Character Table for Point Group D240d

Character table for the D_240d point group

D240d E 2 S480 2 C240 2 S160 2 C120 2 S96 2 C80 2 S480^7 2 C60 2 S160^3 2 C48 2 S480^11 2 C40 2 S480^13 2 C240^7 2 S32 2 C30 2 S480^17 2 C80^3 2 S480^19 2 C24 2 S160^7 2 C240^11 2 S480^23 2 C20 2 S96^5 2 C240^13 2 S160^9 2 C120^7 2 S480^29 2 C16 2 S480^31 2 C15 2 S160^11 2 C240^17 2 S96^7 2 C40^3 2 S480^37 2 C240^19 2 S160^13 2 C12 2 S480^41 2 C80^7 2 S480^43 2 C120^11 2 S32^3 2 C240^23 2 S480^47 2 C10 2 S480^49 2 C48^5 2 S160^17 2 C120^13 2 S480^53 2 C80^9 2 S96^11 2 C60^7 2 S160^19 2 C240^29 2 S480^59 2 C8 2 S480^61 2 C240^31 2 S160^21 2 C15^2 2 S96^13 2 C80^11 2 S480^67 2 C120^17 2 S160^23 2 C48^7 2 S480^71 2 C20^3 2 S480^73 2 C240^37 2 S32^5 2 C120^19 2 S480^77 2 C80^13 2 S480^79 2 C6 2 S160^27 2 C240^41 2 S480^83 2 C40^7 2 S96^17 2 C240^43 2 S160^29 2 C60^11 2 S480^89 2 C16^3 2 S480^91 2 C120^23 2 S160^31 2 C240^47 2 S96^19 2 C5 2 S480^97 2 C240^49 2 S160^33 2 C24^5 2 S480^101 2 C80^17 2 S480^103 2 C60^13 2 S32^7 2 C240^53 2 S480^107 2 C40^9 2 S480^109 2 C48^11 2 S160^37 2 C30^7 2 S480^113 2 C80^19 2 S96^23 2 C120^29 2 S160^39 2 C240^59 2 S480^119 2 C4 2 S480^121 2 C240^61 2 S160^41 2 C120^31 2 S96^25 2 C80^21 2 S480^127 2 C15^4 2 S160^43 2 C48^13 2 S480^131 2 C40^11 2 S480^133 2 C240^67 2 S32^9 2 C60^17 2 S480^137 2 C80^23 2 S480^139 2 C24^7 2 S160^47 2 C240^71 2 S480^143 2 C10^3 2 S96^29 2 C240^73 2 S160^49 2 C120^37 2 S480^149 2 C16^5 2 S480^151 2 C60^19 2 S160^51 2 C240^77 2 S96^31 2 C40^13 2 S480^157 2 C240^79 2 S160^53 2 C3 2 S480^161 2 C80^27 2 S480^163 2 C120^41 2 S32^11 2 C240^83 2 S480^167 2 C20^7 2 S480^169 2 C48^17 2 S160^57 2 C120^43 2 S480^173 2 C80^29 2 S96^35 2 C30^11 2 S160^59 2 C240^89 2 S480^179 2 C8^3 2 S480^181 2 C240^91 2 S160^61 2 C60^23 2 S96^37 2 C80^31 2 S480^187 2 C120^47 2 S160^63 2 C48^19 2 S480^191 2 C5^2 2 S480^193 2 C240^97 2 S32^13 2 C120^49 2 S480^197 2 C80^33 2 S480^199 2 C12^5 2 S160^67 2 C240^101 2 S480^203 2 C40^17 2 S96^41 2 C240^103 2 S160^69 2 C30^13 2 S480^209 2 C16^7 2 S480^211 2 C120^53 2 S160^71 2 C240^107 2 S96^43 2 C20^9 2 S480^217 2 C240^109 2 S160^73 2 C24^11 2 S480^221 2 C80^37 2 S480^223 2 C15^7 2 S32^15 2 C240^113 2 S480^227 2 C40^19 2 S480^229 2 C48^23 2 S160^77 2 C60^29 2 S480^233 2 C80^39 2 S96^47 2 C120^59 2 S160^79 2 C240^119 2 S480^239 C2 240 C2' 240 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 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 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1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 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 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 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1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 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 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -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 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -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 1.99983 1.99931 1.99846 1.99726 1.99572 1.99383 1.99161 1.98904 1.98614 1.98289 1.97930 1.97538 1.97111 1.96651 1.96157 1.95630 1.95068 1.94474 1.93846 1.93185 1.92491 1.91764 1.91004 1.90211 1.89386 1.88528 1.87638 1.86716 1.85762 1.84776 1.83758 1.82709 1.81629 1.80517 1.79375 1.78201 1.76998 1.75763 1.74499 1.73205 1.71881 1.70528 1.69146 1.67734 1.66294 1.64825 1.63328 1.61803 1.60251 1.58671 1.57063 1.55429 1.53768 1.52081 1.50368 1.48629 1.46865 1.45075 1.43260 1.41421 1.39558 1.37671 1.35760 1.33826 1.31869 1.29890 1.27888 1.25864 1.23819 1.21752 1.19665 1.17557 1.15429 1.13281 1.11114 1.08928 1.06723 1.04500 1.02259 1.00000 0.97724 0.95432 0.93123 0.90798 0.88458 0.86102 0.83732 0.81347 0.78949 0.76537 0.74111 0.71674 0.69223 0.66761 0.64288 0.61803 0.59308 0.56803 0.54288 0.51764 0.49231 0.46689 0.44139 0.41582 0.39018 0.36447 0.33870 0.31287 0.28699 0.26105 0.23507 0.20906 0.18300 0.15692 0.13081 0.10467 0.07852 0.05235 0.02618 0.00000 -0.02618 -0.05235 -0.07852 -0.10467 -0.13081 -0.15692 -0.18300 -0.20906 -0.23507 -0.26105 -0.28699 -0.31287 -0.33870 -0.36447 -0.39018 -0.41582 -0.44139 -0.46689 -0.49231 -0.51764 -0.54288 -0.56803 -0.59308 -0.61803 -0.64288 -0.66761 -0.69223 -0.71674 -0.74111 -0.76537 -0.78949 -0.81347 -0.83732 -0.86102 -0.88458 -0.90798 -0.93123 -0.95432 -0.97724 -1.00000 -1.02259 -1.04500 -1.06723 -1.08928 -1.11114 -1.13281 -1.15429 -1.17557 -1.19665 -1.21752 -1.23819 -1.25864 -1.27888 -1.29890 -1.31869 -1.33826 -1.35760 -1.37671 -1.39558 -1.41421 -1.43260 -1.45075 -1.46865 -1.48629 -1.50368 -1.52081 -1.53768 -1.55429 -1.57063 -1.58671 -1.60251 -1.61803 -1.63328 -1.64825 -1.66294 -1.67734 -1.69146 -1.70528 -1.71881 -1.73205 -1.74499 -1.75763 -1.76998 -1.78201 -1.79375 -1.80517 -1.81629 -1.82709 -1.83758 -1.84776 -1.85762 -1.86716 -1.87638 -1.88528 -1.89386 -1.90211 -1.91004 -1.91764 -1.92491 -1.93185 -1.93846 -1.94474 -1.95068 -1.95630 -1.96157 -1.96651 -1.97111 -1.97538 -1.97930 -1.98289 -1.98614 -1.98904 -1.99161 -1.99383 -1.99572 -1.99726 -1.99846 -1.99931 -1.99983 -2.00000 0.00000 0.00000 ... TT. ..... ....TT. ......... ........TT. ............. E2 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 -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 0.00000 0.00000 ... ... TT... ....... ....TT... ........... ........TT... 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TT..... ......... ....TT..... ............. 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TT........... 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Irrational character values: 1.999828655148 = 2*cos(2*π/480) = 2*cos(π/240) 1.999314649951 = 2*cos(4*π/480) = 2*cos(π/120) 1.998458072481 = 2*cos(6*π/480) = 2*cos(π/80) = √4+√8+√2+√10+2*√5−√5/√2 1.997259069509 = 2*cos(8*π/480) = 2*cos(π/60) = (√2−√6−√10+√30+2*(1+√3)*(√5+√5))/8 1.995717846477 = 2*cos(10*π/480) = 2*cos(π/48) = √2+√2+√2+√3 1.993834667466 = 2*cos(12*π/480) = 2*cos(π/40) = √8+√2+√10+2*√5−√5/2 1.991609855149 = 2*cos(14*π/480) = 2*cos(7*π/240) 1.989043790737 = 2*cos(16*π/480) = 2*cos(π/30) = (√15+√3+√10−2*√5)/4 1.986136913910 = 2*cos(18*π/480) = 2*cos(3*π/80) = √4+√8−√2+√10+2*√5+√5/√2 1.982889722748 = 2*cos(20*π/480) = 2*cos(π/24) = √2+√2+√3 1.979302773639 = 2*cos(22*π/480) = 2*cos(11*π/240) 1.975376681190 = 2*cos(24*π/480) = 2*cos(π/20) = (√2+√10+2*(√5−√5))/4 1.971112118116 = 2*cos(26*π/480) = 2*cos(13*π/240) 1.966509815128 = 2*cos(28*π/480) = 2*cos(7*π/120) 1.961570560806 = 2*cos(30*π/480) = 2*cos(π/16) = √2+√2+√2 1.956295201468 = 2*cos(32*π/480) = 2*cos(π/15) = (√5−1+√30+6*√5)/4 1.950684641017 = 2*cos(34*π/480) = 2*cos(17*π/240) 1.944739840795 = 2*cos(36*π/480) = 2*cos(3*π/40) = √8−√2+√10+2*√5+√5/2 1.938461819414 = 2*cos(38*π/480) = 2*cos(19*π/240) 1.931851652578 = 2*cos(40*π/480) = 2*cos(π/12) = √2+√3 1.924910472907 = 2*cos(42*π/480) = 2*cos(7*π/80) = √4+√8+√2−√10+2*√5+√5/√2 1.917639469736 = 2*cos(44*π/480) = 2*cos(11*π/120) 1.910039888914 = 2*cos(46*π/480) = 2*cos(23*π/240) 1.902113032590 = 2*cos(48*π/480) = 2*cos(π/10) = (√10+2*√5)/2 1.893860258990 = 2*cos(50*π/480) = 2*cos(5*π/48) = √2+√2+√2−√3 1.885282982184 = 2*cos(52*π/480) = 2*cos(13*π/120) 1.876382671845 = 2*cos(54*π/480) = 2*cos(9*π/80) = √4+√8+√2+√10−2*√5−√5/√2 1.867160852994 = 2*cos(56*π/480) = 2*cos(7*π/60) = (√2+√6+√10+√30−2*(1−√3)*(√5−√5))/8 1.857619105744 = 2*cos(58*π/480) = 2*cos(29*π/240) 1.847759065023 = 2*cos(60*π/480) = 2*cos(π/8) = √2+√2 1.837582420298 = 2*cos(62*π/480) = 2*cos(31*π/240) 1.827090915285 = 2*cos(64*π/480) = 2*cos(2*π/15) = (1+√5+√30−6*√5)/4 1.816286347650 = 2*cos(66*π/480) = 2*cos(11*π/80) = √4+√8−√2−√10+2*√5−√5/√2 1.805170568700 = 2*cos(68*π/480) = 2*cos(17*π/120) 1.793745483065 = 2*cos(70*π/480) = 2*cos(7*π/48) = √2+√2−√2−√3 1.782013048377 = 2*cos(72*π/480) = 2*cos(3*π/20) = (√10−√2+2*√5+√5)/4 1.769975274926 = 2*cos(74*π/480) = 2*cos(37*π/240) 1.757634225324 = 2*cos(76*π/480) = 2*cos(19*π/120) 1.744992014146 = 2*cos(78*π/480) = 2*cos(13*π/80) = √4+√8−√2+√10−2*√5+√5/√2 1.732050807569 = 2*cos(80*π/480) = 2*cos(π/6) = √3 1.718812823003 = 2*cos(82*π/480) = 2*cos(41*π/240) 1.705280328708 = 2*cos(84*π/480) = 2*cos(7*π/40) = √8+√2−√10+2*√5+√5/2 1.691455643408 = 2*cos(86*π/480) = 2*cos(43*π/240) 1.677341135891 = 2*cos(88*π/480) = 2*cos(11*π/60) = (√6−√2+√10−√30+2*(1+√3)*(√5+√5))/8 1.662939224605 = 2*cos(90*π/480) = 2*cos(3*π/16) = √2+√2−√2 1.648252377244 = 2*cos(92*π/480) = 2*cos(23*π/120) 1.633283110323 = 2*cos(94*π/480) = 2*cos(47*π/240) 1.618033988750 = 2*cos(96*π/480) = 2*cos(π/5) = (√5+1)/2 1.602507625382 = 2*cos(98*π/480) = 2*cos(49*π/240) 1.586706680582 = 2*cos(100*π/480) = 2*cos(5*π/24) = √2+√2−√3 1.570633861761 = 2*cos(102*π/480) = 2*cos(17*π/80) = √4+√8+√2−√10−2*√5+√5/√2 1.554291922914 = 2*cos(104*π/480) = 2*cos(13*π/60) = (√6−√2−√10+√30+2*(1+√3)*(√5−√5))/8 1.537683664147 = 2*cos(106*π/480) = 2*cos(53*π/240) 1.520811931200 = 2*cos(108*π/480) = 2*cos(9*π/40) = √8+√2+√10−2*√5−√5/2 1.503679614958 = 2*cos(110*π/480) = 2*cos(11*π/48) = √2+√2−√2+√3 1.486289650955 = 2*cos(112*π/480) = 2*cos(7*π/30) = (√15−√3+√10+2*√5)/4 1.468645018871 = 2*cos(114*π/480) = 2*cos(19*π/80) = √4+√8−√2−√10−2*√5−√5/√2 1.450748742025 = 2*cos(116*π/480) = 2*cos(29*π/120) 1.432603886849 = 2*cos(118*π/480) = 2*cos(59*π/240) 1.414213562373 = 2*cos(120*π/480) = 2*cos(π/4) = √2 1.395580919683 = 2*cos(122*π/480) = 2*cos(61*π/240) 1.376709151388 = 2*cos(124*π/480) = 2*cos(31*π/120) 1.357601491066 = 2*cos(126*π/480) = 2*cos(21*π/80) = √4−√8−√2−√10−2*√5−√5/√2 1.338261212718 = 2*cos(128*π/480) = 2*cos(4*π/15) = (1−√5+√30+6*√5)/4 1.318691630200 = 2*cos(130*π/480) = 2*cos(13*π/48) = √2−√2−√2+√3 1.298896096660 = 2*cos(132*π/480) = 2*cos(11*π/40) = √8−√2−√10+2*√5−√5/2 1.278878003961 = 2*cos(134*π/480) = 2*cos(67*π/240) 1.258640782100 = 2*cos(136*π/480) = 2*cos(17*π/60) = (√2+√6+√10+√30+2*(1−√3)*(√5−√5))/8 1.238187898620 = 2*cos(138*π/480) = 2*cos(23*π/80) = √4−√8+√2−√10−2*√5+√5/√2 1.217522858017 = 2*cos(140*π/480) = 2*cos(7*π/24) = √2−√2−√3 1.196649201141 = 2*cos(142*π/480) = 2*cos(71*π/240) 1.175570504585 = 2*cos(144*π/480) = 2*cos(3*π/10) = (√10−2*√5)/2 1.154290380074 = 2*cos(146*π/480) = 2*cos(73*π/240) 1.132812473850 = 2*cos(148*π/480) = 2*cos(37*π/120) 1.111140466039 = 2*cos(150*π/480) = 2*cos(5*π/16) = √2−√2−√2 1.089278070030 = 2*cos(152*π/480) = 2*cos(19*π/60) = (√10−√2−√6+√30−2*(1−√3)*(√5+√5))/8 1.067229031831 = 2*cos(154*π/480) = 2*cos(77*π/240) 1.044997129432 = 2*cos(156*π/480) = 2*cos(13*π/40) = √8−√2+√10−2*√5+√5/2 1.022586172154 = 2*cos(158*π/480) = 2*cos(79*π/240) 0.977242482994 = 2*cos(162*π/480) = 2*cos(27*π/80) = √4−√8−√2+√10−2*√5+√5/√2 0.954317520519 = 2*cos(164*π/480) = 2*cos(41*π/120) 0.931229040650 = 2*cos(166*π/480) = 2*cos(83*π/240) 0.907980999479 = 2*cos(168*π/480) = 2*cos(7*π/20) = (√2−√10+2*√5+√5)/4 0.884577380438 = 2*cos(170*π/480) = 2*cos(17*π/48) = √2−√2−√2−√3 0.861022193617 = 2*cos(172*π/480) = 2*cos(43*π/120) 0.837319475075 = 2*cos(174*π/480) = 2*cos(29*π/80) = √4−√8−√2−√10+2*√5−√5/√2 0.813473286152 = 2*cos(176*π/480) = 2*cos(11*π/30) = (√15+√3−√10−2*√5)/4 0.789487712769 = 2*cos(178*π/480) = 2*cos(89*π/240) 0.765366864730 = 2*cos(180*π/480) = 2*cos(3*π/8) = √2−√2 0.741114875020 = 2*cos(182*π/480) = 2*cos(91*π/240) 0.716735899091 = 2*cos(184*π/480) = 2*cos(23*π/60) = (√2−√6+√10−√30+2*(1+√3)*(√5−√5))/8 0.692234114155 = 2*cos(186*π/480) = 2*cos(31*π/80) = √4−√8+√2+√10−2*√5−√5/√2 0.667613718468 = 2*cos(188*π/480) = 2*cos(47*π/120) 0.642878930606 = 2*cos(190*π/480) = 2*cos(19*π/48) = √2−√2+√2−√3 0.618033988750 = 2*cos(192*π/480) = 2*cos(2*π/5) = (√5−1)/2 0.593083149951 = 2*cos(194*π/480) = 2*cos(97*π/240) 0.568030689408 = 2*cos(196*π/480) = 2*cos(49*π/120) 0.542880899730 = 2*cos(198*π/480) = 2*cos(33*π/80) = √4−√8+√2−√10+2*√5+√5/√2 0.517638090205 = 2*cos(200*π/480) = 2*cos(5*π/12) = √2−√3 0.492306586058 = 2*cos(202*π/480) = 2*cos(101*π/240) 0.466890727712 = 2*cos(204*π/480) = 2*cos(17*π/40) = √8+√2−√10−2*√5+√5/2 0.441394870043 = 2*cos(206*π/480) = 2*cos(103*π/240) 0.415823381636 = 2*cos(208*π/480) = 2*cos(13*π/30) = (√3−√15+√10+2*√5)/4 0.390180644032 = 2*cos(210*π/480) = 2*cos(7*π/16) = √2−√2+√2 0.364471050984 = 2*cos(212*π/480) = 2*cos(53*π/120) 0.338699007698 = 2*cos(214*π/480) = 2*cos(107*π/240) 0.312868930080 = 2*cos(216*π/480) = 2*cos(9*π/20) = (√2+√10−2*(√5−√5))/4 0.286985243982 = 2*cos(218*π/480) = 2*cos(109*π/240) 0.261052384440 = 2*cos(220*π/480) = 2*cos(11*π/24) = √2−√2+√3 0.235074794916 = 2*cos(222*π/480) = 2*cos(37*π/80) = √4−√8−√2+√10+2*√5+√5/√2 0.209056926535 = 2*cos(224*π/480) = 2*cos(7*π/15) = (√30−6*√5−√5−1)/4 0.183003237327 = 2*cos(226*π/480) = 2*cos(113*π/240) 0.156918191456 = 2*cos(228*π/480) = 2*cos(19*π/40) = √8−√2−√10−2*√5−√5/2 0.130806258460 = 2*cos(230*π/480) = 2*cos(23*π/48) = √2−√2+√2+√3 0.104671912486 = 2*cos(232*π/480) = 2*cos(29*π/60) = (√10−√2−√6+√30+2*(1−√3)*(√5+√5))/8 0.078519631518 = 2*cos(234*π/480) = 2*cos(39*π/80) = √4−√8+√2+√10+2*√5−√5/√2 0.052353896616 = 2*cos(236*π/480) = 2*cos(59*π/120) 0.026179191143 = 2*cos(238*π/480) = 2*cos(119*π/240) Symmetry of Rotations and Cartesian products A1 d+g+i+k+m z², z⁴, z⁶ A2 R R_z B2 p+f+h+j+l z, z³, z⁵ E1 p+f+h+j+l {x, y}, {xz², yz²}, {xz⁴, yz⁴} E2 d+g+i+k+m {x²−y², xy}, {z²(x²−y²), xyz²}, {z⁴(x²−y²), xyz⁴} E3 f+h+j+l {x(x²−3y²), y(3x²−y²)}, {xz²(x²−3y²), yz²(3x²−y²)} E4 g+i+k+m {(x²−y²)²−4x²y², xy(x²−y²)}, {z²((x²−y²)²−4x²y²), xyz²(x²−y²)} E5 h+j+l {x(x²−(5+2√5)y²)(x²−(5−2√5)y²), y((5+2√5)x²−y²)((5−2√5)x²−y²)} E6 i+k+m {x²(x²−3y²)²−y²(3x²−y²)², xy(x²−3y²)(3x²−y²)} E7 j+l E8 k+m E9 l E10 m E231 m E232 l E233 k+m E234 j+l E235 i+k+m {xz(x²−(5+2√5)y²)(x²−(5−2√5)y²), yz((5+2√5)x²−y²)((5−2√5)x²−y²)} E236 h+j+l {z((x²−y²)²−4x²y²), xyz(x²−y²)} E237 g+i+k+m {xz(x²−3y²), yz(3x²−y²)}, {xz³(x²−3y²), yz³(3x²−y²)} E238 f+h+j+l {z(x²−y²), xyz}, {z³(x²−y²), xyz³} E239 R+d+g+i+k+m {R_x, R_y}, {xz, yz}, {xz³, yz³}, {xz⁵, yz⁵} Notes: α The order of the D_240d point group is 960, and the order of the principal axis (S₄₈₀) is 480. The group has 243 irreducible representations. β The D_240d point group is isomorphic to C_480v and D₄₈₀. γ The D_240d point group is generated by two symmetry elements, S₄₈₀ and either a perpendicular C₂^′ or a vertical σ_d. Also, the group may be generated from a C₂^′ plus a σ_d (some pairs will yield smaller groups, though; choosing a minimum angle is safe). δ The lowest nonvanishing multipole moment in D_240d is 4 (quadrupole moment). ε This point group is non-Abelian (some symmetry operations are not commutative). Therefore, the character table contains multi-membered classes and degenerate irreducible representations. 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 much less than half the order of the principal axis. 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. θ 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 D_240d point group was created by Gernot Katzer.

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

	D_192d
	D_204d
C₂₄₀ C_240v C_240h D₂₄₀ D_240h	D_240d	S₂₄₀
	D_255d
	D_256d

Character table for the D240d point group

Character table for the D_240d point group