Character Table for Point Group D160d

Character table for the D_160d point group

D160d E 2 S320 2 C160 2 S320^3 2 C80 2 S64 2 C160^3 2 S320^7 2 C40 2 S320^9 2 C32 2 S320^11 2 C80^3 2 S320^13 2 C160^7 2 S64^3 2 C20 2 S320^17 2 C160^9 2 S320^19 2 C16 2 S320^21 2 C160^11 2 S320^23 2 C40^3 2 S64^5 2 C160^13 2 S320^27 2 C80^7 2 S320^29 2 C32^3 2 S320^31 2 C10 2 S320^33 2 C160^17 2 S64^7 2 C80^9 2 S320^37 2 C160^19 2 S320^39 2 C8 2 S320^41 2 C160^21 2 S320^43 2 C80^11 2 S64^9 2 C160^23 2 S320^47 2 C20^3 2 S320^49 2 C32^5 2 S320^51 2 C80^13 2 S320^53 2 C160^27 2 S64^11 2 C40^7 2 S320^57 2 C160^29 2 S320^59 2 C16^3 2 S320^61 2 C160^31 2 S320^63 2 C5 2 S64^13 2 C160^33 2 S320^67 2 C80^17 2 S320^69 2 C32^7 2 S320^71 2 C40^9 2 S320^73 2 C160^37 2 S64^15 2 C80^19 2 S320^77 2 C160^39 2 S320^79 2 C4 2 S320^81 2 C160^41 2 S320^83 2 C80^21 2 S64^17 2 C160^43 2 S320^87 2 C40^11 2 S320^89 2 C32^9 2 S320^91 2 C80^23 2 S320^93 2 C160^47 2 S64^19 2 C10^3 2 S320^97 2 C160^49 2 S320^99 2 C16^5 2 S320^101 2 C160^51 2 S320^103 2 C40^13 2 S64^21 2 C160^53 2 S320^107 2 C80^27 2 S320^109 2 C32^11 2 S320^111 2 C20^7 2 S320^113 2 C160^57 2 S64^23 2 C80^29 2 S320^117 2 C160^59 2 S320^119 2 C8^3 2 S320^121 2 C160^61 2 S320^123 2 C80^31 2 S64^25 2 C160^63 2 S320^127 2 C5^2 2 S320^129 2 C32^13 2 S320^131 2 C80^33 2 S320^133 2 C160^67 2 S64^27 2 C40^17 2 S320^137 2 C160^69 2 S320^139 2 C16^7 2 S320^141 2 C160^71 2 S320^143 2 C20^9 2 S64^29 2 C160^73 2 S320^147 2 C80^37 2 S320^149 2 C32^15 2 S320^151 2 C40^19 2 S320^153 2 C160^77 2 S64^31 2 C80^39 2 S320^157 2 C160^79 2 S320^159 C2 160 C2' 160 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 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 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 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 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 ... ... ..... ....... ......... ........... ............. 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 ... ..T ..... ......T ......... ..........T ............. E1 2.00000 1.99961 1.99846 1.99653 1.99383 1.99037 1.98614 1.98114 1.97538 1.96885 1.96157 1.95353 1.94474 1.93520 1.92491 1.91388 1.90211 1.88961 1.87638 1.86243 1.84776 1.83238 1.81629 1.79950 1.78201 1.76384 1.74499 1.72547 1.70528 1.68443 1.66294 1.64080 1.61803 1.59464 1.57063 1.54602 1.52081 1.49502 1.46865 1.44171 1.41421 1.38617 1.35760 1.32850 1.29890 1.26879 1.23819 1.20711 1.17557 1.14358 1.11114 1.07828 1.04500 1.01131 0.97724 0.94279 0.90798 0.87282 0.83732 0.80150 0.76537 0.72894 0.69223 0.65526 0.61803 0.58057 0.54288 0.50498 0.46689 0.42862 0.39018 0.35159 0.31287 0.27402 0.23507 0.19603 0.15692 0.11774 0.07852 0.03927 0.00000 -0.03927 -0.07852 -0.11774 -0.15692 -0.19603 -0.23507 -0.27402 -0.31287 -0.35159 -0.39018 -0.42862 -0.46689 -0.50498 -0.54288 -0.58057 -0.61803 -0.65526 -0.69223 -0.72894 -0.76537 -0.80150 -0.83732 -0.87282 -0.90798 -0.94279 -0.97724 -1.01131 -1.04500 -1.07828 -1.11114 -1.14358 -1.17557 -1.20711 -1.23819 -1.26879 -1.29890 -1.32850 -1.35760 -1.38617 -1.41421 -1.44171 -1.46865 -1.49502 -1.52081 -1.54602 -1.57063 -1.59464 -1.61803 -1.64080 -1.66294 -1.68443 -1.70528 -1.72547 -1.74499 -1.76384 -1.78201 -1.79950 -1.81629 -1.83238 -1.84776 -1.86243 -1.87638 -1.88961 -1.90211 -1.91388 -1.92491 -1.93520 -1.94474 -1.95353 -1.96157 -1.96885 -1.97538 -1.98114 -1.98614 -1.99037 -1.99383 -1.99653 -1.99846 -1.99961 -2.00000 0.00000 0.00000 ... TT. ..... ....TT. ......... ........TT. ............. E2 2.00000 1.99846 1.99383 1.98614 1.97538 1.96157 1.94474 1.92491 1.90211 1.87638 1.84776 1.81629 1.78201 1.74499 1.70528 1.66294 1.61803 1.57063 1.52081 1.46865 1.41421 1.35760 1.29890 1.23819 1.17557 1.11114 1.04500 0.97724 0.90798 0.83732 0.76537 0.69223 0.61803 0.54288 0.46689 0.39018 0.31287 0.23507 0.15692 0.07852 0.00000 -0.07852 -0.15692 -0.23507 -0.31287 -0.39018 -0.46689 -0.54288 -0.61803 -0.69223 -0.76537 -0.83732 -0.90798 -0.97724 -1.04500 -1.11114 -1.17557 -1.23819 -1.29890 -1.35760 -1.41421 -1.46865 -1.52081 -1.57063 -1.61803 -1.66294 -1.70528 -1.74499 -1.78201 -1.81629 -1.84776 -1.87638 -1.90211 -1.92491 -1.94474 -1.96157 -1.97538 -1.98614 -1.99383 -1.99846 -2.00000 -1.99846 -1.99383 -1.98614 -1.97538 -1.96157 -1.94474 -1.92491 -1.90211 -1.87638 -1.84776 -1.81629 -1.78201 -1.74499 -1.70528 -1.66294 -1.61803 -1.57063 -1.52081 -1.46865 -1.41421 -1.35760 -1.29890 -1.23819 -1.17557 -1.11114 -1.04500 -0.97724 -0.90798 -0.83732 -0.76537 -0.69223 -0.61803 -0.54288 -0.46689 -0.39018 -0.31287 -0.23507 -0.15692 -0.07852 0.00000 0.07852 0.15692 0.23507 0.31287 0.39018 0.46689 0.54288 0.61803 0.69223 0.76537 0.83732 0.90798 0.97724 1.04500 1.11114 1.17557 1.23819 1.29890 1.35760 1.41421 1.46865 1.52081 1.57063 1.61803 1.66294 1.70528 1.74499 1.78201 1.81629 1.84776 1.87638 1.90211 1.92491 1.94474 1.96157 1.97538 1.98614 1.99383 1.99846 2.00000 0.00000 0.00000 ... ... TT... ....... ....TT... ........... ........TT... E3 2.00000 1.99653 1.98614 1.96885 1.94474 1.91388 1.87638 1.83238 1.78201 1.72547 1.66294 1.59464 1.52081 1.44171 1.35760 1.26879 1.17557 1.07828 0.97724 0.87282 0.76537 0.65526 0.54288 0.42862 0.31287 0.19603 0.07852 -0.03927 -0.15692 -0.27402 -0.39018 -0.50498 -0.61803 -0.72894 -0.83732 -0.94279 -1.04500 -1.14358 -1.23819 -1.32850 -1.41421 -1.49502 -1.57063 -1.64080 -1.70528 -1.76384 -1.81629 -1.86243 -1.90211 -1.93520 -1.96157 -1.98114 -1.99383 -1.99961 -1.99846 -1.99037 -1.97538 -1.95353 -1.92491 -1.88961 -1.84776 -1.79950 -1.74499 -1.68443 -1.61803 -1.54602 -1.46865 -1.38617 -1.29890 -1.20711 -1.11114 -1.01131 -0.90798 -0.80150 -0.69223 -0.58057 -0.46689 -0.35159 -0.23507 -0.11774 0.00000 0.11774 0.23507 0.35159 0.46689 0.58057 0.69223 0.80150 0.90798 1.01131 1.11114 1.20711 1.29890 1.38617 1.46865 1.54602 1.61803 1.68443 1.74499 1.79950 1.84776 1.88961 1.92491 1.95353 1.97538 1.99037 1.99846 1.99961 1.99383 1.98114 1.96157 1.93520 1.90211 1.86243 1.81629 1.76384 1.70528 1.64080 1.57063 1.49502 1.41421 1.32850 1.23819 1.14358 1.04500 0.94279 0.83732 0.72894 0.61803 0.50498 0.39018 0.27402 0.15692 0.03927 -0.07852 -0.19603 -0.31287 -0.42862 -0.54288 -0.65526 -0.76537 -0.87282 -0.97724 -1.07828 -1.17557 -1.26879 -1.35760 -1.44171 -1.52081 -1.59464 -1.66294 -1.72547 -1.78201 -1.83238 -1.87638 -1.91388 -1.94474 -1.96885 -1.98614 -1.99653 -2.00000 0.00000 0.00000 ... ... ..... TT..... ......... ....TT..... ............. E4 2.00000 1.99383 1.97538 1.94474 1.90211 1.84776 1.78201 1.70528 1.61803 1.52081 1.41421 1.29890 1.17557 1.04500 0.90798 0.76537 0.61803 0.46689 0.31287 0.15692 0.00000 -0.15692 -0.31287 -0.46689 -0.61803 -0.76537 -0.90798 -1.04500 -1.17557 -1.29890 -1.41421 -1.52081 -1.61803 -1.70528 -1.78201 -1.84776 -1.90211 -1.94474 -1.97538 -1.99383 -2.00000 -1.99383 -1.97538 -1.94474 -1.90211 -1.84776 -1.78201 -1.70528 -1.61803 -1.52081 -1.41421 -1.29890 -1.17557 -1.04500 -0.90798 -0.76537 -0.61803 -0.46689 -0.31287 -0.15692 0.00000 0.15692 0.31287 0.46689 0.61803 0.76537 0.90798 1.04500 1.17557 1.29890 1.41421 1.52081 1.61803 1.70528 1.78201 1.84776 1.90211 1.94474 1.97538 1.99383 2.00000 1.99383 1.97538 1.94474 1.90211 1.84776 1.78201 1.70528 1.61803 1.52081 1.41421 1.29890 1.17557 1.04500 0.90798 0.76537 0.61803 0.46689 0.31287 0.15692 0.00000 -0.15692 -0.31287 -0.46689 -0.61803 -0.76537 -0.90798 -1.04500 -1.17557 -1.29890 -1.41421 -1.52081 -1.61803 -1.70528 -1.78201 -1.84776 -1.90211 -1.94474 -1.97538 -1.99383 -2.00000 -1.99383 -1.97538 -1.94474 -1.90211 -1.84776 -1.78201 -1.70528 -1.61803 -1.52081 -1.41421 -1.29890 -1.17557 -1.04500 -0.90798 -0.76537 -0.61803 -0.46689 -0.31287 -0.15692 0.00000 0.15692 0.31287 0.46689 0.61803 0.76537 0.90798 1.04500 1.17557 1.29890 1.41421 1.52081 1.61803 1.70528 1.78201 1.84776 1.90211 1.94474 1.97538 1.99383 2.00000 0.00000 0.00000 ... ... ..... ....... TT....... ........... ....TT....... E5 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 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......... ............. E6 2.00000 1.98614 1.94474 1.87638 1.78201 1.66294 1.52081 1.35760 1.17557 0.97724 0.76537 0.54288 0.31287 0.07852 -0.15692 -0.39018 -0.61803 -0.83732 -1.04500 -1.23819 -1.41421 -1.57063 -1.70528 -1.81629 -1.90211 -1.96157 -1.99383 -1.99846 -1.97538 -1.92491 -1.84776 -1.74499 -1.61803 -1.46865 -1.29890 -1.11114 -0.90798 -0.69223 -0.46689 -0.23507 0.00000 0.23507 0.46689 0.69223 0.90798 1.11114 1.29890 1.46865 1.61803 1.74499 1.84776 1.92491 1.97538 1.99846 1.99383 1.96157 1.90211 1.81629 1.70528 1.57063 1.41421 1.23819 1.04500 0.83732 0.61803 0.39018 0.15692 -0.07852 -0.31287 -0.54288 -0.76537 -0.97724 -1.17557 -1.35760 -1.52081 -1.66294 -1.78201 -1.87638 -1.94474 -1.98614 -2.00000 -1.98614 -1.94474 -1.87638 -1.78201 -1.66294 -1.52081 -1.35760 -1.17557 -0.97724 -0.76537 -0.54288 -0.31287 -0.07852 0.15692 0.39018 0.61803 0.83732 1.04500 1.23819 1.41421 1.57063 1.70528 1.81629 1.90211 1.96157 1.99383 1.99846 1.97538 1.92491 1.84776 1.74499 1.61803 1.46865 1.29890 1.11114 0.90798 0.69223 0.46689 0.23507 0.00000 -0.23507 -0.46689 -0.69223 -0.90798 -1.11114 -1.29890 -1.46865 -1.61803 -1.74499 -1.84776 -1.92491 -1.97538 -1.99846 -1.99383 -1.96157 -1.90211 -1.81629 -1.70528 -1.57063 -1.41421 -1.23819 -1.04500 -0.83732 -0.61803 -0.39018 -0.15692 0.07852 0.31287 0.54288 0.76537 0.97724 1.17557 1.35760 1.52081 1.66294 1.78201 1.87638 1.94474 1.98614 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... TT........... 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E155 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 -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......... E156 2.00000 -1.99383 1.97538 -1.94474 1.90211 -1.84776 1.78201 -1.70528 1.61803 -1.52081 1.41421 -1.29890 1.17557 -1.04500 0.90798 -0.76537 0.61803 -0.46689 0.31287 -0.15692 0.00000 0.15692 -0.31287 0.46689 -0.61803 0.76537 -0.90798 1.04500 -1.17557 1.29890 -1.41421 1.52081 -1.61803 1.70528 -1.78201 1.84776 -1.90211 1.94474 -1.97538 1.99383 -2.00000 1.99383 -1.97538 1.94474 -1.90211 1.84776 -1.78201 1.70528 -1.61803 1.52081 -1.41421 1.29890 -1.17557 1.04500 -0.90798 0.76537 -0.61803 0.46689 -0.31287 0.15692 0.00000 -0.15692 0.31287 -0.46689 0.61803 -0.76537 0.90798 -1.04500 1.17557 -1.29890 1.41421 -1.52081 1.61803 -1.70528 1.78201 -1.84776 1.90211 -1.94474 1.97538 -1.99383 2.00000 -1.99383 1.97538 -1.94474 1.90211 -1.84776 1.78201 -1.70528 1.61803 -1.52081 1.41421 -1.29890 1.17557 -1.04500 0.90798 -0.76537 0.61803 -0.46689 0.31287 -0.15692 0.00000 0.15692 -0.31287 0.46689 -0.61803 0.76537 -0.90798 1.04500 -1.17557 1.29890 -1.41421 1.52081 -1.61803 1.70528 -1.78201 1.84776 -1.90211 1.94474 -1.97538 1.99383 -2.00000 1.99383 -1.97538 1.94474 -1.90211 1.84776 -1.78201 1.70528 -1.61803 1.52081 -1.41421 1.29890 -1.17557 1.04500 -0.90798 0.76537 -0.61803 0.46689 -0.31287 0.15692 0.00000 -0.15692 0.31287 -0.46689 0.61803 -0.76537 0.90798 -1.04500 1.17557 -1.29890 1.41421 -1.52081 1.61803 -1.70528 1.78201 -1.84776 1.90211 -1.94474 1.97538 -1.99383 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ..TT....... ............. E157 2.00000 -1.99653 1.98614 -1.96885 1.94474 -1.91388 1.87638 -1.83238 1.78201 -1.72547 1.66294 -1.59464 1.52081 -1.44171 1.35760 -1.26879 1.17557 -1.07828 0.97724 -0.87282 0.76537 -0.65526 0.54288 -0.42862 0.31287 -0.19603 0.07852 0.03927 -0.15692 0.27402 -0.39018 0.50498 -0.61803 0.72894 -0.83732 0.94279 -1.04500 1.14358 -1.23819 1.32850 -1.41421 1.49502 -1.57063 1.64080 -1.70528 1.76384 -1.81629 1.86243 -1.90211 1.93520 -1.96157 1.98114 -1.99383 1.99961 -1.99846 1.99037 -1.97538 1.95353 -1.92491 1.88961 -1.84776 1.79950 -1.74499 1.68443 -1.61803 1.54602 -1.46865 1.38617 -1.29890 1.20711 -1.11114 1.01131 -0.90798 0.80150 -0.69223 0.58057 -0.46689 0.35159 -0.23507 0.11774 0.00000 -0.11774 0.23507 -0.35159 0.46689 -0.58057 0.69223 -0.80150 0.90798 -1.01131 1.11114 -1.20711 1.29890 -1.38617 1.46865 -1.54602 1.61803 -1.68443 1.74499 -1.79950 1.84776 -1.88961 1.92491 -1.95353 1.97538 -1.99037 1.99846 -1.99961 1.99383 -1.98114 1.96157 -1.93520 1.90211 -1.86243 1.81629 -1.76384 1.70528 -1.64080 1.57063 -1.49502 1.41421 -1.32850 1.23819 -1.14358 1.04500 -0.94279 0.83732 -0.72894 0.61803 -0.50498 0.39018 -0.27402 0.15692 -0.03927 -0.07852 0.19603 -0.31287 0.42862 -0.54288 0.65526 -0.76537 0.87282 -0.97724 1.07828 -1.17557 1.26879 -1.35760 1.44171 -1.52081 1.59464 -1.66294 1.72547 -1.78201 1.83238 -1.87638 1.91388 -1.94474 1.96885 -1.98614 1.99653 -2.00000 0.00000 0.00000 ... ... ..... ....... ..TT..... ........... ......TT..... E158 2.00000 -1.99846 1.99383 -1.98614 1.97538 -1.96157 1.94474 -1.92491 1.90211 -1.87638 1.84776 -1.81629 1.78201 -1.74499 1.70528 -1.66294 1.61803 -1.57063 1.52081 -1.46865 1.41421 -1.35760 1.29890 -1.23819 1.17557 -1.11114 1.04500 -0.97724 0.90798 -0.83732 0.76537 -0.69223 0.61803 -0.54288 0.46689 -0.39018 0.31287 -0.23507 0.15692 -0.07852 0.00000 0.07852 -0.15692 0.23507 -0.31287 0.39018 -0.46689 0.54288 -0.61803 0.69223 -0.76537 0.83732 -0.90798 0.97724 -1.04500 1.11114 -1.17557 1.23819 -1.29890 1.35760 -1.41421 1.46865 -1.52081 1.57063 -1.61803 1.66294 -1.70528 1.74499 -1.78201 1.81629 -1.84776 1.87638 -1.90211 1.92491 -1.94474 1.96157 -1.97538 1.98614 -1.99383 1.99846 -2.00000 1.99846 -1.99383 1.98614 -1.97538 1.96157 -1.94474 1.92491 -1.90211 1.87638 -1.84776 1.81629 -1.78201 1.74499 -1.70528 1.66294 -1.61803 1.57063 -1.52081 1.46865 -1.41421 1.35760 -1.29890 1.23819 -1.17557 1.11114 -1.04500 0.97724 -0.90798 0.83732 -0.76537 0.69223 -0.61803 0.54288 -0.46689 0.39018 -0.31287 0.23507 -0.15692 0.07852 0.00000 -0.07852 0.15692 -0.23507 0.31287 -0.39018 0.46689 -0.54288 0.61803 -0.69223 0.76537 -0.83732 0.90798 -0.97724 1.04500 -1.11114 1.17557 -1.23819 1.29890 -1.35760 1.41421 -1.46865 1.52081 -1.57063 1.61803 -1.66294 1.70528 -1.74499 1.78201 -1.81629 1.84776 -1.87638 1.90211 -1.92491 1.94474 -1.96157 1.97538 -1.98614 1.99383 -1.99846 2.00000 0.00000 0.00000 ... ... ..... ..TT... ......... ......TT... ............. E159 2.00000 -1.99961 1.99846 -1.99653 1.99383 -1.99037 1.98614 -1.98114 1.97538 -1.96885 1.96157 -1.95353 1.94474 -1.93520 1.92491 -1.91388 1.90211 -1.88961 1.87638 -1.86243 1.84776 -1.83238 1.81629 -1.79950 1.78201 -1.76384 1.74499 -1.72547 1.70528 -1.68443 1.66294 -1.64080 1.61803 -1.59464 1.57063 -1.54602 1.52081 -1.49502 1.46865 -1.44171 1.41421 -1.38617 1.35760 -1.32850 1.29890 -1.26879 1.23819 -1.20711 1.17557 -1.14358 1.11114 -1.07828 1.04500 -1.01131 0.97724 -0.94279 0.90798 -0.87282 0.83732 -0.80150 0.76537 -0.72894 0.69223 -0.65526 0.61803 -0.58057 0.54288 -0.50498 0.46689 -0.42862 0.39018 -0.35159 0.31287 -0.27402 0.23507 -0.19603 0.15692 -0.11774 0.07852 -0.03927 0.00000 0.03927 -0.07852 0.11774 -0.15692 0.19603 -0.23507 0.27402 -0.31287 0.35159 -0.39018 0.42862 -0.46689 0.50498 -0.54288 0.58057 -0.61803 0.65526 -0.69223 0.72894 -0.76537 0.80150 -0.83732 0.87282 -0.90798 0.94279 -0.97724 1.01131 -1.04500 1.07828 -1.11114 1.14358 -1.17557 1.20711 -1.23819 1.26879 -1.29890 1.32850 -1.35760 1.38617 -1.41421 1.44171 -1.46865 1.49502 -1.52081 1.54602 -1.57063 1.59464 -1.61803 1.64080 -1.66294 1.68443 -1.70528 1.72547 -1.74499 1.76384 -1.78201 1.79950 -1.81629 1.83238 -1.84776 1.86243 -1.87638 1.88961 -1.90211 1.91388 -1.92491 1.93520 -1.94474 1.95353 -1.96157 1.96885 -1.97538 1.98114 -1.98614 1.99037 -1.99383 1.99653 -1.99846 1.99961 -2.00000 0.00000 0.00000 TT. ... ..TT. ....... ......TT. ........... ..........TT. Irrational character values: 1.999614480964 = 2*cos(2*π/320) = 2*cos(π/160) 1.998458072481 = 2*cos(4*π/320) = 2*cos(π/80) = √4+√8+√2+√10+2*√5−√5/√2 1.996531220369 = 2*cos(6*π/320) = 2*cos(3*π/160) 1.993834667466 = 2*cos(8*π/320) = 2*cos(π/40) = √8+√2+√10+2*√5−√5/2 1.990369453344 = 2*cos(10*π/320) = 2*cos(π/32) = √2+√2+√2+√2 1.986136913910 = 2*cos(12*π/320) = 2*cos(3*π/80) = √4+√8−√2+√10+2*√5+√5/√2 1.981138680887 = 2*cos(14*π/320) = 2*cos(7*π/160) 1.975376681190 = 2*cos(16*π/320) = 2*cos(π/20) = (√2+√10+2*(√5−√5))/4 1.968853136180 = 2*cos(18*π/320) = 2*cos(9*π/160) 1.961570560806 = 2*cos(20*π/320) = 2*cos(π/16) = √2+√2+√2 1.953531762642 = 2*cos(22*π/320) = 2*cos(11*π/160) 1.944739840795 = 2*cos(24*π/320) = 2*cos(3*π/40) = √8−√2+√10+2*√5+√5/2 1.935198184721 = 2*cos(26*π/320) = 2*cos(13*π/160) 1.924910472907 = 2*cos(28*π/320) = 2*cos(7*π/80) = √4+√8+√2−√10+2*√5+√5/√2 1.913880671464 = 2*cos(30*π/320) = 2*cos(3*π/32) = √2+√2+√2−√2 1.902113032590 = 2*cos(32*π/320) = 2*cos(π/10) = (√10+2*√5)/2 1.889612092934 = 2*cos(34*π/320) = 2*cos(17*π/160) 1.876382671845 = 2*cos(36*π/320) = 2*cos(9*π/80) = √4+√8+√2+√10−2*√5−√5/√2 1.862429869518 = 2*cos(38*π/320) = 2*cos(19*π/160) 1.847759065023 = 2*cos(40*π/320) = 2*cos(π/8) = √2+√2 1.832375914234 = 2*cos(42*π/320) = 2*cos(21*π/160) 1.816286347650 = 2*cos(44*π/320) = 2*cos(11*π/80) = √4+√8−√2−√10+2*√5−√5/√2 1.799496568104 = 2*cos(46*π/320) = 2*cos(23*π/160) 1.782013048377 = 2*cos(48*π/320) = 2*cos(3*π/20) = (√10−√2+2*√5+√5)/4 1.763842528697 = 2*cos(50*π/320) = 2*cos(5*π/32) = √2+√2−√2−√2 1.744992014146 = 2*cos(52*π/320) = 2*cos(13*π/80) = √4+√8−√2+√10−2*√5+√5/√2 1.725468771956 = 2*cos(54*π/320) = 2*cos(27*π/160) 1.705280328708 = 2*cos(56*π/320) = 2*cos(7*π/40) = √8+√2−√10+2*√5+√5/2 1.684434467433 = 2*cos(58*π/320) = 2*cos(29*π/160) 1.662939224605 = 2*cos(60*π/320) = 2*cos(3*π/16) = √2+√2−√2 1.640802887051 = 2*cos(62*π/320) = 2*cos(31*π/160) 1.618033988750 = 2*cos(64*π/320) = 2*cos(π/5) = (√5+1)/2 1.594641307545 = 2*cos(66*π/320) = 2*cos(33*π/160) 1.570633861761 = 2*cos(68*π/320) = 2*cos(17*π/80) = √4+√8+√2−√10−2*√5+√5/√2 1.546020906725 = 2*cos(70*π/320) = 2*cos(7*π/32) = √2+√2−√2+√2 1.520811931200 = 2*cos(72*π/320) = 2*cos(9*π/40) = √8+√2+√10−2*√5−√5/2 1.495016653725 = 2*cos(74*π/320) = 2*cos(37*π/160) 1.468645018871 = 2*cos(76*π/320) = 2*cos(19*π/80) = √4+√8−√2−√10−2*√5−√5/√2 1.441707193406 = 2*cos(78*π/320) = 2*cos(39*π/160) 1.414213562373 = 2*cos(80*π/320) = 2*cos(π/4) = √2 1.386174725091 = 2*cos(82*π/320) = 2*cos(41*π/160) 1.357601491066 = 2*cos(84*π/320) = 2*cos(21*π/80) = √4−√8−√2−√10−2*√5−√5/√2 1.328504875823 = 2*cos(86*π/320) = 2*cos(43*π/160) 1.298896096660 = 2*cos(88*π/320) = 2*cos(11*π/40) = √8−√2−√10+2*√5−√5/2 1.268786568327 = 2*cos(90*π/320) = 2*cos(9*π/32) = √2−√2−√2+√2 1.238187898620 = 2*cos(92*π/320) = 2*cos(23*π/80) = √4−√8+√2−√10−2*√5+√5/√2 1.207111883907 = 2*cos(94*π/320) = 2*cos(47*π/160) 1.175570504585 = 2*cos(96*π/320) = 2*cos(3*π/10) = (√10−2*√5)/2 1.143575920455 = 2*cos(98*π/320) = 2*cos(49*π/160) 1.111140466039 = 2*cos(100*π/320) = 2*cos(5*π/16) = √2−√2−√2 1.078276645822 = 2*cos(102*π/320) = 2*cos(51*π/160) 1.044997129432 = 2*cos(104*π/320) = 2*cos(13*π/40) = √8−√2+√10−2*√5+√5/2 1.011314746756 = 2*cos(106*π/320) = 2*cos(53*π/160) 0.977242482994 = 2*cos(108*π/320) = 2*cos(27*π/80) = √4−√8−√2+√10−2*√5+√5/√2 0.942793473652 = 2*cos(110*π/320) = 2*cos(11*π/32) = √2−√2−√2−√2 0.907980999479 = 2*cos(112*π/320) = 2*cos(7*π/20) = (√2−√10+2*√5+√5)/4 0.872818481347 = 2*cos(114*π/320) = 2*cos(57*π/160) 0.837319475075 = 2*cos(116*π/320) = 2*cos(29*π/80) = √4−√8−√2−√10+2*√5−√5/√2 0.801497666206 = 2*cos(118*π/320) = 2*cos(59*π/160) 0.765366864730 = 2*cos(120*π/320) = 2*cos(3*π/8) = √2−√2 0.728940999758 = 2*cos(122*π/320) = 2*cos(61*π/160) 0.692234114155 = 2*cos(124*π/320) = 2*cos(31*π/80) = √4−√8+√2+√10−2*√5−√5/√2 0.655260359123 = 2*cos(126*π/320) = 2*cos(63*π/160) 0.618033988750 = 2*cos(128*π/320) = 2*cos(2*π/5) = (√5−1)/2 0.580569354509 = 2*cos(130*π/320) = 2*cos(13*π/32) = √2−√2+√2−√2 0.542880899730 = 2*cos(132*π/320) = 2*cos(33*π/80) = √4−√8+√2−√10+2*√5+√5/√2 0.504983154030 = 2*cos(134*π/320) = 2*cos(67*π/160) 0.466890727712 = 2*cos(136*π/320) = 2*cos(17*π/40) = √8+√2−√10−2*√5+√5/2 0.428618306130 = 2*cos(138*π/320) = 2*cos(69*π/160) 0.390180644032 = 2*cos(140*π/320) = 2*cos(7*π/16) = √2−√2+√2 0.351592559869 = 2*cos(142*π/320) = 2*cos(71*π/160) 0.312868930080 = 2*cos(144*π/320) = 2*cos(9*π/20) = (√2+√10−2*(√5−√5))/4 0.274024683364 = 2*cos(146*π/320) = 2*cos(73*π/160) 0.235074794916 = 2*cos(148*π/320) = 2*cos(37*π/80) = √4−√8−√2+√10+2*√5+√5/√2 0.196034280659 = 2*cos(150*π/320) = 2*cos(15*π/32) = √2−√2+√2+√2 0.156918191456 = 2*cos(152*π/320) = 2*cos(19*π/40) = √8−√2−√10−2*√5−√5/2 0.117741607302 = 2*cos(154*π/320) = 2*cos(77*π/160) 0.078519631518 = 2*cos(156*π/320) = 2*cos(39*π/80) = √4−√8+√2+√10+2*√5−√5/√2 0.039267384921 = 2*cos(158*π/320) = 2*cos(79*π/160) 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 E151 m E152 l E153 k+m E154 j+l E155 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)} E156 h+j+l {z((x²−y²)²−4x²y²), xyz(x²−y²)} E157 g+i+k+m {xz(x²−3y²), yz(3x²−y²)}, {xz³(x²−3y²), yz³(3x²−y²)} E158 f+h+j+l {z(x²−y²), xyz}, {z³(x²−y²), xyz³} E159 R+d+g+i+k+m {R_x, R_y}, {xz, yz}, {xz³, yz³}, {xz⁵, yz⁵} Notes: α The order of the D_160d point group is 640, and the order of the principal axis (S₃₂₀) is 320. The group has 163 irreducible representations. β The D_160d point group is isomorphic to C_320v and D₃₂₀. γ The D_160d point group is generated by two symmetry elements, S₃₂₀ 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_160d 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. Some of these algebraic expressions, however, are too complicated to be included in the list above. κ 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 10,20,40,80 etc. are easily derived from the regular pentagon by successive halving of angles.

This Character Table for the D_160d point group was created by Gernot Katzer.

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

	D_144d
	D_153d
C₁₆₀ C_160v C_160h D₁₆₀ D_160h	D_160d	S₁₆₀
	D_170d
	D_192d

Character table for the D160d point group

Character table for the D_160d point group