y(y²–(5+2√5)x²)(y²–(5–2√5)x²)(y²–(5–2√5)x²)} E6 i {x²(x²–3y²)²–y²(3x²–y²)², xy(x²–3y²)(3x²–y²)} E115 i {xz(x²–(5+2√5)y²)(x²–(5–2√5)y²), yz(y²–(5+2√5)x²)(y²–(5–2√5)x²)(y²–(5–2√5)x²)} E116 h {z((x²–y²)²–4x²y²), xyz(x²–y²)} E117 g+i {xz(x²–3y²), yz(3x²–y²)}, {xz³(x²–3y²), yz³(3x²–y²)} E118 f+h {z(x²–y²), xyz}, {z³(x²–y²), xyz³} E119 R+d+g+i {R_x, R_y}, {xz, yz}, {xz³, yz³}, {xz⁵, yz⁵} Notes: α The order of the S₂₄₀ point group is 240, and the order of the principal axis is 240. β The S₂₄₀ point group could also be named C_240i, because the S₂₄₀ axis is identical to a roto-inversion axis of order 240. γ This is an Abelian point group (the commutative law holds between all symmetry operations). The S₂₄₀ 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. δ Because the group is Abelian and the maximum order of rotation is >2, some irreducible representations have complex characters. These 238 cases have been combined into 119 two-dimensional representations that are no longer irreducible but have real-valued characters. Accordingly, 119 pairs of left and right rotations have been combined into one two-membered pseudo-class each. ε The 119 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. ζ 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 less than half the order of the principal axis (and often much less). η 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. Some of these algebraic expressions, however, are too complicated to be included in the list above.

This Character Table for the S₂₄₀ point group was created by Gernot Katzer.

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

	S₂₀₄
C₂₄₀ C_240v C_240h D₂₄₀ D_240h D_240d	S₂₄₀
	S₂₅₆

Character table for the S240 point group

Character table for the S₂₄₀ point group