Character Table for Point Group D38d

Character table for the D_38d point group

D38d E 2 S76 2 C38 2 S76^3 2 C19 2 S76^5 2 C38^3 2 S76^7 2 C19^2 2 S76^9 2 C38^5 2 S76^11 2 C19^3 2 S76^13 2 C38^7 2 S76^15 2 C19^4 2 S76^17 2 C38^9 2 S4 2 C19^5 2 S76^21 2 C38^11 2 S76^23 2 C19^6 2 S76^25 2 C38^13 2 S76^27 2 C19^7 2 S76^29 2 C38^15 2 S76^31 2 C19^8 2 S76^33 2 C38^17 2 S76^35 2 C19^9 2 S76^37 C2 38 C2' 38 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 ... ... ....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 ..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 ... ... ..... ....... ......... ........... ............. 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 ... ..T ..... ......T ......... ..........T ............. E1 2.00000 1.99317 1.97272 1.93880 1.89163 1.83155 1.75895 1.67433 1.57828 1.47145 1.35456 1.22843 1.09390 0.95189 0.80339 0.64940 0.49097 0.32919 0.16516 0.00000 -0.16516 -0.32919 -0.49097 -0.64940 -0.80339 -0.95189 -1.09390 -1.22843 -1.35456 -1.47145 -1.57828 -1.67433 -1.75895 -1.83155 -1.89163 -1.93880 -1.97272 -1.99317 -2.00000 0.00000 0.00000 ... TT. ..... ....TT. ......... ........TT. ............. E2 2.00000 1.97272 1.89163 1.75895 1.57828 1.35456 1.09390 0.80339 0.49097 0.16516 -0.16516 -0.49097 -0.80339 -1.09390 -1.35456 -1.57828 -1.75895 -1.89163 -1.97272 -2.00000 -1.97272 -1.89163 -1.75895 -1.57828 -1.35456 -1.09390 -0.80339 -0.49097 -0.16516 0.16516 0.49097 0.80339 1.09390 1.35456 1.57828 1.75895 1.89163 1.97272 2.00000 0.00000 0.00000 ... ... TT... ....... ....TT... ........... ........TT... E3 2.00000 1.93880 1.75895 1.47145 1.09390 0.64940 0.16516 -0.32919 -0.80339 -1.22843 -1.57828 -1.83155 -1.97272 -1.99317 -1.89163 -1.67433 -1.35456 -0.95189 -0.49097 0.00000 0.49097 0.95189 1.35456 1.67433 1.89163 1.99317 1.97272 1.83155 1.57828 1.22843 0.80339 0.32919 -0.16516 -0.64940 -1.09390 -1.47145 -1.75895 -1.93880 -2.00000 0.00000 0.00000 ... ... ..... TT..... ......... ....TT..... ............. E4 2.00000 1.89163 1.57828 1.09390 0.49097 -0.16516 -0.80339 -1.35456 -1.75895 -1.97272 -1.97272 -1.75895 -1.35456 -0.80339 -0.16516 0.49097 1.09390 1.57828 1.89163 2.00000 1.89163 1.57828 1.09390 0.49097 -0.16516 -0.80339 -1.35456 -1.75895 -1.97272 -1.97272 -1.75895 -1.35456 -0.80339 -0.16516 0.49097 1.09390 1.57828 1.89163 2.00000 0.00000 0.00000 ... ... ..... ....... TT....... ........... ....TT....... E5 2.00000 1.83155 1.35456 0.64940 -0.16516 -0.95189 -1.57828 -1.93880 -1.97272 -1.67433 -1.09390 -0.32919 0.49097 1.22843 1.75895 1.99317 1.89163 1.47145 0.80339 0.00000 -0.80339 -1.47145 -1.89163 -1.99317 -1.75895 -1.22843 -0.49097 0.32919 1.09390 1.67433 1.97272 1.93880 1.57828 0.95189 0.16516 -0.64940 -1.35456 -1.83155 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... TT......... ............. E6 2.00000 1.75895 1.09390 0.16516 -0.80339 -1.57828 -1.97272 -1.89163 -1.35456 -0.49097 0.49097 1.35456 1.89163 1.97272 1.57828 0.80339 -0.16516 -1.09390 -1.75895 -2.00000 -1.75895 -1.09390 -0.16516 0.80339 1.57828 1.97272 1.89163 1.35456 0.49097 -0.49097 -1.35456 -1.89163 -1.97272 -1.57828 -0.80339 0.16516 1.09390 1.75895 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... TT........... E7 2.00000 1.67433 0.80339 -0.32919 -1.35456 -1.93880 -1.89163 -1.22843 -0.16516 0.95189 1.75895 1.99317 1.57828 0.64940 -0.49097 -1.47145 -1.97272 -1.83155 -1.09390 0.00000 1.09390 1.83155 1.97272 1.47145 0.49097 -0.64940 -1.57828 -1.99317 -1.75895 -0.95189 0.16516 1.22843 1.89163 1.93880 1.35456 0.32919 -0.80339 -1.67433 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E8 2.00000 1.57828 0.49097 -0.80339 -1.75895 -1.97272 -1.35456 -0.16516 1.09390 1.89163 1.89163 1.09390 -0.16516 -1.35456 -1.97272 -1.75895 -0.80339 0.49097 1.57828 2.00000 1.57828 0.49097 -0.80339 -1.75895 -1.97272 -1.35456 -0.16516 1.09390 1.89163 1.89163 1.09390 -0.16516 -1.35456 -1.97272 -1.75895 -0.80339 0.49097 1.57828 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E9 2.00000 1.47145 0.16516 -1.22843 -1.97272 -1.67433 -0.49097 0.95189 1.89163 1.83155 0.80339 -0.64940 -1.75895 -1.93880 -1.09390 0.32919 1.57828 1.99317 1.35456 0.00000 -1.35456 -1.99317 -1.57828 -0.32919 1.09390 1.93880 1.75895 0.64940 -0.80339 -1.83155 -1.89163 -0.95189 0.49097 1.67433 1.97272 1.22843 -0.16516 -1.47145 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E10 2.00000 1.35456 -0.16516 -1.57828 -1.97272 -1.09390 0.49097 1.75895 1.89163 0.80339 -0.80339 -1.89163 -1.75895 -0.49097 1.09390 1.97272 1.57828 0.16516 -1.35456 -2.00000 -1.35456 0.16516 1.57828 1.97272 1.09390 -0.49097 -1.75895 -1.89163 -0.80339 0.80339 1.89163 1.75895 0.49097 -1.09390 -1.97272 -1.57828 -0.16516 1.35456 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E11 2.00000 1.22843 -0.49097 -1.83155 -1.75895 -0.32919 1.35456 1.99317 1.09390 -0.64940 -1.89163 -1.67433 -0.16516 1.47145 1.97272 0.95189 -0.80339 -1.93880 -1.57828 0.00000 1.57828 1.93880 0.80339 -0.95189 -1.97272 -1.47145 0.16516 1.67433 1.89163 0.64940 -1.09390 -1.99317 -1.35456 0.32919 1.75895 1.83155 0.49097 -1.22843 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E12 2.00000 1.09390 -0.80339 -1.97272 -1.35456 0.49097 1.89163 1.57828 -0.16516 -1.75895 -1.75895 -0.16516 1.57828 1.89163 0.49097 -1.35456 -1.97272 -0.80339 1.09390 2.00000 1.09390 -0.80339 -1.97272 -1.35456 0.49097 1.89163 1.57828 -0.16516 -1.75895 -1.75895 -0.16516 1.57828 1.89163 0.49097 -1.35456 -1.97272 -0.80339 1.09390 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E13 2.00000 0.95189 -1.09390 -1.99317 -0.80339 1.22843 1.97272 0.64940 -1.35456 -1.93880 -0.49097 1.47145 1.89163 0.32919 -1.57828 -1.83155 -0.16516 1.67433 1.75895 0.00000 -1.75895 -1.67433 0.16516 1.83155 1.57828 -0.32919 -1.89163 -1.47145 0.49097 1.93880 1.35456 -0.64940 -1.97272 -1.22843 0.80339 1.99317 1.09390 -0.95189 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E14 2.00000 0.80339 -1.35456 -1.89163 -0.16516 1.75895 1.57828 -0.49097 -1.97272 -1.09390 1.09390 1.97272 0.49097 -1.57828 -1.75895 0.16516 1.89163 1.35456 -0.80339 -2.00000 -0.80339 1.35456 1.89163 0.16516 -1.75895 -1.57828 0.49097 1.97272 1.09390 -1.09390 -1.97272 -0.49097 1.57828 1.75895 -0.16516 -1.89163 -1.35456 0.80339 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E15 2.00000 0.64940 -1.57828 -1.67433 0.49097 1.99317 0.80339 -1.47145 -1.75895 0.32919 1.97272 0.95189 -1.35456 -1.83155 0.16516 1.93880 1.09390 -1.22843 -1.89163 0.00000 1.89163 1.22843 -1.09390 -1.93880 -0.16516 1.83155 1.35456 -0.95189 -1.97272 -0.32919 1.75895 1.47145 -0.80339 -1.99317 -0.49097 1.67433 1.57828 -0.64940 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E16 2.00000 0.49097 -1.75895 -1.35456 1.09390 1.89163 -0.16516 -1.97272 -0.80339 1.57828 1.57828 -0.80339 -1.97272 -0.16516 1.89163 1.09390 -1.35456 -1.75895 0.49097 2.00000 0.49097 -1.75895 -1.35456 1.09390 1.89163 -0.16516 -1.97272 -0.80339 1.57828 1.57828 -0.80339 -1.97272 -0.16516 1.89163 1.09390 -1.35456 -1.75895 0.49097 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E17 2.00000 0.32919 -1.89163 -0.95189 1.57828 1.47145 -1.09390 -1.83155 0.49097 1.99317 0.16516 -1.93880 -0.80339 1.67433 1.35456 -1.22843 -1.75895 0.64940 1.97272 0.00000 -1.97272 -0.64940 1.75895 1.22843 -1.35456 -1.67433 0.80339 1.93880 -0.16516 -1.99317 -0.49097 1.83155 1.09390 -1.47145 -1.57828 0.95189 1.89163 -0.32919 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E18 2.00000 0.16516 -1.97272 -0.49097 1.89163 0.80339 -1.75895 -1.09390 1.57828 1.35456 -1.35456 -1.57828 1.09390 1.75895 -0.80339 -1.89163 0.49097 1.97272 -0.16516 -2.00000 -0.16516 1.97272 0.49097 -1.89163 -0.80339 1.75895 1.09390 -1.57828 -1.35456 1.35456 1.57828 -1.09390 -1.75895 0.80339 1.89163 -0.49097 -1.97272 0.16516 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E19 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 ... ... ..... ....... ......... ........... ............. E20 2.00000 -0.16516 -1.97272 0.49097 1.89163 -0.80339 -1.75895 1.09390 1.57828 -1.35456 -1.35456 1.57828 1.09390 -1.75895 -0.80339 1.89163 0.49097 -1.97272 -0.16516 2.00000 -0.16516 -1.97272 0.49097 1.89163 -0.80339 -1.75895 1.09390 1.57828 -1.35456 -1.35456 1.57828 1.09390 -1.75895 -0.80339 1.89163 0.49097 -1.97272 -0.16516 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E21 2.00000 -0.32919 -1.89163 0.95189 1.57828 -1.47145 -1.09390 1.83155 0.49097 -1.99317 0.16516 1.93880 -0.80339 -1.67433 1.35456 1.22843 -1.75895 -0.64940 1.97272 0.00000 -1.97272 0.64940 1.75895 -1.22843 -1.35456 1.67433 0.80339 -1.93880 -0.16516 1.99317 -0.49097 -1.83155 1.09390 1.47145 -1.57828 -0.95189 1.89163 0.32919 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E22 2.00000 -0.49097 -1.75895 1.35456 1.09390 -1.89163 -0.16516 1.97272 -0.80339 -1.57828 1.57828 0.80339 -1.97272 0.16516 1.89163 -1.09390 -1.35456 1.75895 0.49097 -2.00000 0.49097 1.75895 -1.35456 -1.09390 1.89163 0.16516 -1.97272 0.80339 1.57828 -1.57828 -0.80339 1.97272 -0.16516 -1.89163 1.09390 1.35456 -1.75895 -0.49097 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E23 2.00000 -0.64940 -1.57828 1.67433 0.49097 -1.99317 0.80339 1.47145 -1.75895 -0.32919 1.97272 -0.95189 -1.35456 1.83155 0.16516 -1.93880 1.09390 1.22843 -1.89163 0.00000 1.89163 -1.22843 -1.09390 1.93880 -0.16516 -1.83155 1.35456 0.95189 -1.97272 0.32919 1.75895 -1.47145 -0.80339 1.99317 -0.49097 -1.67433 1.57828 0.64940 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E24 2.00000 -0.80339 -1.35456 1.89163 -0.16516 -1.75895 1.57828 0.49097 -1.97272 1.09390 1.09390 -1.97272 0.49097 1.57828 -1.75895 -0.16516 1.89163 -1.35456 -0.80339 2.00000 -0.80339 -1.35456 1.89163 -0.16516 -1.75895 1.57828 0.49097 -1.97272 1.09390 1.09390 -1.97272 0.49097 1.57828 -1.75895 -0.16516 1.89163 -1.35456 -0.80339 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E25 2.00000 -0.95189 -1.09390 1.99317 -0.80339 -1.22843 1.97272 -0.64940 -1.35456 1.93880 -0.49097 -1.47145 1.89163 -0.32919 -1.57828 1.83155 -0.16516 -1.67433 1.75895 0.00000 -1.75895 1.67433 0.16516 -1.83155 1.57828 0.32919 -1.89163 1.47145 0.49097 -1.93880 1.35456 0.64940 -1.97272 1.22843 0.80339 -1.99317 1.09390 0.95189 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E26 2.00000 -1.09390 -0.80339 1.97272 -1.35456 -0.49097 1.89163 -1.57828 -0.16516 1.75895 -1.75895 0.16516 1.57828 -1.89163 0.49097 1.35456 -1.97272 0.80339 1.09390 -2.00000 1.09390 0.80339 -1.97272 1.35456 0.49097 -1.89163 1.57828 0.16516 -1.75895 1.75895 -0.16516 -1.57828 1.89163 -0.49097 -1.35456 1.97272 -0.80339 -1.09390 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E27 2.00000 -1.22843 -0.49097 1.83155 -1.75895 0.32919 1.35456 -1.99317 1.09390 0.64940 -1.89163 1.67433 -0.16516 -1.47145 1.97272 -0.95189 -0.80339 1.93880 -1.57828 0.00000 1.57828 -1.93880 0.80339 0.95189 -1.97272 1.47145 0.16516 -1.67433 1.89163 -0.64940 -1.09390 1.99317 -1.35456 -0.32919 1.75895 -1.83155 0.49097 1.22843 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E28 2.00000 -1.35456 -0.16516 1.57828 -1.97272 1.09390 0.49097 -1.75895 1.89163 -0.80339 -0.80339 1.89163 -1.75895 0.49097 1.09390 -1.97272 1.57828 -0.16516 -1.35456 2.00000 -1.35456 -0.16516 1.57828 -1.97272 1.09390 0.49097 -1.75895 1.89163 -0.80339 -0.80339 1.89163 -1.75895 0.49097 1.09390 -1.97272 1.57828 -0.16516 -1.35456 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E29 2.00000 -1.47145 0.16516 1.22843 -1.97272 1.67433 -0.49097 -0.95189 1.89163 -1.83155 0.80339 0.64940 -1.75895 1.93880 -1.09390 -0.32919 1.57828 -1.99317 1.35456 0.00000 -1.35456 1.99317 -1.57828 0.32919 1.09390 -1.93880 1.75895 -0.64940 -0.80339 1.83155 -1.89163 0.95189 0.49097 -1.67433 1.97272 -1.22843 -0.16516 1.47145 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E30 2.00000 -1.57828 0.49097 0.80339 -1.75895 1.97272 -1.35456 0.16516 1.09390 -1.89163 1.89163 -1.09390 -0.16516 1.35456 -1.97272 1.75895 -0.80339 -0.49097 1.57828 -2.00000 1.57828 -0.49097 -0.80339 1.75895 -1.97272 1.35456 -0.16516 -1.09390 1.89163 -1.89163 1.09390 0.16516 -1.35456 1.97272 -1.75895 0.80339 0.49097 -1.57828 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E31 2.00000 -1.67433 0.80339 0.32919 -1.35456 1.93880 -1.89163 1.22843 -0.16516 -0.95189 1.75895 -1.99317 1.57828 -0.64940 -0.49097 1.47145 -1.97272 1.83155 -1.09390 0.00000 1.09390 -1.83155 1.97272 -1.47145 0.49097 0.64940 -1.57828 1.99317 -1.75895 0.95189 0.16516 -1.22843 1.89163 -1.93880 1.35456 -0.32919 -0.80339 1.67433 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E32 2.00000 -1.75895 1.09390 -0.16516 -0.80339 1.57828 -1.97272 1.89163 -1.35456 0.49097 0.49097 -1.35456 1.89163 -1.97272 1.57828 -0.80339 -0.16516 1.09390 -1.75895 2.00000 -1.75895 1.09390 -0.16516 -0.80339 1.57828 -1.97272 1.89163 -1.35456 0.49097 0.49097 -1.35456 1.89163 -1.97272 1.57828 -0.80339 -0.16516 1.09390 -1.75895 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E33 2.00000 -1.83155 1.35456 -0.64940 -0.16516 0.95189 -1.57828 1.93880 -1.97272 1.67433 -1.09390 0.32919 0.49097 -1.22843 1.75895 -1.99317 1.89163 -1.47145 0.80339 0.00000 -0.80339 1.47145 -1.89163 1.99317 -1.75895 1.22843 -0.49097 -0.32919 1.09390 -1.67433 1.97272 -1.93880 1.57828 -0.95189 0.16516 0.64940 -1.35456 1.83155 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ..TT......... E34 2.00000 -1.89163 1.57828 -1.09390 0.49097 0.16516 -0.80339 1.35456 -1.75895 1.97272 -1.97272 1.75895 -1.35456 0.80339 -0.16516 -0.49097 1.09390 -1.57828 1.89163 -2.00000 1.89163 -1.57828 1.09390 -0.49097 -0.16516 0.80339 -1.35456 1.75895 -1.97272 1.97272 -1.75895 1.35456 -0.80339 0.16516 0.49097 -1.09390 1.57828 -1.89163 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ..TT....... ............. E35 2.00000 -1.93880 1.75895 -1.47145 1.09390 -0.64940 0.16516 0.32919 -0.80339 1.22843 -1.57828 1.83155 -1.97272 1.99317 -1.89163 1.67433 -1.35456 0.95189 -0.49097 0.00000 0.49097 -0.95189 1.35456 -1.67433 1.89163 -1.99317 1.97272 -1.83155 1.57828 -1.22843 0.80339 -0.32919 -0.16516 0.64940 -1.09390 1.47145 -1.75895 1.93880 -2.00000 0.00000 0.00000 ... ... ..... ....... ..TT..... ........... ......TT..... E36 2.00000 -1.97272 1.89163 -1.75895 1.57828 -1.35456 1.09390 -0.80339 0.49097 -0.16516 -0.16516 0.49097 -0.80339 1.09390 -1.35456 1.57828 -1.75895 1.89163 -1.97272 2.00000 -1.97272 1.89163 -1.75895 1.57828 -1.35456 1.09390 -0.80339 0.49097 -0.16516 -0.16516 0.49097 -0.80339 1.09390 -1.35456 1.57828 -1.75895 1.89163 -1.97272 2.00000 0.00000 0.00000 ... ... ..... ..TT... ......... ......TT... ............. E37 2.00000 -1.99317 1.97272 -1.93880 1.89163 -1.83155 1.75895 -1.67433 1.57828 -1.47145 1.35456 -1.22843 1.09390 -0.95189 0.80339 -0.64940 0.49097 -0.32919 0.16516 0.00000 -0.16516 0.32919 -0.49097 0.64940 -0.80339 0.95189 -1.09390 1.22843 -1.35456 1.47145 -1.57828 1.67433 -1.75895 1.83155 -1.89163 1.93880 -1.97272 1.99317 -2.00000 0.00000 0.00000 TT. ... ..TT. ....... ......TT. ........... ..........TT. Irrational character values: 1.993168986013 = 2*cos(2*π/76) = 2*cos(π/38) 1.972722606805 = 2*cos(4*π/76) = 2*cos(π/19) 1.938800531879 = 2*cos(6*π/76) = 2*cos(3*π/38) 1.891634483401 = 2*cos(8*π/76) = 2*cos(2*π/19) 1.831546653310 = 2*cos(10*π/76) = 2*cos(5*π/38) 1.758947502413 = 2*cos(12*π/76) = 2*cos(3*π/19) 1.674332956525 = 2*cos(14*π/76) = 2*cos(7*π/38) 1.578281018793 = 2*cos(16*π/76) = 2*cos(4*π/19) 1.471447821346 = 2*cos(18*π/76) = 2*cos(9*π/38) 1.354563143251 = 2*cos(20*π/76) = 2*cos(5*π/19) 1.228425425379 = 2*cos(22*π/76) = 2*cos(11*π/38) 1.093896316245 = 2*cos(24*π/76) = 2*cos(6*π/19) 0.951894786074 = 2*cos(26*π/76) = 2*cos(13*π/38) 0.803390849306 = 2*cos(28*π/76) = 2*cos(7*π/19) 0.649398938409 = 2*cos(30*π/76) = 2*cos(15*π/38) 0.490970974282 = 2*cos(32*π/76) = 2*cos(8*π/19) 0.329189180561 = 2*cos(34*π/76) = 2*cos(17*π/38) 0.165158690945 = 2*cos(36*π/76) = 2*cos(9*π/19) 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 E29 m E30 l E31 k+m E32 j+l E33 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)} E34 h+j+l {z((x²−y²)²−4x²y²), xyz(x²−y²)} E35 g+i+k+m {xz(x²−3y²), yz(3x²−y²)}, {xz³(x²−3y²), yz³(3x²−y²)} E36 f+h+j+l {z(x²−y²), xyz}, {z³(x²−y²), xyz³} E37 R+d+g+i+k+m {R_x, R_y}, {xz, yz}, {xz³, yz³}, {xz⁵, yz⁵} Notes: α The order of the D_38d point group is 152, and the order of the principal axis (S₇₆) is 76. The group has 41 irreducible representations. β The D_38d point group is isomorphic to C_76v and D₇₆. γ The D_38d point group is generated by two symmetry elements, S₇₆ 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_38d is 4 (quadrupole moment). η This point group is non-Abelian (some symmetry operations are not commutative). Therefore, the character table contains multi-membered classes and degenerate irreducible representations. θ Some of the characters in the table are irrational because the order of the principal axis is neither 1,2,3,4 nor 6. These irrational values can be expressed as cosine values, or as solutions of algebraic equations with a leading coefficient of 1. All characters are algebraic integers of a degree much less than half the order of the principal axis. ι The point group corresponds to a polygon inconstructible by the classical means of ruler and compass. Yet it becomes constructible if angle trisection is allowed, e.g., with neusis construction or origami. This is because the order of the principal axis is given by a product of any number of different Pierpont primes (...,5,7,13,17,19,37,73,97,109,163,...) times arbitrary powers of two and three. All characters of this group can be expressed using complex numbers, elementary arithmetic operations, square roots and third roots.

This Character Table for the D_38d point group was created by Gernot Katzer.

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

	D_36d
	D_37d
C₃₈ C_38v C_38h D₃₈ D_38h	D_38d	S₃₈
	D_39d
	D_40d

Character table for the D38d point group

Character table for the D_38d point group