Character Table for Point Group D128d

Character table for the D_128d point group

D128d E 2 S256 2 C128 2 S256^3 2 C64 2 S256^5 2 C128^3 2 S256^7 2 C32 2 S256^9 2 C128^5 2 S256^11 2 C64^3 2 S256^13 2 C128^7 2 S256^15 2 C16 2 S256^17 2 C128^9 2 S256^19 2 C64^5 2 S256^21 2 C128^11 2 S256^23 2 C32^3 2 S256^25 2 C128^13 2 S256^27 2 C64^7 2 S256^29 2 C128^15 2 S256^31 2 C8 2 S256^33 2 C128^17 2 S256^35 2 C64^9 2 S256^37 2 C128^19 2 S256^39 2 C32^5 2 S256^41 2 C128^21 2 S256^43 2 C64^11 2 S256^45 2 C128^23 2 S256^47 2 C16^3 2 S256^49 2 C128^25 2 S256^51 2 C64^13 2 S256^53 2 C128^27 2 S256^55 2 C32^7 2 S256^57 2 C128^29 2 S256^59 2 C64^15 2 S256^61 2 C128^31 2 S256^63 2 C4 2 S256^65 2 C128^33 2 S256^67 2 C64^17 2 S256^69 2 C128^35 2 S256^71 2 C32^9 2 S256^73 2 C128^37 2 S256^75 2 C64^19 2 S256^77 2 C128^39 2 S256^79 2 C16^5 2 S256^81 2 C128^41 2 S256^83 2 C64^21 2 S256^85 2 C128^43 2 S256^87 2 C32^11 2 S256^89 2 C128^45 2 S256^91 2 C64^23 2 S256^93 2 C128^47 2 S256^95 2 C8^3 2 S256^97 2 C128^49 2 S256^99 2 C64^25 2 S256^101 2 C128^51 2 S256^103 2 C32^13 2 S256^105 2 C128^53 2 S256^107 2 C64^27 2 S256^109 2 C128^55 2 S256^111 2 C16^7 2 S256^113 2 C128^57 2 S256^115 2 C64^29 2 S256^117 2 C128^59 2 S256^119 2 C32^15 2 S256^121 2 C128^61 2 S256^123 2 C64^31 2 S256^125 2 C128^63 2 S256^127 C2 128 C2' 128 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 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 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 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 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 ... ... ..... ....... ......... ........... ............. 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 ... ..T ..... ......T ......... ..........T ............. E1 2.00000 1.99940 1.99759 1.99458 1.99037 1.98496 1.97835 1.97056 1.96157 1.95140 1.94006 1.92755 1.91388 1.89906 1.88309 1.86599 1.84776 1.82842 1.80798 1.78645 1.76384 1.74017 1.71546 1.68971 1.66294 1.63517 1.60642 1.57669 1.54602 1.51442 1.48190 1.44849 1.41421 1.37908 1.34312 1.30635 1.26879 1.23046 1.19140 1.15162 1.11114 1.07000 1.02821 0.98580 0.94279 0.89922 0.85511 0.81048 0.76537 0.71979 0.67378 0.62736 0.58057 0.53343 0.48596 0.43820 0.39018 0.34192 0.29346 0.24482 0.19603 0.14713 0.09814 0.04908 0.00000 -0.04908 -0.09814 -0.14713 -0.19603 -0.24482 -0.29346 -0.34192 -0.39018 -0.43820 -0.48596 -0.53343 -0.58057 -0.62736 -0.67378 -0.71979 -0.76537 -0.81048 -0.85511 -0.89922 -0.94279 -0.98580 -1.02821 -1.07000 -1.11114 -1.15162 -1.19140 -1.23046 -1.26879 -1.30635 -1.34312 -1.37908 -1.41421 -1.44849 -1.48190 -1.51442 -1.54602 -1.57669 -1.60642 -1.63517 -1.66294 -1.68971 -1.71546 -1.74017 -1.76384 -1.78645 -1.80798 -1.82842 -1.84776 -1.86599 -1.88309 -1.89906 -1.91388 -1.92755 -1.94006 -1.95140 -1.96157 -1.97056 -1.97835 -1.98496 -1.99037 -1.99458 -1.99759 -1.99940 -2.00000 0.00000 0.00000 ... TT. ..... ....TT. ......... ........TT. ............. E2 2.00000 1.99759 1.99037 1.97835 1.96157 1.94006 1.91388 1.88309 1.84776 1.80798 1.76384 1.71546 1.66294 1.60642 1.54602 1.48190 1.41421 1.34312 1.26879 1.19140 1.11114 1.02821 0.94279 0.85511 0.76537 0.67378 0.58057 0.48596 0.39018 0.29346 0.19603 0.09814 0.00000 -0.09814 -0.19603 -0.29346 -0.39018 -0.48596 -0.58057 -0.67378 -0.76537 -0.85511 -0.94279 -1.02821 -1.11114 -1.19140 -1.26879 -1.34312 -1.41421 -1.48190 -1.54602 -1.60642 -1.66294 -1.71546 -1.76384 -1.80798 -1.84776 -1.88309 -1.91388 -1.94006 -1.96157 -1.97835 -1.99037 -1.99759 -2.00000 -1.99759 -1.99037 -1.97835 -1.96157 -1.94006 -1.91388 -1.88309 -1.84776 -1.80798 -1.76384 -1.71546 -1.66294 -1.60642 -1.54602 -1.48190 -1.41421 -1.34312 -1.26879 -1.19140 -1.11114 -1.02821 -0.94279 -0.85511 -0.76537 -0.67378 -0.58057 -0.48596 -0.39018 -0.29346 -0.19603 -0.09814 0.00000 0.09814 0.19603 0.29346 0.39018 0.48596 0.58057 0.67378 0.76537 0.85511 0.94279 1.02821 1.11114 1.19140 1.26879 1.34312 1.41421 1.48190 1.54602 1.60642 1.66294 1.71546 1.76384 1.80798 1.84776 1.88309 1.91388 1.94006 1.96157 1.97835 1.99037 1.99759 2.00000 0.00000 0.00000 ... ... TT... ....... ....TT... ........... ........TT... E3 2.00000 1.99458 1.97835 1.95140 1.91388 1.86599 1.80798 1.74017 1.66294 1.57669 1.48190 1.37908 1.26879 1.15162 1.02821 0.89922 0.76537 0.62736 0.48596 0.34192 0.19603 0.04908 -0.09814 -0.24482 -0.39018 -0.53343 -0.67378 -0.81048 -0.94279 -1.07000 -1.19140 -1.30635 -1.41421 -1.51442 -1.60642 -1.68971 -1.76384 -1.82842 -1.88309 -1.92755 -1.96157 -1.98496 -1.99759 -1.99940 -1.99037 -1.97056 -1.94006 -1.89906 -1.84776 -1.78645 -1.71546 -1.63517 -1.54602 -1.44849 -1.34312 -1.23046 -1.11114 -0.98580 -0.85511 -0.71979 -0.58057 -0.43820 -0.29346 -0.14713 0.00000 0.14713 0.29346 0.43820 0.58057 0.71979 0.85511 0.98580 1.11114 1.23046 1.34312 1.44849 1.54602 1.63517 1.71546 1.78645 1.84776 1.89906 1.94006 1.97056 1.99037 1.99940 1.99759 1.98496 1.96157 1.92755 1.88309 1.82842 1.76384 1.68971 1.60642 1.51442 1.41421 1.30635 1.19140 1.07000 0.94279 0.81048 0.67378 0.53343 0.39018 0.24482 0.09814 -0.04908 -0.19603 -0.34192 -0.48596 -0.62736 -0.76537 -0.89922 -1.02821 -1.15162 -1.26879 -1.37908 -1.48190 -1.57669 -1.66294 -1.74017 -1.80798 -1.86599 -1.91388 -1.95140 -1.97835 -1.99458 -2.00000 0.00000 0.00000 ... ... ..... TT..... ......... ....TT..... ............. E4 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 -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....... E5 2.00000 1.98496 1.94006 1.86599 1.76384 1.63517 1.48190 1.30635 1.11114 0.89922 0.67378 0.43820 0.19603 -0.04908 -0.29346 -0.53343 -0.76537 -0.98580 -1.19140 -1.37908 -1.54602 -1.68971 -1.80798 -1.89906 -1.96157 -1.99458 -1.99759 -1.97056 -1.91388 -1.82842 -1.71546 -1.57669 -1.41421 -1.23046 -1.02821 -0.81048 -0.58057 -0.34192 -0.09814 0.14713 0.39018 0.62736 0.85511 1.07000 1.26879 1.44849 1.60642 1.74017 1.84776 1.92755 1.97835 1.99940 1.99037 1.95140 1.88309 1.78645 1.66294 1.51442 1.34312 1.15162 0.94279 0.71979 0.48596 0.24482 0.00000 -0.24482 -0.48596 -0.71979 -0.94279 -1.15162 -1.34312 -1.51442 -1.66294 -1.78645 -1.88309 -1.95140 -1.99037 -1.99940 -1.97835 -1.92755 -1.84776 -1.74017 -1.60642 -1.44849 -1.26879 -1.07000 -0.85511 -0.62736 -0.39018 -0.14713 0.09814 0.34192 0.58057 0.81048 1.02821 1.23046 1.41421 1.57669 1.71546 1.82842 1.91388 1.97056 1.99759 1.99458 1.96157 1.89906 1.80798 1.68971 1.54602 1.37908 1.19140 0.98580 0.76537 0.53343 0.29346 0.04908 -0.19603 -0.43820 -0.67378 -0.89922 -1.11114 -1.30635 -1.48190 -1.63517 -1.76384 -1.86599 -1.94006 -1.98496 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... TT......... ............. E6 2.00000 1.97835 1.91388 1.80798 1.66294 1.48190 1.26879 1.02821 0.76537 0.48596 0.19603 -0.09814 -0.39018 -0.67378 -0.94279 -1.19140 -1.41421 -1.60642 -1.76384 -1.88309 -1.96157 -1.99759 -1.99037 -1.94006 -1.84776 -1.71546 -1.54602 -1.34312 -1.11114 -0.85511 -0.58057 -0.29346 0.00000 0.29346 0.58057 0.85511 1.11114 1.34312 1.54602 1.71546 1.84776 1.94006 1.99037 1.99759 1.96157 1.88309 1.76384 1.60642 1.41421 1.19140 0.94279 0.67378 0.39018 0.09814 -0.19603 -0.48596 -0.76537 -1.02821 -1.26879 -1.48190 -1.66294 -1.80798 -1.91388 -1.97835 -2.00000 -1.97835 -1.91388 -1.80798 -1.66294 -1.48190 -1.26879 -1.02821 -0.76537 -0.48596 -0.19603 0.09814 0.39018 0.67378 0.94279 1.19140 1.41421 1.60642 1.76384 1.88309 1.96157 1.99759 1.99037 1.94006 1.84776 1.71546 1.54602 1.34312 1.11114 0.85511 0.58057 0.29346 0.00000 -0.29346 -0.58057 -0.85511 -1.11114 -1.34312 -1.54602 -1.71546 -1.84776 -1.94006 -1.99037 -1.99759 -1.96157 -1.88309 -1.76384 -1.60642 -1.41421 -1.19140 -0.94279 -0.67378 -0.39018 -0.09814 0.19603 0.48596 0.76537 1.02821 1.26879 1.48190 1.66294 1.80798 1.91388 1.97835 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... TT........... E7 2.00000 1.97056 1.88309 1.74017 1.54602 1.30635 1.02821 0.71979 0.39018 0.04908 -0.29346 -0.62736 -0.94279 -1.23046 -1.48190 -1.68971 -1.84776 -1.95140 -1.99759 -1.98496 -1.91388 -1.78645 -1.60642 -1.37908 -1.11114 -0.81048 -0.48596 -0.14713 0.19603 0.53343 0.85511 1.15162 1.41421 1.63517 1.80798 1.92755 1.99037 1.99458 1.94006 1.82842 1.66294 1.44849 1.19140 0.89922 0.58057 0.24482 -0.09814 -0.43820 -0.76537 -1.07000 -1.34312 -1.57669 -1.76384 -1.89906 -1.97835 -1.99940 -1.96157 -1.86599 -1.71546 -1.51442 -1.26879 -0.98580 -0.67378 -0.34192 0.00000 0.34192 0.67378 0.98580 1.26879 1.51442 1.71546 1.86599 1.96157 1.99940 1.97835 1.89906 1.76384 1.57669 1.34312 1.07000 0.76537 0.43820 0.09814 -0.24482 -0.58057 -0.89922 -1.19140 -1.44849 -1.66294 -1.82842 -1.94006 -1.99458 -1.99037 -1.92755 -1.80798 -1.63517 -1.41421 -1.15162 -0.85511 -0.53343 -0.19603 0.14713 0.48596 0.81048 1.11114 1.37908 1.60642 1.78645 1.91388 1.98496 1.99759 1.95140 1.84776 1.68971 1.48190 1.23046 0.94279 0.62736 0.29346 -0.04908 -0.39018 -0.71979 -1.02821 -1.30635 -1.54602 -1.74017 -1.88309 -1.97056 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. 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Irrational character values: 1.999397637392 = 2*cos(2*π/256) = 2*cos(π/128) = √2+√2+√2+√2+√2+√2 1.997590912410 = 2*cos(4*π/256) = 2*cos(π/64) = √2+√2+√2+√2+√2 1.994580913357 = 2*cos(6*π/256) = 2*cos(3*π/128) = √2+√2+√2+√2+√2−√2 1.990369453344 = 2*cos(8*π/256) = 2*cos(π/32) = √2+√2+√2+√2 1.984959069197 = 2*cos(10*π/256) = 2*cos(5*π/128) = √2+√2+√2+√2−√2−√2 1.978353019930 = 2*cos(12*π/256) = 2*cos(3*π/64) = √2+√2+√2+√2−√2 1.970555284778 = 2*cos(14*π/256) = 2*cos(7*π/128) = √2+√2+√2+√2−√2+√2 1.961570560806 = 2*cos(16*π/256) = 2*cos(π/16) = √2+√2+√2 1.951404260077 = 2*cos(18*π/256) = 2*cos(9*π/128) = √2+√2+√2−√2−√2+√2 1.940062506389 = 2*cos(20*π/256) = 2*cos(5*π/64) = √2+√2+√2−√2−√2 1.927552131591 = 2*cos(22*π/256) = 2*cos(11*π/128) = √2+√2+√2−√2−√2−√2 1.913880671464 = 2*cos(24*π/256) = 2*cos(3*π/32) = √2+√2+√2−√2 1.899056361186 = 2*cos(26*π/256) = 2*cos(13*π/128) = √2+√2+√2−√2+√2−√2 1.883088130366 = 2*cos(28*π/256) = 2*cos(7*π/64) = √2+√2+√2−√2+√2 1.865985597669 = 2*cos(30*π/256) = 2*cos(15*π/128) = √2+√2+√2−√2+√2+√2 1.847759065023 = 2*cos(32*π/256) = 2*cos(π/8) = √2+√2 1.828419511407 = 2*cos(34*π/256) = 2*cos(17*π/128) = √2+√2−√2−√2+√2+√2 1.807978586247 = 2*cos(36*π/256) = 2*cos(9*π/64) = √2+√2−√2−√2+√2 1.786448602391 = 2*cos(38*π/256) = 2*cos(19*π/128) = √2+√2−√2−√2+√2−√2 1.763842528697 = 2*cos(40*π/256) = 2*cos(5*π/32) = √2+√2−√2−√2 1.740173982217 = 2*cos(42*π/256) = 2*cos(21*π/128) = √2+√2−√2−√2−√2−√2 1.715457220001 = 2*cos(44*π/256) = 2*cos(11*π/64) = √2+√2−√2−√2−√2 1.689707130499 = 2*cos(46*π/256) = 2*cos(23*π/128) = √2+√2−√2−√2−√2+√2 1.662939224605 = 2*cos(48*π/256) = 2*cos(3*π/16) = √2+√2−√2 1.635169626303 = 2*cos(50*π/256) = 2*cos(25*π/128) = √2+√2−√2+√2−√2+√2 1.606415062961 = 2*cos(52*π/256) = 2*cos(13*π/64) = √2+√2−√2+√2−√2 1.576692855253 = 2*cos(54*π/256) = 2*cos(27*π/128) = √2+√2−√2+√2−√2−√2 1.546020906725 = 2*cos(56*π/256) = 2*cos(7*π/32) = √2+√2−√2+√2 1.514417693013 = 2*cos(58*π/256) = 2*cos(29*π/128) = √2+√2−√2+√2+√2−√2 1.481902250710 = 2*cos(60*π/256) = 2*cos(15*π/64) = √2+√2−√2+√2+√2 1.448494165903 = 2*cos(62*π/256) = 2*cos(31*π/128) = √2+√2−√2+√2+√2+√2 1.414213562373 = 2*cos(64*π/256) = 2*cos(π/4) = √2 1.379081089474 = 2*cos(66*π/256) = 2*cos(33*π/128) = √2−√2−√2+√2+√2+√2 1.343117909694 = 2*cos(68*π/256) = 2*cos(17*π/64) = √2−√2−√2+√2+√2 1.306345685908 = 2*cos(70*π/256) = 2*cos(35*π/128) = √2−√2−√2+√2+√2−√2 1.268786568327 = 2*cos(72*π/256) = 2*cos(9*π/32) = √2−√2−√2+√2 1.230463181161 = 2*cos(74*π/256) = 2*cos(37*π/128) = √2−√2−√2+√2−√2−√2 1.191398608985 = 2*cos(76*π/256) = 2*cos(19*π/64) = √2−√2−√2+√2−√2 1.151616382836 = 2*cos(78*π/256) = 2*cos(39*π/128) = √2−√2−√2+√2−√2+√2 1.111140466039 = 2*cos(80*π/256) = 2*cos(5*π/16) = √2−√2−√2 1.069995239774 = 2*cos(82*π/256) = 2*cos(41*π/128) = √2−√2−√2−√2−√2+√2 1.028205488386 = 2*cos(84*π/256) = 2*cos(21*π/64) = √2−√2−√2−√2−√2 0.985796384460 = 2*cos(86*π/256) = 2*cos(43*π/128) = √2−√2−√2−√2−√2−√2 0.942793473652 = 2*cos(88*π/256) = 2*cos(11*π/32) = √2−√2−√2−√2 0.899222659309 = 2*cos(90*π/256) = 2*cos(45*π/128) = √2−√2−√2−√2+√2−√2 0.855110186861 = 2*cos(92*π/256) = 2*cos(23*π/64) = √2−√2−√2−√2+√2 0.810482628010 = 2*cos(94*π/256) = 2*cos(47*π/128) = √2−√2−√2−√2+√2+√2 0.765366864730 = 2*cos(96*π/256) = 2*cos(3*π/8) = √2−√2 0.719790073070 = 2*cos(98*π/256) = 2*cos(49*π/128) = √2−√2+√2−√2+√2+√2 0.673779706784 = 2*cos(100*π/256) = 2*cos(25*π/64) = √2−√2+√2−√2+√2 0.627363480798 = 2*cos(102*π/256) = 2*cos(51*π/128) = √2−√2+√2−√2+√2−√2 0.580569354509 = 2*cos(104*π/256) = 2*cos(13*π/32) = √2−√2+√2−√2 0.533425514950 = 2*cos(106*π/256) = 2*cos(53*π/128) = √2−√2+√2−√2−√2−√2 0.485960359807 = 2*cos(108*π/256) = 2*cos(27*π/64) = √2−√2+√2−√2−√2 0.438202480314 = 2*cos(110*π/256) = 2*cos(55*π/128) = √2−√2+√2−√2−√2+√2 0.390180644032 = 2*cos(112*π/256) = 2*cos(7*π/16) = √2−√2+√2 0.341923777521 = 2*cos(114*π/256) = 2*cos(57*π/128) = √2−√2+√2+√2−√2+√2 0.293460948911 = 2*cos(116*π/256) = 2*cos(29*π/64) = √2−√2+√2+√2−√2 0.244821350398 = 2*cos(118*π/256) = 2*cos(59*π/128) = √2−√2+√2+√2−√2−√2 0.196034280659 = 2*cos(120*π/256) = 2*cos(15*π/32) = √2−√2+√2+√2 0.147129127199 = 2*cos(122*π/256) = 2*cos(61*π/128) = √2−√2+√2+√2+√2−√2 0.098135348655 = 2*cos(124*π/256) = 2*cos(31*π/64) = √2−√2+√2+√2+√2 0.049082457046 = 2*cos(126*π/256) = 2*cos(63*π/128) = √2−√2+√2+√2+√2+√2 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 E119 m E120 l E121 k+m E122 j+l E123 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)} E124 h+j+l {z((x²−y²)²−4x²y²), xyz(x²−y²)} E125 g+i+k+m {xz(x²−3y²), yz(3x²−y²)}, {xz³(x²−3y²), yz³(3x²−y²)} E126 f+h+j+l {z(x²−y²), xyz}, {z³(x²−y²), xyz³} E127 R+d+g+i+k+m {R_x, R_y}, {xz, yz}, {xz³, yz³}, {xz⁵, yz⁵} Notes: α The order of the D_128d point group is 512, and the order of the principal axis (S₂₅₆) is 256. The group has 131 irreducible representations. β The D_128d point group is isomorphic to C_256v and D₂₅₆. γ The D_128d point group is generated by two symmetry elements, S₂₅₆ and either a perpendicular C₂^′ or a vertical σ_d. Also, the group may be generated from any C₂^′ plus any σ_d plane. 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. ζ The lowest nonvanishing multipole moment in D_128d 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 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 D_128d point group was created by Gernot Katzer.

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

	D_126d
	D_127d
C₁₂₈ C_128v C_128h D₁₂₈ D_128h	D_128d	S₁₂₈
	D_136d
	D_144d

Character table for the D128d point group

Character table for the D_128d point group