Character Table for Point Group D76d

Character table for the D_76d point group

D76d E 2 S152 2 C76 2 S152^3 2 C38 2 S152^5 2 C76^3 2 S152^7 2 C19 2 S152^9 2 C76^5 2 S152^11 2 C38^3 2 S152^13 2 C76^7 2 S152^15 2 C19^2 2 S152^17 2 C76^9 2 S8 2 C38^5 2 S152^21 2 C76^11 2 S152^23 2 C19^3 2 S152^25 2 C76^13 2 S152^27 2 C38^7 2 S152^29 2 C76^15 2 S152^31 2 C19^4 2 S152^33 2 C76^17 2 S152^35 2 C38^9 2 S152^37 2 C4 2 S152^39 2 C19^5 2 S152^41 2 C76^21 2 S152^43 2 C38^11 2 S152^45 2 C76^23 2 S152^47 2 C19^6 2 S152^49 2 C76^25 2 S152^51 2 C38^13 2 S152^53 2 C76^27 2 S152^55 2 C19^7 2 S8^3 2 C76^29 2 S152^59 2 C38^15 2 S152^61 2 C76^31 2 S152^63 2 C19^8 2 S152^65 2 C76^33 2 S152^67 2 C38^17 2 S152^69 2 C76^35 2 S152^71 2 C19^9 2 S152^73 2 C76^37 2 S152^75 C2 76 C2' 76 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 ... ... ....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 ..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 ... ... ..... ....... ......... ........... ............. 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 ... ..T ..... ......T ......... ..........T ............. E1 2.00000 1.99829 1.99317 1.98464 1.97272 1.95743 1.93880 1.91685 1.89163 1.86318 1.83155 1.79678 1.75895 1.71811 1.67433 1.62770 1.57828 1.52617 1.47145 1.41421 1.35456 1.29260 1.22843 1.16215 1.09390 1.02377 0.95189 0.87839 0.80339 0.72702 0.64940 0.57067 0.49097 0.41043 0.32919 0.24739 0.16516 0.08265 0.00000 -0.08265 -0.16516 -0.24739 -0.32919 -0.41043 -0.49097 -0.57067 -0.64940 -0.72702 -0.80339 -0.87839 -0.95189 -1.02377 -1.09390 -1.16215 -1.22843 -1.29260 -1.35456 -1.41421 -1.47145 -1.52617 -1.57828 -1.62770 -1.67433 -1.71811 -1.75895 -1.79678 -1.83155 -1.86318 -1.89163 -1.91685 -1.93880 -1.95743 -1.97272 -1.98464 -1.99317 -1.99829 -2.00000 0.00000 0.00000 ... TT. ..... ....TT. ......... ........TT. ............. E2 2.00000 1.99317 1.97272 1.93880 1.89163 1.83155 1.75895 1.67433 1.57828 1.47145 1.35456 1.22843 1.09390 0.95189 0.80339 0.64940 0.49097 0.32919 0.16516 0.00000 -0.16516 -0.32919 -0.49097 -0.64940 -0.80339 -0.95189 -1.09390 -1.22843 -1.35456 -1.47145 -1.57828 -1.67433 -1.75895 -1.83155 -1.89163 -1.93880 -1.97272 -1.99317 -2.00000 -1.99317 -1.97272 -1.93880 -1.89163 -1.83155 -1.75895 -1.67433 -1.57828 -1.47145 -1.35456 -1.22843 -1.09390 -0.95189 -0.80339 -0.64940 -0.49097 -0.32919 -0.16516 0.00000 0.16516 0.32919 0.49097 0.64940 0.80339 0.95189 1.09390 1.22843 1.35456 1.47145 1.57828 1.67433 1.75895 1.83155 1.89163 1.93880 1.97272 1.99317 2.00000 0.00000 0.00000 ... ... TT... ....... ....TT... ........... ........TT... E3 2.00000 1.98464 1.93880 1.86318 1.75895 1.62770 1.47145 1.29260 1.09390 0.87839 0.64940 0.41043 0.16516 -0.08265 -0.32919 -0.57067 -0.80339 -1.02377 -1.22843 -1.41421 -1.57828 -1.71811 -1.83155 -1.91685 -1.97272 -1.99829 -1.99317 -1.95743 -1.89163 -1.79678 -1.67433 -1.52617 -1.35456 -1.16215 -0.95189 -0.72702 -0.49097 -0.24739 0.00000 0.24739 0.49097 0.72702 0.95189 1.16215 1.35456 1.52617 1.67433 1.79678 1.89163 1.95743 1.99317 1.99829 1.97272 1.91685 1.83155 1.71811 1.57828 1.41421 1.22843 1.02377 0.80339 0.57067 0.32919 0.08265 -0.16516 -0.41043 -0.64940 -0.87839 -1.09390 -1.29260 -1.47145 -1.62770 -1.75895 -1.86318 -1.93880 -1.98464 -2.00000 0.00000 0.00000 ... ... ..... TT..... ......... ....TT..... ............. E4 2.00000 1.97272 1.89163 1.75895 1.57828 1.35456 1.09390 0.80339 0.49097 0.16516 -0.16516 -0.49097 -0.80339 -1.09390 -1.35456 -1.57828 -1.75895 -1.89163 -1.97272 -2.00000 -1.97272 -1.89163 -1.75895 -1.57828 -1.35456 -1.09390 -0.80339 -0.49097 -0.16516 0.16516 0.49097 0.80339 1.09390 1.35456 1.57828 1.75895 1.89163 1.97272 2.00000 1.97272 1.89163 1.75895 1.57828 1.35456 1.09390 0.80339 0.49097 0.16516 -0.16516 -0.49097 -0.80339 -1.09390 -1.35456 -1.57828 -1.75895 -1.89163 -1.97272 -2.00000 -1.97272 -1.89163 -1.75895 -1.57828 -1.35456 -1.09390 -0.80339 -0.49097 -0.16516 0.16516 0.49097 0.80339 1.09390 1.35456 1.57828 1.75895 1.89163 1.97272 2.00000 0.00000 0.00000 ... ... ..... ....... TT....... ........... ....TT....... E5 2.00000 1.95743 1.83155 1.62770 1.35456 1.02377 0.64940 0.24739 -0.16516 -0.57067 -0.95189 -1.29260 -1.57828 -1.79678 -1.93880 -1.99829 -1.97272 -1.86318 -1.67433 -1.41421 -1.09390 -0.72702 -0.32919 0.08265 0.49097 0.87839 1.22843 1.52617 1.75895 1.91685 1.99317 1.98464 1.89163 1.71811 1.47145 1.16215 0.80339 0.41043 0.00000 -0.41043 -0.80339 -1.16215 -1.47145 -1.71811 -1.89163 -1.98464 -1.99317 -1.91685 -1.75895 -1.52617 -1.22843 -0.87839 -0.49097 -0.08265 0.32919 0.72702 1.09390 1.41421 1.67433 1.86318 1.97272 1.99829 1.93880 1.79678 1.57828 1.29260 0.95189 0.57067 0.16516 -0.24739 -0.64940 -1.02377 -1.35456 -1.62770 -1.83155 -1.95743 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... TT......... ............. E6 2.00000 1.93880 1.75895 1.47145 1.09390 0.64940 0.16516 -0.32919 -0.80339 -1.22843 -1.57828 -1.83155 -1.97272 -1.99317 -1.89163 -1.67433 -1.35456 -0.95189 -0.49097 0.00000 0.49097 0.95189 1.35456 1.67433 1.89163 1.99317 1.97272 1.83155 1.57828 1.22843 0.80339 0.32919 -0.16516 -0.64940 -1.09390 -1.47145 -1.75895 -1.93880 -2.00000 -1.93880 -1.75895 -1.47145 -1.09390 -0.64940 -0.16516 0.32919 0.80339 1.22843 1.57828 1.83155 1.97272 1.99317 1.89163 1.67433 1.35456 0.95189 0.49097 0.00000 -0.49097 -0.95189 -1.35456 -1.67433 -1.89163 -1.99317 -1.97272 -1.83155 -1.57828 -1.22843 -0.80339 -0.32919 0.16516 0.64940 1.09390 1.47145 1.75895 1.93880 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... TT........... E7 2.00000 1.91685 1.67433 1.29260 0.80339 0.24739 -0.32919 -0.87839 -1.35456 -1.71811 -1.93880 -1.99829 -1.89163 -1.62770 -1.22843 -0.72702 -0.16516 0.41043 0.95189 1.41421 1.75895 1.95743 1.99317 1.86318 1.57828 1.16215 0.64940 0.08265 -0.49097 -1.02377 -1.47145 -1.79678 -1.97272 -1.98464 -1.83155 -1.52617 -1.09390 -0.57067 0.00000 0.57067 1.09390 1.52617 1.83155 1.98464 1.97272 1.79678 1.47145 1.02377 0.49097 -0.08265 -0.64940 -1.16215 -1.57828 -1.86318 -1.99317 -1.95743 -1.75895 -1.41421 -0.95189 -0.41043 0.16516 0.72702 1.22843 1.62770 1.89163 1.99829 1.93880 1.71811 1.35456 0.87839 0.32919 -0.24739 -0.80339 -1.29260 -1.67433 -1.91685 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E8 2.00000 1.89163 1.57828 1.09390 0.49097 -0.16516 -0.80339 -1.35456 -1.75895 -1.97272 -1.97272 -1.75895 -1.35456 -0.80339 -0.16516 0.49097 1.09390 1.57828 1.89163 2.00000 1.89163 1.57828 1.09390 0.49097 -0.16516 -0.80339 -1.35456 -1.75895 -1.97272 -1.97272 -1.75895 -1.35456 -0.80339 -0.16516 0.49097 1.09390 1.57828 1.89163 2.00000 1.89163 1.57828 1.09390 0.49097 -0.16516 -0.80339 -1.35456 -1.75895 -1.97272 -1.97272 -1.75895 -1.35456 -0.80339 -0.16516 0.49097 1.09390 1.57828 1.89163 2.00000 1.89163 1.57828 1.09390 0.49097 -0.16516 -0.80339 -1.35456 -1.75895 -1.97272 -1.97272 -1.75895 -1.35456 -0.80339 -0.16516 0.49097 1.09390 1.57828 1.89163 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E9 2.00000 1.86318 1.47145 0.87839 0.16516 -0.57067 -1.22843 -1.71811 -1.97272 -1.95743 -1.67433 -1.16215 -0.49097 0.24739 0.95189 1.52617 1.89163 1.99829 1.83155 1.41421 0.80339 0.08265 -0.64940 -1.29260 -1.75895 -1.98464 -1.93880 -1.62770 -1.09390 -0.41043 0.32919 1.02377 1.57828 1.91685 1.99317 1.79678 1.35456 0.72702 0.00000 -0.72702 -1.35456 -1.79678 -1.99317 -1.91685 -1.57828 -1.02377 -0.32919 0.41043 1.09390 1.62770 1.93880 1.98464 1.75895 1.29260 0.64940 -0.08265 -0.80339 -1.41421 -1.83155 -1.99829 -1.89163 -1.52617 -0.95189 -0.24739 0.49097 1.16215 1.67433 1.95743 1.97272 1.71811 1.22843 0.57067 -0.16516 -0.87839 -1.47145 -1.86318 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E10 2.00000 1.83155 1.35456 0.64940 -0.16516 -0.95189 -1.57828 -1.93880 -1.97272 -1.67433 -1.09390 -0.32919 0.49097 1.22843 1.75895 1.99317 1.89163 1.47145 0.80339 0.00000 -0.80339 -1.47145 -1.89163 -1.99317 -1.75895 -1.22843 -0.49097 0.32919 1.09390 1.67433 1.97272 1.93880 1.57828 0.95189 0.16516 -0.64940 -1.35456 -1.83155 -2.00000 -1.83155 -1.35456 -0.64940 0.16516 0.95189 1.57828 1.93880 1.97272 1.67433 1.09390 0.32919 -0.49097 -1.22843 -1.75895 -1.99317 -1.89163 -1.47145 -0.80339 0.00000 0.80339 1.47145 1.89163 1.99317 1.75895 1.22843 0.49097 -0.32919 -1.09390 -1.67433 -1.97272 -1.93880 -1.57828 -0.95189 -0.16516 0.64940 1.35456 1.83155 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E11 2.00000 1.79678 1.22843 0.41043 -0.49097 -1.29260 -1.83155 -1.99829 -1.75895 -1.16215 -0.32919 0.57067 1.35456 1.86318 1.99317 1.71811 1.09390 0.24739 -0.64940 -1.41421 -1.89163 -1.98464 -1.67433 -1.02377 -0.16516 0.72702 1.47145 1.91685 1.97272 1.62770 0.95189 0.08265 -0.80339 -1.52617 -1.93880 -1.95743 -1.57828 -0.87839 0.00000 0.87839 1.57828 1.95743 1.93880 1.52617 0.80339 -0.08265 -0.95189 -1.62770 -1.97272 -1.91685 -1.47145 -0.72702 0.16516 1.02377 1.67433 1.98464 1.89163 1.41421 0.64940 -0.24739 -1.09390 -1.71811 -1.99317 -1.86318 -1.35456 -0.57067 0.32919 1.16215 1.75895 1.99829 1.83155 1.29260 0.49097 -0.41043 -1.22843 -1.79678 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E12 2.00000 1.75895 1.09390 0.16516 -0.80339 -1.57828 -1.97272 -1.89163 -1.35456 -0.49097 0.49097 1.35456 1.89163 1.97272 1.57828 0.80339 -0.16516 -1.09390 -1.75895 -2.00000 -1.75895 -1.09390 -0.16516 0.80339 1.57828 1.97272 1.89163 1.35456 0.49097 -0.49097 -1.35456 -1.89163 -1.97272 -1.57828 -0.80339 0.16516 1.09390 1.75895 2.00000 1.75895 1.09390 0.16516 -0.80339 -1.57828 -1.97272 -1.89163 -1.35456 -0.49097 0.49097 1.35456 1.89163 1.97272 1.57828 0.80339 -0.16516 -1.09390 -1.75895 -2.00000 -1.75895 -1.09390 -0.16516 0.80339 1.57828 1.97272 1.89163 1.35456 0.49097 -0.49097 -1.35456 -1.89163 -1.97272 -1.57828 -0.80339 0.16516 1.09390 1.75895 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E13 2.00000 1.71811 0.95189 -0.08265 -1.09390 -1.79678 -1.99317 -1.62770 -0.80339 0.24739 1.22843 1.86318 1.97272 1.52617 0.64940 -0.41043 -1.35456 -1.91685 -1.93880 -1.41421 -0.49097 0.57067 1.47145 1.95743 1.89163 1.29260 0.32919 -0.72702 -1.57828 -1.98464 -1.83155 -1.16215 -0.16516 0.87839 1.67433 1.99829 1.75895 1.02377 0.00000 -1.02377 -1.75895 -1.99829 -1.67433 -0.87839 0.16516 1.16215 1.83155 1.98464 1.57828 0.72702 -0.32919 -1.29260 -1.89163 -1.95743 -1.47145 -0.57067 0.49097 1.41421 1.93880 1.91685 1.35456 0.41043 -0.64940 -1.52617 -1.97272 -1.86318 -1.22843 -0.24739 0.80339 1.62770 1.99317 1.79678 1.09390 0.08265 -0.95189 -1.71811 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E14 2.00000 1.67433 0.80339 -0.32919 -1.35456 -1.93880 -1.89163 -1.22843 -0.16516 0.95189 1.75895 1.99317 1.57828 0.64940 -0.49097 -1.47145 -1.97272 -1.83155 -1.09390 0.00000 1.09390 1.83155 1.97272 1.47145 0.49097 -0.64940 -1.57828 -1.99317 -1.75895 -0.95189 0.16516 1.22843 1.89163 1.93880 1.35456 0.32919 -0.80339 -1.67433 -2.00000 -1.67433 -0.80339 0.32919 1.35456 1.93880 1.89163 1.22843 0.16516 -0.95189 -1.75895 -1.99317 -1.57828 -0.64940 0.49097 1.47145 1.97272 1.83155 1.09390 0.00000 -1.09390 -1.83155 -1.97272 -1.47145 -0.49097 0.64940 1.57828 1.99317 1.75895 0.95189 -0.16516 -1.22843 -1.89163 -1.93880 -1.35456 -0.32919 0.80339 1.67433 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E15 2.00000 1.62770 0.64940 -0.57067 -1.57828 -1.99829 -1.67433 -0.72702 0.49097 1.52617 1.99317 1.71811 0.80339 -0.41043 -1.47145 -1.98464 -1.75895 -0.87839 0.32919 1.41421 1.97272 1.79678 0.95189 -0.24739 -1.35456 -1.95743 -1.83155 -1.02377 0.16516 1.29260 1.93880 1.86318 1.09390 -0.08265 -1.22843 -1.91685 -1.89163 -1.16215 0.00000 1.16215 1.89163 1.91685 1.22843 0.08265 -1.09390 -1.86318 -1.93880 -1.29260 -0.16516 1.02377 1.83155 1.95743 1.35456 0.24739 -0.95189 -1.79678 -1.97272 -1.41421 -0.32919 0.87839 1.75895 1.98464 1.47145 0.41043 -0.80339 -1.71811 -1.99317 -1.52617 -0.49097 0.72702 1.67433 1.99829 1.57828 0.57067 -0.64940 -1.62770 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. 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E74 2.00000 -1.99317 1.97272 -1.93880 1.89163 -1.83155 1.75895 -1.67433 1.57828 -1.47145 1.35456 -1.22843 1.09390 -0.95189 0.80339 -0.64940 0.49097 -0.32919 0.16516 0.00000 -0.16516 0.32919 -0.49097 0.64940 -0.80339 0.95189 -1.09390 1.22843 -1.35456 1.47145 -1.57828 1.67433 -1.75895 1.83155 -1.89163 1.93880 -1.97272 1.99317 -2.00000 1.99317 -1.97272 1.93880 -1.89163 1.83155 -1.75895 1.67433 -1.57828 1.47145 -1.35456 1.22843 -1.09390 0.95189 -0.80339 0.64940 -0.49097 0.32919 -0.16516 0.00000 0.16516 -0.32919 0.49097 -0.64940 0.80339 -0.95189 1.09390 -1.22843 1.35456 -1.47145 1.57828 -1.67433 1.75895 -1.83155 1.89163 -1.93880 1.97272 -1.99317 2.00000 0.00000 0.00000 ... ... ..... ..TT... ......... ......TT... ............. E75 2.00000 -1.99829 1.99317 -1.98464 1.97272 -1.95743 1.93880 -1.91685 1.89163 -1.86318 1.83155 -1.79678 1.75895 -1.71811 1.67433 -1.62770 1.57828 -1.52617 1.47145 -1.41421 1.35456 -1.29260 1.22843 -1.16215 1.09390 -1.02377 0.95189 -0.87839 0.80339 -0.72702 0.64940 -0.57067 0.49097 -0.41043 0.32919 -0.24739 0.16516 -0.08265 0.00000 0.08265 -0.16516 0.24739 -0.32919 0.41043 -0.49097 0.57067 -0.64940 0.72702 -0.80339 0.87839 -0.95189 1.02377 -1.09390 1.16215 -1.22843 1.29260 -1.35456 1.41421 -1.47145 1.52617 -1.57828 1.62770 -1.67433 1.71811 -1.75895 1.79678 -1.83155 1.86318 -1.89163 1.91685 -1.93880 1.95743 -1.97272 1.98464 -1.99317 1.99829 -2.00000 0.00000 0.00000 TT. ... ..TT. ....... ......TT. ........... ..........TT. Irrational character values: 1.998291516775 = 2*cos(2*π/152) = 2*cos(π/76) 1.993168986013 = 2*cos(4*π/152) = 2*cos(π/38) 1.984641159474 = 2*cos(6*π/152) = 2*cos(3*π/76) 1.972722606805 = 2*cos(8*π/152) = 2*cos(π/19) 1.957433690655 = 2*cos(10*π/152) = 2*cos(5*π/76) 1.938800531879 = 2*cos(12*π/152) = 2*cos(3*π/38) 1.916854964917 = 2*cos(14*π/152) = 2*cos(7*π/76) 1.891634483401 = 2*cos(16*π/152) = 2*cos(2*π/19) 1.863182176103 = 2*cos(18*π/152) = 2*cos(9*π/76) 1.831546653310 = 2*cos(20*π/152) = 2*cos(5*π/38) 1.796781963784 = 2*cos(22*π/152) = 2*cos(11*π/76) 1.758947502413 = 2*cos(24*π/152) = 2*cos(3*π/19) 1.718107908740 = 2*cos(26*π/152) = 2*cos(13*π/76) 1.674332956525 = 2*cos(28*π/152) = 2*cos(7*π/38) 1.627697434540 = 2*cos(30*π/152) = 2*cos(15*π/76) 1.578281018793 = 2*cos(32*π/152) = 2*cos(4*π/19) 1.526168136400 = 2*cos(34*π/152) = 2*cos(17*π/76) 1.471447821346 = 2*cos(36*π/152) = 2*cos(9*π/38) 1.414213562373 = 2*cos(38*π/152) = 2*cos(π/4) = √2 1.354563143251 = 2*cos(40*π/152) = 2*cos(5*π/19) 1.292598475722 = 2*cos(42*π/152) = 2*cos(21*π/76) 1.228425425379 = 2*cos(44*π/152) = 2*cos(11*π/38) 1.162153630804 = 2*cos(46*π/152) = 2*cos(23*π/76) 1.093896316245 = 2*cos(48*π/152) = 2*cos(6*π/19) 1.023770098179 = 2*cos(50*π/152) = 2*cos(25*π/76) 0.951894786074 = 2*cos(52*π/152) = 2*cos(13*π/38) 0.878393177695 = 2*cos(54*π/152) = 2*cos(27*π/76) 0.803390849306 = 2*cos(56*π/152) = 2*cos(7*π/19) 0.727015941128 = 2*cos(58*π/152) = 2*cos(29*π/76) 0.649398938409 = 2*cos(60*π/152) = 2*cos(15*π/38) 0.570672448498 = 2*cos(62*π/152) = 2*cos(31*π/76) 0.490970974282 = 2*cos(64*π/152) = 2*cos(8*π/19) 0.410430684391 = 2*cos(66*π/152) = 2*cos(33*π/76) 0.329189180561 = 2*cos(68*π/152) = 2*cos(17*π/38) 0.247385262539 = 2*cos(70*π/152) = 2*cos(35*π/76) 0.165158690945 = 2*cos(72*π/152) = 2*cos(9*π/19) 0.082649948498 = 2*cos(74*π/152) = 2*cos(37*π/76) 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 E67 m E68 l E69 k+m E70 j+l E71 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)} E72 h+j+l {z((x²−y²)²−4x²y²), xyz(x²−y²)} E73 g+i+k+m {xz(x²−3y²), yz(3x²−y²)}, {xz³(x²−3y²), yz³(3x²−y²)} E74 f+h+j+l {z(x²−y²), xyz}, {z³(x²−y²), xyz³} E75 R+d+g+i+k+m {R_x, R_y}, {xz, yz}, {xz³, yz³}, {xz⁵, yz⁵} Notes: α The order of the D_76d point group is 304, and the order of the principal axis (S₁₅₂) is 152. The group has 79 irreducible representations. β The D_76d point group is isomorphic to C_152v and D₁₅₂. γ The D_76d point group is generated by two symmetry elements, S₁₅₂ 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 group contains one set of twofold symmetry axes (C₂^′) perpendicular to the principal (z) axis. Both x and y axes are members of that set. ε The single σ_d set of symmetry planes contains neither the xz nor the yz planes; but it contains the median plane (x+y)z. ζ The lowest nonvanishing multipole moment in D_76d is 4 (quadrupole moment). η This point group is non-Abelian (some symmetry operations are not commutative). Therefore, the character table contains multi-membered classes and degenerate irreducible representations. θ Some of the characters in the table are irrational because the order of the principal axis is neither 1,2,3,4 nor 6. These irrational values can be expressed as cosine values, or as solutions of algebraic equations with a leading coefficient of 1. All characters are algebraic integers of a degree much less than half the order of the principal axis. ι The point group corresponds to a polygon inconstructible by the classical means of ruler and compass. Yet it becomes constructible if angle trisection is allowed, e.g., with neusis construction or origami. This is because the order of the principal axis is given by a product of any number of different Pierpont primes (...,5,7,13,17,19,37,73,97,109,163,...) times arbitrary powers of two and three. While some characters of this group can be expressed using real integers and square roots alone, others need complex numbers and third roots.

This Character Table for the D_76d point group was created by Gernot Katzer.

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

	D_74d
	D_75d
C₇₆ C_76v C_76h D₇₆ D_76h	D_76d	S₇₆
	D_77d
	D_78d

Character table for the D76d point group

Character table for the D_76d point group