Character Table for Point Group D40d

Character table for the D_40d point group

D40d E 2 S80 2 C40 2 S80^3 2 C20 2 S16 2 C40^3 2 S80^7 2 C10 2 S80^9 2 C8 2 S80^11 2 C20^3 2 S80^13 2 C40^7 2 S16^3 2 C5 2 S80^17 2 C40^9 2 S80^19 2 C4 2 S80^21 2 C40^11 2 S80^23 2 C10^3 2 S16^5 2 C40^13 2 S80^27 2 C20^7 2 S80^29 2 C8^3 2 S80^31 2 C5^2 2 S80^33 2 C40^17 2 S16^7 2 C20^9 2 S80^37 2 C40^19 2 S80^39 C2 40 C2' 40 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 ... ... ....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 ..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 ... ... ..... ....... ......... ........... ............. 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 ... ..T ..... ......T ......... ..........T ............. E1 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. ......... ........TT. ............. E2 2.00000 1.97538 1.90211 1.78201 1.61803 1.41421 1.17557 0.90798 0.61803 0.31287 0.00000 -0.31287 -0.61803 -0.90798 -1.17557 -1.41421 -1.61803 -1.78201 -1.90211 -1.97538 -2.00000 -1.97538 -1.90211 -1.78201 -1.61803 -1.41421 -1.17557 -0.90798 -0.61803 -0.31287 0.00000 0.31287 0.61803 0.90798 1.17557 1.41421 1.61803 1.78201 1.90211 1.97538 2.00000 0.00000 0.00000 ... ... TT... ....... ....TT... ........... ........TT... E3 2.00000 1.94474 1.78201 1.52081 1.17557 0.76537 0.31287 -0.15692 -0.61803 -1.04500 -1.41421 -1.70528 -1.90211 -1.99383 -1.97538 -1.84776 -1.61803 -1.29890 -0.90798 -0.46689 0.00000 0.46689 0.90798 1.29890 1.61803 1.84776 1.97538 1.99383 1.90211 1.70528 1.41421 1.04500 0.61803 0.15692 -0.31287 -0.76537 -1.17557 -1.52081 -1.78201 -1.94474 -2.00000 0.00000 0.00000 ... ... ..... TT..... ......... ....TT..... ............. E4 2.00000 1.90211 1.61803 1.17557 0.61803 0.00000 -0.61803 -1.17557 -1.61803 -1.90211 -2.00000 -1.90211 -1.61803 -1.17557 -0.61803 0.00000 0.61803 1.17557 1.61803 1.90211 2.00000 1.90211 1.61803 1.17557 0.61803 0.00000 -0.61803 -1.17557 -1.61803 -1.90211 -2.00000 -1.90211 -1.61803 -1.17557 -0.61803 0.00000 0.61803 1.17557 1.61803 1.90211 2.00000 0.00000 0.00000 ... ... ..... ....... TT....... ........... ....TT....... E5 2.00000 1.84776 1.41421 0.76537 0.00000 -0.76537 -1.41421 -1.84776 -2.00000 -1.84776 -1.41421 -0.76537 0.00000 0.76537 1.41421 1.84776 2.00000 1.84776 1.41421 0.76537 0.00000 -0.76537 -1.41421 -1.84776 -2.00000 -1.84776 -1.41421 -0.76537 0.00000 0.76537 1.41421 1.84776 2.00000 1.84776 1.41421 0.76537 0.00000 -0.76537 -1.41421 -1.84776 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... TT......... ............. E6 2.00000 1.78201 1.17557 0.31287 -0.61803 -1.41421 -1.90211 -1.97538 -1.61803 -0.90798 0.00000 0.90798 1.61803 1.97538 1.90211 1.41421 0.61803 -0.31287 -1.17557 -1.78201 -2.00000 -1.78201 -1.17557 -0.31287 0.61803 1.41421 1.90211 1.97538 1.61803 0.90798 0.00000 -0.90798 -1.61803 -1.97538 -1.90211 -1.41421 -0.61803 0.31287 1.17557 1.78201 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... TT........... E7 2.00000 1.70528 0.90798 -0.15692 -1.17557 -1.84776 -1.97538 -1.52081 -0.61803 0.46689 1.41421 1.94474 1.90211 1.29890 0.31287 -0.76537 -1.61803 -1.99383 -1.78201 -1.04500 0.00000 1.04500 1.78201 1.99383 1.61803 0.76537 -0.31287 -1.29890 -1.90211 -1.94474 -1.41421 -0.46689 0.61803 1.52081 1.97538 1.84776 1.17557 0.15692 -0.90798 -1.70528 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E8 2.00000 1.61803 0.61803 -0.61803 -1.61803 -2.00000 -1.61803 -0.61803 0.61803 1.61803 2.00000 1.61803 0.61803 -0.61803 -1.61803 -2.00000 -1.61803 -0.61803 0.61803 1.61803 2.00000 1.61803 0.61803 -0.61803 -1.61803 -2.00000 -1.61803 -0.61803 0.61803 1.61803 2.00000 1.61803 0.61803 -0.61803 -1.61803 -2.00000 -1.61803 -0.61803 0.61803 1.61803 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E9 2.00000 1.52081 0.31287 -1.04500 -1.90211 -1.84776 -0.90798 0.46689 1.61803 1.99383 1.41421 0.15692 -1.17557 -1.94474 -1.78201 -0.76537 0.61803 1.70528 1.97538 1.29890 0.00000 -1.29890 -1.97538 -1.70528 -0.61803 0.76537 1.78201 1.94474 1.17557 -0.15692 -1.41421 -1.99383 -1.61803 -0.46689 0.90798 1.84776 1.90211 1.04500 -0.31287 -1.52081 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E10 2.00000 1.41421 0.00000 -1.41421 -2.00000 -1.41421 0.00000 1.41421 2.00000 1.41421 0.00000 -1.41421 -2.00000 -1.41421 0.00000 1.41421 2.00000 1.41421 0.00000 -1.41421 -2.00000 -1.41421 0.00000 1.41421 2.00000 1.41421 0.00000 -1.41421 -2.00000 -1.41421 0.00000 1.41421 2.00000 1.41421 0.00000 -1.41421 -2.00000 -1.41421 0.00000 1.41421 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E11 2.00000 1.29890 -0.31287 -1.70528 -1.90211 -0.76537 0.90798 1.94474 1.61803 0.15692 -1.41421 -1.99383 -1.17557 0.46689 1.78201 1.84776 0.61803 -1.04500 -1.97538 -1.52081 0.00000 1.52081 1.97538 1.04500 -0.61803 -1.84776 -1.78201 -0.46689 1.17557 1.99383 1.41421 -0.15692 -1.61803 -1.94474 -0.90798 0.76537 1.90211 1.70528 0.31287 -1.29890 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E12 2.00000 1.17557 -0.61803 -1.90211 -1.61803 0.00000 1.61803 1.90211 0.61803 -1.17557 -2.00000 -1.17557 0.61803 1.90211 1.61803 0.00000 -1.61803 -1.90211 -0.61803 1.17557 2.00000 1.17557 -0.61803 -1.90211 -1.61803 0.00000 1.61803 1.90211 0.61803 -1.17557 -2.00000 -1.17557 0.61803 1.90211 1.61803 0.00000 -1.61803 -1.90211 -0.61803 1.17557 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E13 2.00000 1.04500 -0.90798 -1.99383 -1.17557 0.76537 1.97538 1.29890 -0.61803 -1.94474 -1.41421 0.46689 1.90211 1.52081 -0.31287 -1.84776 -1.61803 0.15692 1.78201 1.70528 0.00000 -1.70528 -1.78201 -0.15692 1.61803 1.84776 0.31287 -1.52081 -1.90211 -0.46689 1.41421 1.94474 0.61803 -1.29890 -1.97538 -0.76537 1.17557 1.99383 0.90798 -1.04500 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E14 2.00000 0.90798 -1.17557 -1.97538 -0.61803 1.41421 1.90211 0.31287 -1.61803 -1.78201 0.00000 1.78201 1.61803 -0.31287 -1.90211 -1.41421 0.61803 1.97538 1.17557 -0.90798 -2.00000 -0.90798 1.17557 1.97538 0.61803 -1.41421 -1.90211 -0.31287 1.61803 1.78201 0.00000 -1.78201 -1.61803 0.31287 1.90211 1.41421 -0.61803 -1.97538 -1.17557 0.90798 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E15 2.00000 0.76537 -1.41421 -1.84776 0.00000 1.84776 1.41421 -0.76537 -2.00000 -0.76537 1.41421 1.84776 0.00000 -1.84776 -1.41421 0.76537 2.00000 0.76537 -1.41421 -1.84776 0.00000 1.84776 1.41421 -0.76537 -2.00000 -0.76537 1.41421 1.84776 0.00000 -1.84776 -1.41421 0.76537 2.00000 0.76537 -1.41421 -1.84776 0.00000 1.84776 1.41421 -0.76537 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E16 2.00000 0.61803 -1.61803 -1.61803 0.61803 2.00000 0.61803 -1.61803 -1.61803 0.61803 2.00000 0.61803 -1.61803 -1.61803 0.61803 2.00000 0.61803 -1.61803 -1.61803 0.61803 2.00000 0.61803 -1.61803 -1.61803 0.61803 2.00000 0.61803 -1.61803 -1.61803 0.61803 2.00000 0.61803 -1.61803 -1.61803 0.61803 2.00000 0.61803 -1.61803 -1.61803 0.61803 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E17 2.00000 0.46689 -1.78201 -1.29890 1.17557 1.84776 -0.31287 -1.99383 -0.61803 1.70528 1.41421 -1.04500 -1.90211 0.15692 1.97538 0.76537 -1.61803 -1.52081 0.90798 1.94474 0.00000 -1.94474 -0.90798 1.52081 1.61803 -0.76537 -1.97538 -0.15692 1.90211 1.04500 -1.41421 -1.70528 0.61803 1.99383 0.31287 -1.84776 -1.17557 1.29890 1.78201 -0.46689 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E18 2.00000 0.31287 -1.90211 -0.90798 1.61803 1.41421 -1.17557 -1.78201 0.61803 1.97538 0.00000 -1.97538 -0.61803 1.78201 1.17557 -1.41421 -1.61803 0.90798 1.90211 -0.31287 -2.00000 -0.31287 1.90211 0.90798 -1.61803 -1.41421 1.17557 1.78201 -0.61803 -1.97538 0.00000 1.97538 0.61803 -1.78201 -1.17557 1.41421 1.61803 -0.90798 -1.90211 0.31287 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E19 2.00000 0.15692 -1.97538 -0.46689 1.90211 0.76537 -1.78201 -1.04500 1.61803 1.29890 -1.41421 -1.52081 1.17557 1.70528 -0.90798 -1.84776 0.61803 1.94474 -0.31287 -1.99383 0.00000 1.99383 0.31287 -1.94474 -0.61803 1.84776 0.90798 -1.70528 -1.17557 1.52081 1.41421 -1.29890 -1.61803 1.04500 1.78201 -0.76537 -1.90211 0.46689 1.97538 -0.15692 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E20 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E21 2.00000 -0.15692 -1.97538 0.46689 1.90211 -0.76537 -1.78201 1.04500 1.61803 -1.29890 -1.41421 1.52081 1.17557 -1.70528 -0.90798 1.84776 0.61803 -1.94474 -0.31287 1.99383 0.00000 -1.99383 0.31287 1.94474 -0.61803 -1.84776 0.90798 1.70528 -1.17557 -1.52081 1.41421 1.29890 -1.61803 -1.04500 1.78201 0.76537 -1.90211 -0.46689 1.97538 0.15692 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E22 2.00000 -0.31287 -1.90211 0.90798 1.61803 -1.41421 -1.17557 1.78201 0.61803 -1.97538 0.00000 1.97538 -0.61803 -1.78201 1.17557 1.41421 -1.61803 -0.90798 1.90211 0.31287 -2.00000 0.31287 1.90211 -0.90798 -1.61803 1.41421 1.17557 -1.78201 -0.61803 1.97538 0.00000 -1.97538 0.61803 1.78201 -1.17557 -1.41421 1.61803 0.90798 -1.90211 -0.31287 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E23 2.00000 -0.46689 -1.78201 1.29890 1.17557 -1.84776 -0.31287 1.99383 -0.61803 -1.70528 1.41421 1.04500 -1.90211 -0.15692 1.97538 -0.76537 -1.61803 1.52081 0.90798 -1.94474 0.00000 1.94474 -0.90798 -1.52081 1.61803 0.76537 -1.97538 0.15692 1.90211 -1.04500 -1.41421 1.70528 0.61803 -1.99383 0.31287 1.84776 -1.17557 -1.29890 1.78201 0.46689 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E24 2.00000 -0.61803 -1.61803 1.61803 0.61803 -2.00000 0.61803 1.61803 -1.61803 -0.61803 2.00000 -0.61803 -1.61803 1.61803 0.61803 -2.00000 0.61803 1.61803 -1.61803 -0.61803 2.00000 -0.61803 -1.61803 1.61803 0.61803 -2.00000 0.61803 1.61803 -1.61803 -0.61803 2.00000 -0.61803 -1.61803 1.61803 0.61803 -2.00000 0.61803 1.61803 -1.61803 -0.61803 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E25 2.00000 -0.76537 -1.41421 1.84776 0.00000 -1.84776 1.41421 0.76537 -2.00000 0.76537 1.41421 -1.84776 0.00000 1.84776 -1.41421 -0.76537 2.00000 -0.76537 -1.41421 1.84776 0.00000 -1.84776 1.41421 0.76537 -2.00000 0.76537 1.41421 -1.84776 0.00000 1.84776 -1.41421 -0.76537 2.00000 -0.76537 -1.41421 1.84776 0.00000 -1.84776 1.41421 0.76537 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E26 2.00000 -0.90798 -1.17557 1.97538 -0.61803 -1.41421 1.90211 -0.31287 -1.61803 1.78201 0.00000 -1.78201 1.61803 0.31287 -1.90211 1.41421 0.61803 -1.97538 1.17557 0.90798 -2.00000 0.90798 1.17557 -1.97538 0.61803 1.41421 -1.90211 0.31287 1.61803 -1.78201 0.00000 1.78201 -1.61803 -0.31287 1.90211 -1.41421 -0.61803 1.97538 -1.17557 -0.90798 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E27 2.00000 -1.04500 -0.90798 1.99383 -1.17557 -0.76537 1.97538 -1.29890 -0.61803 1.94474 -1.41421 -0.46689 1.90211 -1.52081 -0.31287 1.84776 -1.61803 -0.15692 1.78201 -1.70528 0.00000 1.70528 -1.78201 0.15692 1.61803 -1.84776 0.31287 1.52081 -1.90211 0.46689 1.41421 -1.94474 0.61803 1.29890 -1.97538 0.76537 1.17557 -1.99383 0.90798 1.04500 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E28 2.00000 -1.17557 -0.61803 1.90211 -1.61803 0.00000 1.61803 -1.90211 0.61803 1.17557 -2.00000 1.17557 0.61803 -1.90211 1.61803 0.00000 -1.61803 1.90211 -0.61803 -1.17557 2.00000 -1.17557 -0.61803 1.90211 -1.61803 0.00000 1.61803 -1.90211 0.61803 1.17557 -2.00000 1.17557 0.61803 -1.90211 1.61803 0.00000 -1.61803 1.90211 -0.61803 -1.17557 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E29 2.00000 -1.29890 -0.31287 1.70528 -1.90211 0.76537 0.90798 -1.94474 1.61803 -0.15692 -1.41421 1.99383 -1.17557 -0.46689 1.78201 -1.84776 0.61803 1.04500 -1.97538 1.52081 0.00000 -1.52081 1.97538 -1.04500 -0.61803 1.84776 -1.78201 0.46689 1.17557 -1.99383 1.41421 0.15692 -1.61803 1.94474 -0.90798 -0.76537 1.90211 -1.70528 0.31287 1.29890 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E30 2.00000 -1.41421 0.00000 1.41421 -2.00000 1.41421 0.00000 -1.41421 2.00000 -1.41421 0.00000 1.41421 -2.00000 1.41421 0.00000 -1.41421 2.00000 -1.41421 0.00000 1.41421 -2.00000 1.41421 0.00000 -1.41421 2.00000 -1.41421 0.00000 1.41421 -2.00000 1.41421 0.00000 -1.41421 2.00000 -1.41421 0.00000 1.41421 -2.00000 1.41421 0.00000 -1.41421 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E31 2.00000 -1.52081 0.31287 1.04500 -1.90211 1.84776 -0.90798 -0.46689 1.61803 -1.99383 1.41421 -0.15692 -1.17557 1.94474 -1.78201 0.76537 0.61803 -1.70528 1.97538 -1.29890 0.00000 1.29890 -1.97538 1.70528 -0.61803 -0.76537 1.78201 -1.94474 1.17557 0.15692 -1.41421 1.99383 -1.61803 0.46689 0.90798 -1.84776 1.90211 -1.04500 -0.31287 1.52081 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E32 2.00000 -1.61803 0.61803 0.61803 -1.61803 2.00000 -1.61803 0.61803 0.61803 -1.61803 2.00000 -1.61803 0.61803 0.61803 -1.61803 2.00000 -1.61803 0.61803 0.61803 -1.61803 2.00000 -1.61803 0.61803 0.61803 -1.61803 2.00000 -1.61803 0.61803 0.61803 -1.61803 2.00000 -1.61803 0.61803 0.61803 -1.61803 2.00000 -1.61803 0.61803 0.61803 -1.61803 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E33 2.00000 -1.70528 0.90798 0.15692 -1.17557 1.84776 -1.97538 1.52081 -0.61803 -0.46689 1.41421 -1.94474 1.90211 -1.29890 0.31287 0.76537 -1.61803 1.99383 -1.78201 1.04500 0.00000 -1.04500 1.78201 -1.99383 1.61803 -0.76537 -0.31287 1.29890 -1.90211 1.94474 -1.41421 0.46689 0.61803 -1.52081 1.97538 -1.84776 1.17557 -0.15692 -0.90798 1.70528 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E34 2.00000 -1.78201 1.17557 -0.31287 -0.61803 1.41421 -1.90211 1.97538 -1.61803 0.90798 0.00000 -0.90798 1.61803 -1.97538 1.90211 -1.41421 0.61803 0.31287 -1.17557 1.78201 -2.00000 1.78201 -1.17557 0.31287 0.61803 -1.41421 1.90211 -1.97538 1.61803 -0.90798 0.00000 0.90798 -1.61803 1.97538 -1.90211 1.41421 -0.61803 -0.31287 1.17557 -1.78201 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E35 2.00000 -1.84776 1.41421 -0.76537 0.00000 0.76537 -1.41421 1.84776 -2.00000 1.84776 -1.41421 0.76537 0.00000 -0.76537 1.41421 -1.84776 2.00000 -1.84776 1.41421 -0.76537 0.00000 0.76537 -1.41421 1.84776 -2.00000 1.84776 -1.41421 0.76537 0.00000 -0.76537 1.41421 -1.84776 2.00000 -1.84776 1.41421 -0.76537 0.00000 0.76537 -1.41421 1.84776 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ..TT......... E36 2.00000 -1.90211 1.61803 -1.17557 0.61803 0.00000 -0.61803 1.17557 -1.61803 1.90211 -2.00000 1.90211 -1.61803 1.17557 -0.61803 0.00000 0.61803 -1.17557 1.61803 -1.90211 2.00000 -1.90211 1.61803 -1.17557 0.61803 0.00000 -0.61803 1.17557 -1.61803 1.90211 -2.00000 1.90211 -1.61803 1.17557 -0.61803 0.00000 0.61803 -1.17557 1.61803 -1.90211 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ..TT....... ............. E37 2.00000 -1.94474 1.78201 -1.52081 1.17557 -0.76537 0.31287 0.15692 -0.61803 1.04500 -1.41421 1.70528 -1.90211 1.99383 -1.97538 1.84776 -1.61803 1.29890 -0.90798 0.46689 0.00000 -0.46689 0.90798 -1.29890 1.61803 -1.84776 1.97538 -1.99383 1.90211 -1.70528 1.41421 -1.04500 0.61803 -0.15692 -0.31287 0.76537 -1.17557 1.52081 -1.78201 1.94474 -2.00000 0.00000 0.00000 ... ... ..... ....... ..TT..... ........... ......TT..... E38 2.00000 -1.97538 1.90211 -1.78201 1.61803 -1.41421 1.17557 -0.90798 0.61803 -0.31287 0.00000 0.31287 -0.61803 0.90798 -1.17557 1.41421 -1.61803 1.78201 -1.90211 1.97538 -2.00000 1.97538 -1.90211 1.78201 -1.61803 1.41421 -1.17557 0.90798 -0.61803 0.31287 0.00000 -0.31287 0.61803 -0.90798 1.17557 -1.41421 1.61803 -1.78201 1.90211 -1.97538 2.00000 0.00000 0.00000 ... ... ..... ..TT... ......... ......TT... ............. E39 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. ....... ......TT. ........... ..........TT. Irrational character values: 1.993834667466 = 2*cos(2*π/80) = 2*cos(π/40) = √8+√2+√10+2*√5−√5/2 1.975376681190 = 2*cos(4*π/80) = 2*cos(π/20) = (√2+√10+2*(√5−√5))/4 1.944739840795 = 2*cos(6*π/80) = 2*cos(3*π/40) = √8−√2+√10+2*√5+√5/2 1.902113032590 = 2*cos(8*π/80) = 2*cos(π/10) = (√10+2*√5)/2 1.847759065023 = 2*cos(10*π/80) = 2*cos(π/8) = √2+√2 1.782013048377 = 2*cos(12*π/80) = 2*cos(3*π/20) = (√10−√2+2*√5+√5)/4 1.705280328708 = 2*cos(14*π/80) = 2*cos(7*π/40) = √8+√2−√10+2*√5+√5/2 1.618033988750 = 2*cos(16*π/80) = 2*cos(π/5) = (√5+1)/2 1.520811931200 = 2*cos(18*π/80) = 2*cos(9*π/40) = √8+√2+√10−2*√5−√5/2 1.414213562373 = 2*cos(20*π/80) = 2*cos(π/4) = √2 1.298896096660 = 2*cos(22*π/80) = 2*cos(11*π/40) = √8−√2−√10+2*√5−√5/2 1.175570504585 = 2*cos(24*π/80) = 2*cos(3*π/10) = (√10−2*√5)/2 1.044997129432 = 2*cos(26*π/80) = 2*cos(13*π/40) = √8−√2+√10−2*√5+√5/2 0.907980999479 = 2*cos(28*π/80) = 2*cos(7*π/20) = (√2−√10+2*√5+√5)/4 0.765366864730 = 2*cos(30*π/80) = 2*cos(3*π/8) = √2−√2 0.618033988750 = 2*cos(32*π/80) = 2*cos(2*π/5) = (√5−1)/2 0.466890727712 = 2*cos(34*π/80) = 2*cos(17*π/40) = √8+√2−√10−2*√5+√5/2 0.312868930080 = 2*cos(36*π/80) = 2*cos(9*π/20) = (√2+√10−2*(√5−√5))/4 0.156918191456 = 2*cos(38*π/80) = 2*cos(19*π/40) = √8−√2−√10−2*√5−√5/2 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 E31 m E32 l E33 k+m E34 j+l E35 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)} E36 h+j+l {z((x²−y²)²−4x²y²), xyz(x²−y²)} E37 g+i+k+m {xz(x²−3y²), yz(3x²−y²)}, {xz³(x²−3y²), yz³(3x²−y²)} E38 f+h+j+l {z(x²−y²), xyz}, {z³(x²−y²), xyz³} E39 R+d+g+i+k+m {R_x, R_y}, {xz, yz}, {xz³, yz³}, {xz⁵, yz⁵} Notes: α The order of the D_40d point group is 160, and the order of the principal axis (S₈₀) is 80. The group has 43 irreducible representations. β The D_40d point group is isomorphic to C_80v and D₈₀. γ The D_40d point group is generated by two symmetry elements, S₈₀ 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_40d 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. 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_40d point group was created by Gernot Katzer.

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

	D_38d
	D_39d
C₄₀ C_40v C_40h D₄₀ D_40h	D_40d	S₄₀
	D_41d
	D_42d

Character table for the D40d point group

Character table for the D_40d point group