Character Table for Point Group C96v

Character table for the C_96v point group

C96v E 2 C96 2 C48 2 C32 2 C24 2 C96^5 2 C16 2 C96^7 2 C12 2 C32^3 2 C48^5 2 C96^11 2 C8 2 C96^13 2 C48^7 2 C32^5 2 C6 2 C96^17 2 C16^3 2 C96^19 2 C24^5 2 C32^7 2 C48^11 2 C96^23 2 C4 2 C96^25 2 C48^13 2 C32^9 2 C24^7 2 C96^29 2 C16^5 2 C96^31 2 C3 2 C32^11 2 C48^17 2 C96^35 2 C8^3 2 C96^37 2 C48^19 2 C32^13 2 C12^5 2 C96^41 2 C16^7 2 C96^43 2 C24^11 2 C32^15 2 C48^23 2 C96^47 C2 48 sv 48 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 ... ..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 ..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 ... ... ..... ....... ......... ........... ............. 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 ... ... ..... ....... ......... ........... ............. E1 2.00000 1.99572 1.98289 1.96157 1.93185 1.89386 1.84776 1.79375 1.73205 1.66294 1.58671 1.50368 1.41421 1.31869 1.21752 1.11114 1.00000 0.88458 0.76537 0.64288 0.51764 0.39018 0.26105 0.13081 0.00000 -0.13081 -0.26105 -0.39018 -0.51764 -0.64288 -0.76537 -0.88458 -1.00000 -1.11114 -1.21752 -1.31869 -1.41421 -1.50368 -1.58671 -1.66294 -1.73205 -1.79375 -1.84776 -1.89386 -1.93185 -1.96157 -1.98289 -1.99572 -2.00000 0.00000 0.00000 TT. TT. ..TT. ....TT. ......TT. ........TT. ..........TT. E2 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 0.00000 0.00000 ... ... TT... ..TT... ....TT... ......TT... ........TT... E3 2.00000 1.96157 1.84776 1.66294 1.41421 1.11114 0.76537 0.39018 0.00000 -0.39018 -0.76537 -1.11114 -1.41421 -1.66294 -1.84776 -1.96157 -2.00000 -1.96157 -1.84776 -1.66294 -1.41421 -1.11114 -0.76537 -0.39018 0.00000 0.39018 0.76537 1.11114 1.41421 1.66294 1.84776 1.96157 2.00000 1.96157 1.84776 1.66294 1.41421 1.11114 0.76537 0.39018 0.00000 -0.39018 -0.76537 -1.11114 -1.41421 -1.66294 -1.84776 -1.96157 -2.00000 0.00000 0.00000 ... ... ..... TT..... ..TT..... ....TT..... ......TT..... E4 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....... ....TT....... E5 2.00000 1.89386 1.58671 1.11114 0.51764 -0.13081 -0.76537 -1.31869 -1.73205 -1.96157 -1.98289 -1.79375 -1.41421 -0.88458 -0.26105 0.39018 1.00000 1.50368 1.84776 1.99572 1.93185 1.66294 1.21752 0.64288 0.00000 -0.64288 -1.21752 -1.66294 -1.93185 -1.99572 -1.84776 -1.50368 -1.00000 -0.39018 0.26105 0.88458 1.41421 1.79375 1.98289 1.96157 1.73205 1.31869 0.76537 0.13081 -0.51764 -1.11114 -1.58671 -1.89386 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... TT......... ..TT......... E6 2.00000 1.84776 1.41421 0.76537 0.00000 -0.76537 -1.41421 -1.84776 -2.00000 -1.84776 -1.41421 -0.76537 0.00000 0.76537 1.41421 1.84776 2.00000 1.84776 1.41421 0.76537 0.00000 -0.76537 -1.41421 -1.84776 -2.00000 -1.84776 -1.41421 -0.76537 0.00000 0.76537 1.41421 1.84776 2.00000 1.84776 1.41421 0.76537 0.00000 -0.76537 -1.41421 -1.84776 -2.00000 -1.84776 -1.41421 -0.76537 0.00000 0.76537 1.41421 1.84776 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... TT........... E7 2.00000 1.79375 1.21752 0.39018 -0.51764 -1.31869 -1.84776 -1.99572 -1.73205 -1.11114 -0.26105 0.64288 1.41421 1.89386 1.98289 1.66294 1.00000 0.13081 -0.76537 -1.50368 -1.93185 -1.96157 -1.58671 -0.88458 0.00000 0.88458 1.58671 1.96157 1.93185 1.50368 0.76537 -0.13081 -1.00000 -1.66294 -1.98289 -1.89386 -1.41421 -0.64288 0.26105 1.11114 1.73205 1.99572 1.84776 1.31869 0.51764 -0.39018 -1.21752 -1.79375 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E8 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 ... ... ..... ....... ......... ........... ............. E9 2.00000 1.66294 0.76537 -0.39018 -1.41421 -1.96157 -1.84776 -1.11114 0.00000 1.11114 1.84776 1.96157 1.41421 0.39018 -0.76537 -1.66294 -2.00000 -1.66294 -0.76537 0.39018 1.41421 1.96157 1.84776 1.11114 0.00000 -1.11114 -1.84776 -1.96157 -1.41421 -0.39018 0.76537 1.66294 2.00000 1.66294 0.76537 -0.39018 -1.41421 -1.96157 -1.84776 -1.11114 0.00000 1.11114 1.84776 1.96157 1.41421 0.39018 -0.76537 -1.66294 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E10 2.00000 1.58671 0.51764 -0.76537 -1.73205 -1.98289 -1.41421 -0.26105 1.00000 1.84776 1.93185 1.21752 0.00000 -1.21752 -1.93185 -1.84776 -1.00000 0.26105 1.41421 1.98289 1.73205 0.76537 -0.51764 -1.58671 -2.00000 -1.58671 -0.51764 0.76537 1.73205 1.98289 1.41421 0.26105 -1.00000 -1.84776 -1.93185 -1.21752 0.00000 1.21752 1.93185 1.84776 1.00000 -0.26105 -1.41421 -1.98289 -1.73205 -0.76537 0.51764 1.58671 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E11 2.00000 1.50368 0.26105 -1.11114 -1.93185 -1.79375 -0.76537 0.64288 1.73205 1.96157 1.21752 -0.13081 -1.41421 -1.99572 -1.58671 -0.39018 1.00000 1.89386 1.84776 0.88458 -0.51764 -1.66294 -1.98289 -1.31869 0.00000 1.31869 1.98289 1.66294 0.51764 -0.88458 -1.84776 -1.89386 -1.00000 0.39018 1.58671 1.99572 1.41421 0.13081 -1.21752 -1.96157 -1.73205 -0.64288 0.76537 1.79375 1.93185 1.11114 -0.26105 -1.50368 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E12 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 ... ... ..... ....... ......... ........... ............. E13 2.00000 1.31869 -0.26105 -1.66294 -1.93185 -0.88458 0.76537 1.89386 1.73205 0.39018 -1.21752 -1.99572 -1.41421 0.13081 1.58671 1.96157 1.00000 -0.64288 -1.84776 -1.79375 -0.51764 1.11114 1.98289 1.50368 0.00000 -1.50368 -1.98289 -1.11114 0.51764 1.79375 1.84776 0.64288 -1.00000 -1.96157 -1.58671 -0.13081 1.41421 1.99572 1.21752 -0.39018 -1.73205 -1.89386 -0.76537 0.88458 1.93185 1.66294 0.26105 -1.31869 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E14 2.00000 1.21752 -0.51764 -1.84776 -1.73205 -0.26105 1.41421 1.98289 1.00000 -0.76537 -1.93185 -1.58671 0.00000 1.58671 1.93185 0.76537 -1.00000 -1.98289 -1.41421 0.26105 1.73205 1.84776 0.51764 -1.21752 -2.00000 -1.21752 0.51764 1.84776 1.73205 0.26105 -1.41421 -1.98289 -1.00000 0.76537 1.93185 1.58671 0.00000 -1.58671 -1.93185 -0.76537 1.00000 1.98289 1.41421 -0.26105 -1.73205 -1.84776 -0.51764 1.21752 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E15 2.00000 1.11114 -0.76537 -1.96157 -1.41421 0.39018 1.84776 1.66294 0.00000 -1.66294 -1.84776 -0.39018 1.41421 1.96157 0.76537 -1.11114 -2.00000 -1.11114 0.76537 1.96157 1.41421 -0.39018 -1.84776 -1.66294 0.00000 1.66294 1.84776 0.39018 -1.41421 -1.96157 -0.76537 1.11114 2.00000 1.11114 -0.76537 -1.96157 -1.41421 0.39018 1.84776 1.66294 0.00000 -1.66294 -1.84776 -0.39018 1.41421 1.96157 0.76537 -1.11114 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E16 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 ... ... ..... ....... ......... ........... ............. E17 2.00000 0.88458 -1.21752 -1.96157 -0.51764 1.50368 1.84776 0.13081 -1.73205 -1.66294 0.26105 1.89386 1.41421 -0.64288 -1.98289 -1.11114 1.00000 1.99572 0.76537 -1.31869 -1.93185 -0.39018 1.58671 1.79375 0.00000 -1.79375 -1.58671 0.39018 1.93185 1.31869 -0.76537 -1.99572 -1.00000 1.11114 1.98289 0.64288 -1.41421 -1.89386 -0.26105 1.66294 1.73205 -0.13081 -1.84776 -1.50368 0.51764 1.96157 1.21752 -0.88458 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E18 2.00000 0.76537 -1.41421 -1.84776 0.00000 1.84776 1.41421 -0.76537 -2.00000 -0.76537 1.41421 1.84776 0.00000 -1.84776 -1.41421 0.76537 2.00000 0.76537 -1.41421 -1.84776 0.00000 1.84776 1.41421 -0.76537 -2.00000 -0.76537 1.41421 1.84776 0.00000 -1.84776 -1.41421 0.76537 2.00000 0.76537 -1.41421 -1.84776 0.00000 1.84776 1.41421 -0.76537 -2.00000 -0.76537 1.41421 1.84776 0.00000 -1.84776 -1.41421 0.76537 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E19 2.00000 0.64288 -1.58671 -1.66294 0.51764 1.99572 0.76537 -1.50368 -1.73205 0.39018 1.98289 0.88458 -1.41421 -1.79375 0.26105 1.96157 1.00000 -1.31869 -1.84776 0.13081 1.93185 1.11114 -1.21752 -1.89386 0.00000 1.89386 1.21752 -1.11114 -1.93185 -0.13081 1.84776 1.31869 -1.00000 -1.96157 -0.26105 1.79375 1.41421 -0.88458 -1.98289 -0.39018 1.73205 1.50368 -0.76537 -1.99572 -0.51764 1.66294 1.58671 -0.64288 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E20 2.00000 0.51764 -1.73205 -1.41421 1.00000 1.93185 0.00000 -1.93185 -1.00000 1.41421 1.73205 -0.51764 -2.00000 -0.51764 1.73205 1.41421 -1.00000 -1.93185 0.00000 1.93185 1.00000 -1.41421 -1.73205 0.51764 2.00000 0.51764 -1.73205 -1.41421 1.00000 1.93185 0.00000 -1.93185 -1.00000 1.41421 1.73205 -0.51764 -2.00000 -0.51764 1.73205 1.41421 -1.00000 -1.93185 0.00000 1.93185 1.00000 -1.41421 -1.73205 0.51764 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E21 2.00000 0.39018 -1.84776 -1.11114 1.41421 1.66294 -0.76537 -1.96157 0.00000 1.96157 0.76537 -1.66294 -1.41421 1.11114 1.84776 -0.39018 -2.00000 -0.39018 1.84776 1.11114 -1.41421 -1.66294 0.76537 1.96157 0.00000 -1.96157 -0.76537 1.66294 1.41421 -1.11114 -1.84776 0.39018 2.00000 0.39018 -1.84776 -1.11114 1.41421 1.66294 -0.76537 -1.96157 0.00000 1.96157 0.76537 -1.66294 -1.41421 1.11114 1.84776 -0.39018 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E22 2.00000 0.26105 -1.93185 -0.76537 1.73205 1.21752 -1.41421 -1.58671 1.00000 1.84776 -0.51764 -1.98289 0.00000 1.98289 0.51764 -1.84776 -1.00000 1.58671 1.41421 -1.21752 -1.73205 0.76537 1.93185 -0.26105 -2.00000 -0.26105 1.93185 0.76537 -1.73205 -1.21752 1.41421 1.58671 -1.00000 -1.84776 0.51764 1.98289 0.00000 -1.98289 -0.51764 1.84776 1.00000 -1.58671 -1.41421 1.21752 1.73205 -0.76537 -1.93185 0.26105 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E23 2.00000 0.13081 -1.98289 -0.39018 1.93185 0.64288 -1.84776 -0.88458 1.73205 1.11114 -1.58671 -1.31869 1.41421 1.50368 -1.21752 -1.66294 1.00000 1.79375 -0.76537 -1.89386 0.51764 1.96157 -0.26105 -1.99572 0.00000 1.99572 0.26105 -1.96157 -0.51764 1.89386 0.76537 -1.79375 -1.00000 1.66294 1.21752 -1.50368 -1.41421 1.31869 1.58671 -1.11114 -1.73205 0.88458 1.84776 -0.64288 -1.93185 0.39018 1.98289 -0.13081 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E24 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E25 2.00000 -0.13081 -1.98289 0.39018 1.93185 -0.64288 -1.84776 0.88458 1.73205 -1.11114 -1.58671 1.31869 1.41421 -1.50368 -1.21752 1.66294 1.00000 -1.79375 -0.76537 1.89386 0.51764 -1.96157 -0.26105 1.99572 0.00000 -1.99572 0.26105 1.96157 -0.51764 -1.89386 0.76537 1.79375 -1.00000 -1.66294 1.21752 1.50368 -1.41421 -1.31869 1.58671 1.11114 -1.73205 -0.88458 1.84776 0.64288 -1.93185 -0.39018 1.98289 0.13081 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E26 2.00000 -0.26105 -1.93185 0.76537 1.73205 -1.21752 -1.41421 1.58671 1.00000 -1.84776 -0.51764 1.98289 0.00000 -1.98289 0.51764 1.84776 -1.00000 -1.58671 1.41421 1.21752 -1.73205 -0.76537 1.93185 0.26105 -2.00000 0.26105 1.93185 -0.76537 -1.73205 1.21752 1.41421 -1.58671 -1.00000 1.84776 0.51764 -1.98289 0.00000 1.98289 -0.51764 -1.84776 1.00000 1.58671 -1.41421 -1.21752 1.73205 0.76537 -1.93185 -0.26105 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E27 2.00000 -0.39018 -1.84776 1.11114 1.41421 -1.66294 -0.76537 1.96157 0.00000 -1.96157 0.76537 1.66294 -1.41421 -1.11114 1.84776 0.39018 -2.00000 0.39018 1.84776 -1.11114 -1.41421 1.66294 0.76537 -1.96157 0.00000 1.96157 -0.76537 -1.66294 1.41421 1.11114 -1.84776 -0.39018 2.00000 -0.39018 -1.84776 1.11114 1.41421 -1.66294 -0.76537 1.96157 0.00000 -1.96157 0.76537 1.66294 -1.41421 -1.11114 1.84776 0.39018 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E28 2.00000 -0.51764 -1.73205 1.41421 1.00000 -1.93185 0.00000 1.93185 -1.00000 -1.41421 1.73205 0.51764 -2.00000 0.51764 1.73205 -1.41421 -1.00000 1.93185 0.00000 -1.93185 1.00000 1.41421 -1.73205 -0.51764 2.00000 -0.51764 -1.73205 1.41421 1.00000 -1.93185 0.00000 1.93185 -1.00000 -1.41421 1.73205 0.51764 -2.00000 0.51764 1.73205 -1.41421 -1.00000 1.93185 0.00000 -1.93185 1.00000 1.41421 -1.73205 -0.51764 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E29 2.00000 -0.64288 -1.58671 1.66294 0.51764 -1.99572 0.76537 1.50368 -1.73205 -0.39018 1.98289 -0.88458 -1.41421 1.79375 0.26105 -1.96157 1.00000 1.31869 -1.84776 -0.13081 1.93185 -1.11114 -1.21752 1.89386 0.00000 -1.89386 1.21752 1.11114 -1.93185 0.13081 1.84776 -1.31869 -1.00000 1.96157 -0.26105 -1.79375 1.41421 0.88458 -1.98289 0.39018 1.73205 -1.50368 -0.76537 1.99572 -0.51764 -1.66294 1.58671 0.64288 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E30 2.00000 -0.76537 -1.41421 1.84776 0.00000 -1.84776 1.41421 0.76537 -2.00000 0.76537 1.41421 -1.84776 0.00000 1.84776 -1.41421 -0.76537 2.00000 -0.76537 -1.41421 1.84776 0.00000 -1.84776 1.41421 0.76537 -2.00000 0.76537 1.41421 -1.84776 0.00000 1.84776 -1.41421 -0.76537 2.00000 -0.76537 -1.41421 1.84776 0.00000 -1.84776 1.41421 0.76537 -2.00000 0.76537 1.41421 -1.84776 0.00000 1.84776 -1.41421 -0.76537 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. 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E37 2.00000 -1.50368 0.26105 1.11114 -1.93185 1.79375 -0.76537 -0.64288 1.73205 -1.96157 1.21752 0.13081 -1.41421 1.99572 -1.58671 0.39018 1.00000 -1.89386 1.84776 -0.88458 -0.51764 1.66294 -1.98289 1.31869 0.00000 -1.31869 1.98289 -1.66294 0.51764 0.88458 -1.84776 1.89386 -1.00000 -0.39018 1.58671 -1.99572 1.41421 -0.13081 -1.21752 1.96157 -1.73205 0.64288 0.76537 -1.79375 1.93185 -1.11114 -0.26105 1.50368 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E38 2.00000 -1.58671 0.51764 0.76537 -1.73205 1.98289 -1.41421 0.26105 1.00000 -1.84776 1.93185 -1.21752 0.00000 1.21752 -1.93185 1.84776 -1.00000 -0.26105 1.41421 -1.98289 1.73205 -0.76537 -0.51764 1.58671 -2.00000 1.58671 -0.51764 -0.76537 1.73205 -1.98289 1.41421 -0.26105 -1.00000 1.84776 -1.93185 1.21752 0.00000 -1.21752 1.93185 -1.84776 1.00000 0.26105 -1.41421 1.98289 -1.73205 0.76537 0.51764 -1.58671 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E39 2.00000 -1.66294 0.76537 0.39018 -1.41421 1.96157 -1.84776 1.11114 0.00000 -1.11114 1.84776 -1.96157 1.41421 -0.39018 -0.76537 1.66294 -2.00000 1.66294 -0.76537 -0.39018 1.41421 -1.96157 1.84776 -1.11114 0.00000 1.11114 -1.84776 1.96157 -1.41421 0.39018 0.76537 -1.66294 2.00000 -1.66294 0.76537 0.39018 -1.41421 1.96157 -1.84776 1.11114 0.00000 -1.11114 1.84776 -1.96157 1.41421 -0.39018 -0.76537 1.66294 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E40 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 ... ... ..... ....... ......... ........... ............. E41 2.00000 -1.79375 1.21752 -0.39018 -0.51764 1.31869 -1.84776 1.99572 -1.73205 1.11114 -0.26105 -0.64288 1.41421 -1.89386 1.98289 -1.66294 1.00000 -0.13081 -0.76537 1.50368 -1.93185 1.96157 -1.58671 0.88458 0.00000 -0.88458 1.58671 -1.96157 1.93185 -1.50368 0.76537 0.13081 -1.00000 1.66294 -1.98289 1.89386 -1.41421 0.64288 0.26105 -1.11114 1.73205 -1.99572 1.84776 -1.31869 0.51764 0.39018 -1.21752 1.79375 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E42 2.00000 -1.84776 1.41421 -0.76537 0.00000 0.76537 -1.41421 1.84776 -2.00000 1.84776 -1.41421 0.76537 0.00000 -0.76537 1.41421 -1.84776 2.00000 -1.84776 1.41421 -0.76537 0.00000 0.76537 -1.41421 1.84776 -2.00000 1.84776 -1.41421 0.76537 0.00000 -0.76537 1.41421 -1.84776 2.00000 -1.84776 1.41421 -0.76537 0.00000 0.76537 -1.41421 1.84776 -2.00000 1.84776 -1.41421 0.76537 0.00000 -0.76537 1.41421 -1.84776 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E43 2.00000 -1.89386 1.58671 -1.11114 0.51764 0.13081 -0.76537 1.31869 -1.73205 1.96157 -1.98289 1.79375 -1.41421 0.88458 -0.26105 -0.39018 1.00000 -1.50368 1.84776 -1.99572 1.93185 -1.66294 1.21752 -0.64288 0.00000 0.64288 -1.21752 1.66294 -1.93185 1.99572 -1.84776 1.50368 -1.00000 0.39018 0.26105 -0.88458 1.41421 -1.79375 1.98289 -1.96157 1.73205 -1.31869 0.76537 -0.13081 -0.51764 1.11114 -1.58671 1.89386 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E44 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 ... ... ..... ....... ......... ........... ............. E45 2.00000 -1.96157 1.84776 -1.66294 1.41421 -1.11114 0.76537 -0.39018 0.00000 0.39018 -0.76537 1.11114 -1.41421 1.66294 -1.84776 1.96157 -2.00000 1.96157 -1.84776 1.66294 -1.41421 1.11114 -0.76537 0.39018 0.00000 -0.39018 0.76537 -1.11114 1.41421 -1.66294 1.84776 -1.96157 2.00000 -1.96157 1.84776 -1.66294 1.41421 -1.11114 0.76537 -0.39018 0.00000 0.39018 -0.76537 1.11114 -1.41421 1.66294 -1.84776 1.96157 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E46 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E47 2.00000 -1.99572 1.98289 -1.96157 1.93185 -1.89386 1.84776 -1.79375 1.73205 -1.66294 1.58671 -1.50368 1.41421 -1.31869 1.21752 -1.11114 1.00000 -0.88458 0.76537 -0.64288 0.51764 -0.39018 0.26105 -0.13081 0.00000 0.13081 -0.26105 0.39018 -0.51764 0.64288 -0.76537 0.88458 -1.00000 1.11114 -1.21752 1.31869 -1.41421 1.50368 -1.58671 1.66294 -1.73205 1.79375 -1.84776 1.89386 -1.93185 1.96157 -1.98289 1.99572 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.995717846477 = 2*cos(2*π/96) = 2*cos(π/48) = √2+√2+√2+√3 1.982889722748 = 2*cos(4*π/96) = 2*cos(π/24) = √2+√2+√3 1.961570560806 = 2*cos(6*π/96) = 2*cos(π/16) = √2+√2+√2 1.931851652578 = 2*cos(8*π/96) = 2*cos(π/12) = √2+√3 1.893860258990 = 2*cos(10*π/96) = 2*cos(5*π/48) = √2+√2+√2-√3 1.847759065023 = 2*cos(12*π/96) = 2*cos(π/8) = √2+√2 1.793745483065 = 2*cos(14*π/96) = 2*cos(7*π/48) = √2+√2-√2-√3 1.732050807569 = 2*cos(16*π/96) = 2*cos(π/6) = √3 1.662939224605 = 2*cos(18*π/96) = 2*cos(3*π/16) = √2+√2-√2 1.586706680582 = 2*cos(20*π/96) = 2*cos(5*π/24) = √2+√2-√3 1.503679614958 = 2*cos(22*π/96) = 2*cos(11*π/48) = √2+√2-√2+√3 1.414213562373 = 2*cos(24*π/96) = 2*cos(π/4) = √2 1.318691630200 = 2*cos(26*π/96) = 2*cos(13*π/48) = √2-√2-√2+√3 1.217522858017 = 2*cos(28*π/96) = 2*cos(7*π/24) = √2-√2-√3 1.111140466039 = 2*cos(30*π/96) = 2*cos(5*π/16) = √2-√2-√2 0.884577380438 = 2*cos(34*π/96) = 2*cos(17*π/48) = √2-√2-√2-√3 0.765366864730 = 2*cos(36*π/96) = 2*cos(3*π/8) = √2-√2 0.642878930606 = 2*cos(38*π/96) = 2*cos(19*π/48) = √2-√2+√2-√3 0.517638090205 = 2*cos(40*π/96) = 2*cos(5*π/12) = √2-√3 0.390180644032 = 2*cos(42*π/96) = 2*cos(7*π/16) = √2-√2+√2 0.261052384440 = 2*cos(44*π/96) = 2*cos(11*π/24) = √2-√2+√3 0.130806258460 = 2*cos(46*π/96) = 2*cos(23*π/48) = √2-√2+√2+√3 Symmetry of Rotations and Cartesian products A1 p+d+f+g+h+i z, z², z³, z⁴, z⁵, z⁶ A2 R R_z E1 R+p+d+f+g+h+i {x, y}, {R_x, R_y}, {xz, yz}, {xz², yz²}, {xz³, yz³}, {xz⁴, yz⁴}, {xz⁵, yz⁵} E2 d+f+g+h+i {x²–y², xy}, {z(x²–y²), xyz}, {z²(x²–y²), xyz²}, {z³(x²–y²), xyz³}, {z⁴(x²–y²), xyz⁴} E3 f+g+h+i {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 {(x²–y²)²–2x²y², xy(x²–y²)}, {z((x²–y²)²–4x²y²), xyz(x²–y²)}, {z²((x²–y²)²–4x²y²), xyz²(x²–y²)} E5 h+i {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)}, {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)} E6 i {x²(x²–3y²)²–y²(3x²–y²)², xy(x²–3y²)(3x²–y²)} Notes: α The order of the C_96v point group is 192, and the order of the principal axis is 96. β 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. γ This point group is non-Abelian (some symmetry operations are not commutative). Therefore, the character table contains multi-membered classes and degenerate irreducible representations. δ 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). ε 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. ζ The fact that the regular 96-gon is constructible was used by Archimedes of Syracuse to calculate a rational approximation to π. His method consisted of computing (a) the ratio between the height (distance between opposing edges) to the circumference and (b) the ratio between the diameter (distance between opposing vertices) to the circumference as upper and lower bounds to π, respectively. This is is equivalent of approximating a circle with two regular 96-gons circumscribed and inscribed to it. To avoid irrational numbers, Archimedes approximated √3 by simple fractions chosen such to preserve the strict bound properties of the results. He arrived at the result 25344/8069 < π < 29376/9347 (in decimal 3.14091 < π < 3.14283), which he then shortened to the simpler result 3+¹⁰/₇₁ < π < 3+¹/₇ or 3.14085 < π < 3.14286. The exact result would have been 96*sin(2π/192) < π < 96*tan(2π/192) or 3.141032 < π < 3.142715. Using a series involving the regular 96-gon, the Chinese mathematician Liú Huī 劉徽 got slightly tighter bounds, 3.141024 < π < 3.142704.

This Character Table for the C_96v point group was created by Gernot Katzer.

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

	C_95v
C₉₆	C_96v	C_96h D₉₆ D_96h D_96d S₉₆
	C_97v

Character table for the C96v point group

Character table for the C_96v point group