Character Table for Point Group D192

Character table for the D₁₉₂ point group

D192 E 2 C192 2 C96 2 C64 2 C48 2 C192^5 2 C32 2 C192^7 2 C24 2 C64^3 2 C96^5 2 C192^11 2 C16 2 C192^13 2 C96^7 2 C64^5 2 C12 2 C192^17 2 C32^3 2 C192^19 2 C48^5 2 C64^7 2 C96^11 2 C192^23 2 C8 2 C192^25 2 C96^13 2 C64^9 2 C48^7 2 C192^29 2 C32^5 2 C192^31 2 C6 2 C64^11 2 C96^17 2 C192^35 2 C16^3 2 C192^37 2 C96^19 2 C64^13 2 C24^5 2 C192^41 2 C32^7 2 C192^43 2 C48^11 2 C64^15 2 C96^23 2 C192^47 2 C4 2 C192^49 2 C96^25 2 C64^17 2 C48^13 2 C192^53 2 C32^9 2 C192^55 2 C24^7 2 C64^19 2 C96^29 2 C192^59 2 C16^5 2 C192^61 2 C96^31 2 C64^21 2 C3 2 C192^65 2 C32^11 2 C192^67 2 C48^17 2 C64^23 2 C96^35 2 C192^71 2 C8^3 2 C192^73 2 C96^37 2 C64^25 2 C48^19 2 C192^77 2 C32^13 2 C192^79 2 C12^5 2 C64^27 2 C96^41 2 C192^83 2 C16^7 2 C192^85 2 C96^43 2 C64^29 2 C24^11 2 C192^89 2 C32^15 2 C192^91 2 C48^23 2 C64^31 2 C96^47 2 C192^95 C2 96 C2' 96 C2" <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 1.00000 1.00000 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 1.00000 1.00000 1.00000 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 ............. 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 ... ... ..... ....... ......... ........... ............. 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 ... ... ..... ....... ......... ........... ............. E1 2.00000 1.99893 1.99572 1.99037 1.98289 1.97329 1.96157 1.94775 1.93185 1.91388 1.89386 1.87181 1.84776 1.82173 1.79375 1.76384 1.73205 1.69840 1.66294 1.62569 1.58671 1.54602 1.50368 1.45973 1.41421 1.36718 1.31869 1.26879 1.21752 1.16496 1.11114 1.05614 1.00000 0.94279 0.88458 0.82541 0.76537 0.70450 0.64288 0.58057 0.51764 0.45415 0.39018 0.32579 0.26105 0.19603 0.13081 0.06544 0.00000 -0.06544 -0.13081 -0.19603 -0.26105 -0.32579 -0.39018 -0.45415 -0.51764 -0.58057 -0.64288 -0.70450 -0.76537 -0.82541 -0.88458 -0.94279 -1.00000 -1.05614 -1.11114 -1.16496 -1.21752 -1.26879 -1.31869 -1.36718 -1.41421 -1.45973 -1.50368 -1.54602 -1.58671 -1.62569 -1.66294 -1.69840 -1.73205 -1.76384 -1.79375 -1.82173 -1.84776 -1.87181 -1.89386 -1.91388 -1.93185 -1.94775 -1.96157 -1.97329 -1.98289 -1.99037 -1.99572 -1.99893 -2.00000 0.00000 0.00000 TT. TT. ..TT. ....TT. ......TT. ........TT. ..........TT. E2 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 -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... E3 2.00000 1.99037 1.96157 1.91388 1.84776 1.76384 1.66294 1.54602 1.41421 1.26879 1.11114 0.94279 0.76537 0.58057 0.39018 0.19603 0.00000 -0.19603 -0.39018 -0.58057 -0.76537 -0.94279 -1.11114 -1.26879 -1.41421 -1.54602 -1.66294 -1.76384 -1.84776 -1.91388 -1.96157 -1.99037 -2.00000 -1.99037 -1.96157 -1.91388 -1.84776 -1.76384 -1.66294 -1.54602 -1.41421 -1.26879 -1.11114 -0.94279 -0.76537 -0.58057 -0.39018 -0.19603 0.00000 0.19603 0.39018 0.58057 0.76537 0.94279 1.11114 1.26879 1.41421 1.54602 1.66294 1.76384 1.84776 1.91388 1.96157 1.99037 2.00000 1.99037 1.96157 1.91388 1.84776 1.76384 1.66294 1.54602 1.41421 1.26879 1.11114 0.94279 0.76537 0.58057 0.39018 0.19603 0.00000 -0.19603 -0.39018 -0.58057 -0.76537 -0.94279 -1.11114 -1.26879 -1.41421 -1.54602 -1.66294 -1.76384 -1.84776 -1.91388 -1.96157 -1.99037 -2.00000 0.00000 0.00000 ... ... ..... TT..... ..TT..... ....TT..... ......TT..... E4 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 0.00000 0.00000 ... ... ..... ....... TT....... ..TT....... ....TT....... E5 2.00000 1.97329 1.89386 1.76384 1.58671 1.36718 1.11114 0.82541 0.51764 0.19603 -0.13081 -0.45415 -0.76537 -1.05614 -1.31869 -1.54602 -1.73205 -1.87181 -1.96157 -1.99893 -1.98289 -1.91388 -1.79375 -1.62569 -1.41421 -1.16496 -0.88458 -0.58057 -0.26105 0.06544 0.39018 0.70450 1.00000 1.26879 1.50368 1.69840 1.84776 1.94775 1.99572 1.99037 1.93185 1.82173 1.66294 1.45973 1.21752 0.94279 0.64288 0.32579 0.00000 -0.32579 -0.64288 -0.94279 -1.21752 -1.45973 -1.66294 -1.82173 -1.93185 -1.99037 -1.99572 -1.94775 -1.84776 -1.69840 -1.50368 -1.26879 -1.00000 -0.70450 -0.39018 -0.06544 0.26105 0.58057 0.88458 1.16496 1.41421 1.62569 1.79375 1.91388 1.98289 1.99893 1.96157 1.87181 1.73205 1.54602 1.31869 1.05614 0.76537 0.45415 0.13081 -0.19603 -0.51764 -0.82541 -1.11114 -1.36718 -1.58671 -1.76384 -1.89386 -1.97329 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... TT......... ..TT......... E6 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 -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........... E7 2.00000 1.94775 1.79375 1.54602 1.21752 0.82541 0.39018 -0.06544 -0.51764 -0.94279 -1.31869 -1.62569 -1.84776 -1.97329 -1.99572 -1.91388 -1.73205 -1.45973 -1.11114 -0.70450 -0.26105 0.19603 0.64288 1.05614 1.41421 1.69840 1.89386 1.99037 1.98289 1.87181 1.66294 1.36718 1.00000 0.58057 0.13081 -0.32579 -0.76537 -1.16496 -1.50368 -1.76384 -1.93185 -1.99893 -1.96157 -1.82173 -1.58671 -1.26879 -0.88458 -0.45415 0.00000 0.45415 0.88458 1.26879 1.58671 1.82173 1.96157 1.99893 1.93185 1.76384 1.50368 1.16496 0.76537 0.32579 -0.13081 -0.58057 -1.00000 -1.36718 -1.66294 -1.87181 -1.98289 -1.99037 -1.89386 -1.69840 -1.41421 -1.05614 -0.64288 -0.19603 0.26105 0.70450 1.11114 1.45973 1.73205 1.91388 1.99572 1.97329 1.84776 1.62569 1.31869 0.94279 0.51764 0.06544 -0.39018 -0.82541 -1.21752 -1.54602 -1.79375 -1.94775 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E8 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E9 2.00000 1.91388 1.66294 1.26879 0.76537 0.19603 -0.39018 -0.94279 -1.41421 -1.76384 -1.96157 -1.99037 -1.84776 -1.54602 -1.11114 -0.58057 0.00000 0.58057 1.11114 1.54602 1.84776 1.99037 1.96157 1.76384 1.41421 0.94279 0.39018 -0.19603 -0.76537 -1.26879 -1.66294 -1.91388 -2.00000 -1.91388 -1.66294 -1.26879 -0.76537 -0.19603 0.39018 0.94279 1.41421 1.76384 1.96157 1.99037 1.84776 1.54602 1.11114 0.58057 0.00000 -0.58057 -1.11114 -1.54602 -1.84776 -1.99037 -1.96157 -1.76384 -1.41421 -0.94279 -0.39018 0.19603 0.76537 1.26879 1.66294 1.91388 2.00000 1.91388 1.66294 1.26879 0.76537 0.19603 -0.39018 -0.94279 -1.41421 -1.76384 -1.96157 -1.99037 -1.84776 -1.54602 -1.11114 -0.58057 0.00000 0.58057 1.11114 1.54602 1.84776 1.99037 1.96157 1.76384 1.41421 0.94279 0.39018 -0.19603 -0.76537 -1.26879 -1.66294 -1.91388 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E10 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 -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 ... ... ..... ....... ......... ........... ............. E11 2.00000 1.87181 1.50368 0.94279 0.26105 -0.45415 -1.11114 -1.62569 -1.93185 -1.99037 -1.79375 -1.36718 -0.76537 -0.06544 0.64288 1.26879 1.73205 1.97329 1.96157 1.69840 1.21752 0.58057 -0.13081 -0.82541 -1.41421 -1.82173 -1.99572 -1.91388 -1.58671 -1.05614 -0.39018 0.32579 1.00000 1.54602 1.89386 1.99893 1.84776 1.45973 0.88458 0.19603 -0.51764 -1.16496 -1.66294 -1.94775 -1.98289 -1.76384 -1.31869 -0.70450 0.00000 0.70450 1.31869 1.76384 1.98289 1.94775 1.66294 1.16496 0.51764 -0.19603 -0.88458 -1.45973 -1.84776 -1.99893 -1.89386 -1.54602 -1.00000 -0.32579 0.39018 1.05614 1.58671 1.91388 1.99572 1.82173 1.41421 0.82541 0.13081 -0.58057 -1.21752 -1.69840 -1.96157 -1.97329 -1.73205 -1.26879 -0.64288 0.06544 0.76537 1.36718 1.79375 1.99037 1.93185 1.62569 1.11114 0.45415 -0.26105 -0.94279 -1.50368 -1.87181 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. 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E94 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 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 ... ... ..... ....... ......... ........... ............. E95 2.00000 -1.99893 1.99572 -1.99037 1.98289 -1.97329 1.96157 -1.94775 1.93185 -1.91388 1.89386 -1.87181 1.84776 -1.82173 1.79375 -1.76384 1.73205 -1.69840 1.66294 -1.62569 1.58671 -1.54602 1.50368 -1.45973 1.41421 -1.36718 1.31869 -1.26879 1.21752 -1.16496 1.11114 -1.05614 1.00000 -0.94279 0.88458 -0.82541 0.76537 -0.70450 0.64288 -0.58057 0.51764 -0.45415 0.39018 -0.32579 0.26105 -0.19603 0.13081 -0.06544 0.00000 0.06544 -0.13081 0.19603 -0.26105 0.32579 -0.39018 0.45415 -0.51764 0.58057 -0.64288 0.70450 -0.76537 0.82541 -0.88458 0.94279 -1.00000 1.05614 -1.11114 1.16496 -1.21752 1.26879 -1.31869 1.36718 -1.41421 1.45973 -1.50368 1.54602 -1.58671 1.62569 -1.66294 1.69840 -1.73205 1.76384 -1.79375 1.82173 -1.84776 1.87181 -1.89386 1.91388 -1.93185 1.94775 -1.96157 1.97329 -1.98289 1.99037 -1.99572 1.99893 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.998929174953 = 2*cos(2*π/192) = 2*cos(π/96) = √2+√2+√2+√2+√3 1.995717846477 = 2*cos(4*π/192) = 2*cos(π/48) = √2+√2+√2+√3 1.990369453344 = 2*cos(6*π/192) = 2*cos(π/32) = √2+√2+√2+√2 1.982889722748 = 2*cos(8*π/192) = 2*cos(π/24) = √2+√2+√3 1.973286664170 = 2*cos(10*π/192) = 2*cos(5*π/96) = √2+√2+√2+√2−√3 1.961570560806 = 2*cos(12*π/192) = 2*cos(π/16) = √2+√2+√2 1.947753958555 = 2*cos(14*π/192) = 2*cos(7*π/96) = √2+√2+√2−√2−√3 1.931851652578 = 2*cos(16*π/192) = 2*cos(π/12) = √2+√3 1.913880671464 = 2*cos(18*π/192) = 2*cos(3*π/32) = √2+√2+√2−√2 1.893860258990 = 2*cos(20*π/192) = 2*cos(5*π/48) = √2+√2+√2−√3 1.871811853515 = 2*cos(22*π/192) = 2*cos(11*π/96) = √2+√2+√2−√2+√3 1.847759065023 = 2*cos(24*π/192) = 2*cos(π/8) = √2+√2 1.821727649842 = 2*cos(26*π/192) = 2*cos(13*π/96) = √2+√2−√2−√2+√3 1.793745483065 = 2*cos(28*π/192) = 2*cos(7*π/48) = √2+√2−√2−√3 1.763842528697 = 2*cos(30*π/192) = 2*cos(5*π/32) = √2+√2−√2−√2 1.732050807569 = 2*cos(32*π/192) = 2*cos(π/6) = √3 1.698404363053 = 2*cos(34*π/192) = 2*cos(17*π/96) = √2+√2−√2−√2−√3 1.662939224605 = 2*cos(36*π/192) = 2*cos(3*π/16) = √2+√2−√2 1.625693369183 = 2*cos(38*π/192) = 2*cos(19*π/96) = √2+√2−√2+√2−√3 1.586706680582 = 2*cos(40*π/192) = 2*cos(5*π/24) = √2+√2−√3 1.546020906725 = 2*cos(42*π/192) = 2*cos(7*π/32) = √2+√2−√2+√2 1.503679614958 = 2*cos(44*π/192) = 2*cos(11*π/48) = √2+√2−√2+√3 1.459728145396 = 2*cos(46*π/192) = 2*cos(23*π/96) = √2+√2−√2+√2+√3 1.414213562373 = 2*cos(48*π/192) = 2*cos(π/4) = √2 1.367184604046 = 2*cos(50*π/192) = 2*cos(25*π/96) = √2−√2−√2+√2+√3 1.318691630200 = 2*cos(52*π/192) = 2*cos(13*π/48) = √2−√2−√2+√3 1.268786568327 = 2*cos(54*π/192) = 2*cos(9*π/32) = √2−√2−√2+√2 1.217522858017 = 2*cos(56*π/192) = 2*cos(7*π/24) = √2−√2−√3 1.164955393736 = 2*cos(58*π/192) = 2*cos(29*π/96) = √2−√2−√2+√2−√3 1.111140466039 = 2*cos(60*π/192) = 2*cos(5*π/16) = √2−√2−√2 1.056135701301 = 2*cos(62*π/192) = 2*cos(31*π/96) = √2−√2−√2−√2−√3 0.942793473652 = 2*cos(66*π/192) = 2*cos(11*π/32) = √2−√2−√2−√2 0.884577380438 = 2*cos(68*π/192) = 2*cos(17*π/48) = √2−√2−√2−√3 0.825414059609 = 2*cos(70*π/192) = 2*cos(35*π/96) = √2−√2−√2−√2+√3 0.765366864730 = 2*cos(72*π/192) = 2*cos(3*π/8) = √2−√2 0.704500095842 = 2*cos(74*π/192) = 2*cos(37*π/96) = √2−√2+√2−√2+√3 0.642878930606 = 2*cos(76*π/192) = 2*cos(19*π/48) = √2−√2+√2−√3 0.580569354509 = 2*cos(78*π/192) = 2*cos(13*π/32) = √2−√2+√2−√2 0.517638090205 = 2*cos(80*π/192) = 2*cos(5*π/12) = √2−√3 0.454152526069 = 2*cos(82*π/192) = 2*cos(41*π/96) = √2−√2+√2−√2−√3 0.390180644032 = 2*cos(84*π/192) = 2*cos(7*π/16) = √2−√2+√2 0.325790946789 = 2*cos(86*π/192) = 2*cos(43*π/96) = √2−√2+√2+√2−√3 0.261052384440 = 2*cos(88*π/192) = 2*cos(11*π/24) = √2−√2+√3 0.196034280659 = 2*cos(90*π/192) = 2*cos(15*π/32) = √2−√2+√2+√2 0.130806258460 = 2*cos(92*π/192) = 2*cos(23*π/48) = √2−√2+√2+√3 0.065438165644 = 2*cos(94*π/192) = 2*cos(47*π/96) = √2−√2+√2+√2+√3 Symmetry of Rotations and Cartesian products A1 d+g+i+k+m z², z⁴, z⁶ A2 R+p+f+h+j+l R_z, z, z³, z⁵ E1 R+p+d+f+g+h+i+j+k+l+m {R_x, R_y}, {x, y}, {xz, yz}, {xz², yz²}, {xz³, yz³}, {xz⁴, yz⁴}, {xz⁵, yz⁵} E2 d+f+g+h+i+j+k+l+m {x²−y², xy}, {z(x²−y²), xyz}, {z²(x²−y²), xyz²}, {z³(x²−y²), xyz³}, {z⁴(x²−y²), xyz⁴} E3 f+g+h+i+j+k+l+m {x(x²−3y²), y(3x²−y²)}, {xz(x²−3y²), yz(3x²−y²)}, {xz²(x²−3y²), yz²(3x²−y²)}, {xz³(x²−3y²), yz³(3x²−y²)} E4 g+h+i+j+k+l+m {(x²−y²)²−4x²y², xy(x²−y²)}, {z((x²−y²)²−4x²y²), xyz(x²−y²)}, {z²((x²−y²)²−4x²y²), xyz²(x²−y²)} E5 h+i+j+k+l+m {x(x²−(5+2√5)y²)(x²−(5−2√5)y²), y((5+2√5)x²−y²)((5−2√5)x²−y²)}, {xz(x²−(5+2√5)y²)(x²−(5−2√5)y²), yz((5+2√5)x²−y²)((5−2√5)x²−y²)} E6 i+j+k+l+m {x²(x²−3y²)²−y²(3x²−y²)², xy(x²−3y²)(3x²−y²)} E7 j+k+l+m E8 k+l+m E9 l+m E10 m Notes: α The order of the D₁₉₂ point group is 384, and the order of the principal axis (C₁₉₂) is 192. The group has 99 irreducible representations. β The D₁₉₂ point group is isomorphic to D_96d and C_192v. γ The D₁₉₂ point group is generated by two symmetry elements, C₁₉₂ and a perpendicular C₂^′ (or, non-canonically, C₂^″). Also, the group may be generated from a C₂^′ plus a C₂^″ (some pairs will yield smaller groups, though; choosing a minimum angle is safe). δ There are two different sets of twofold symmetry axes perpendicular to the principal axis (z axis in standard orientation). By convention, the set denoted as C₂^′ contains both the x and y axes. ε The lowest nonvanishing multipole moment in D₁₉₂ is 4 (quadrupole moment). ζ This point group is non-Abelian (some symmetry operations are not commutative). Therefore, the character table contains multi-membered classes and degenerate irreducible representations. η The point group is chiral, as it does not contain any mirroring operation. 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 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.

This Character Table for the D₁₉₂ point group was created by Gernot Katzer.

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

	D₁₆₀
	D₁₇₀
C₁₉₂ C_192v C_192h	D₁₉₂	D_192h D_192d S₁₉₂
	D₂₀₄
	D₂₄₀

Character table for the D192 point group

Character table for the D₁₉₂ point group