Character Table for Point Group D40h

Character table for the D_40h point group

D40h E 2 C40 2 C20 2 C40^3 2 C10 2 C8 2 C20^3 2 C40^7 2 C5 2 C40^9 2 C4 2 C40^11 2 C10^3 2 C40^13 2 C20^7 2 C8^3 2 C5^2 2 C40^17 2 C20^9 2 C40^19 C2 20 C2' 20 C2" i 2 S40 2 S20 2 S40^3 2 S10 2 S8 2 S20^3 2 S40^7 2 S5 2 S40^9 2 S4 2 S40^11 2 S10^3 2 S40^13 2 S20^7 2 S8^3 2 S5^2 2 S40^17 2 S20^9 2 S40^19 sh 20 sv 20 sd <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> A1g 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 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 A2g 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 -1.00000 -1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 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 ... ..... ....... ......... ........... ............. B1g 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 1.00000 -1.00000 ... ... ..... ....... ......... ........... ............. B2g 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 ... ... ..... ....... ......... ........... ............. E1g 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 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. ........... ..........TT. E2g 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 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... ........... ........TT... E3g 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 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..... ........... ......TT..... E4g 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 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 ... ... ..... ....... TT....... ........... ....TT....... E5g 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 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 ... ... ..... ....... ......... ........... ..TT......... E6g 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 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 ... ... ..... ....... ......... ........... TT........... E7g 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 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 ... ... ..... ....... ......... ........... ............. E8g 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 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 ... ... ..... ....... ......... ........... ............. E9g 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 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 ... ... ..... ....... ......... ........... ............. E10g 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 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 ... ... ..... ....... ......... ........... ............. E11g 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 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 ... ... ..... ....... ......... ........... ............. E12g 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 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 ... ... ..... ....... ......... ........... ............. E13g 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 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 ... ... ..... ....... ......... ........... ............. E14g 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 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 ... ... ..... ....... ......... ........... ............. E15g 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 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 ... ... ..... ....... ......... ........... ............. E16g 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 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 ... ... ..... ....... ......... ........... ............. E17g 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 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 ... ... ..... ....... ......... ........... ............. E18g 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 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 ... ... ..... ....... ......... ........... ............. E19g 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 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 ... ... ..... ....... ......... ........... ............. A1u 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 ... ... ..... ....... ......... ........... ............. A2u 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -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 ............. B1u 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 1.00000 -1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 -1.00000 1.00000 ... ... ..... ....... ......... ........... ............. B2u 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 ... ... ..... ....... ......... ........... ............. E1u 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 -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. ............. E2u 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 -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... ............. E3u 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 -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..... ......... ....TT..... ............. E4u 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 -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 ... ... ..... ....... ......... ..TT....... ............. E5u 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 -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 ... ... ..... ....... ......... TT......... ............. E6u 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 -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 ... ... ..... ....... ......... ........... ............. E7u 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 -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 ... ... ..... ....... ......... ........... ............. E8u 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 -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 ... ... ..... ....... ......... ........... ............. E9u 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 -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 ... ... ..... ....... ......... ........... ............. E10u 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 -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 ... ... ..... ....... ......... ........... ............. E11u 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 -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 ... ... ..... ....... ......... ........... ............. E12u 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 -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 ... ... ..... ....... ......... ........... ............. E13u 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 -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 ... ... ..... ....... ......... ........... ............. E14u 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 -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 ... ... ..... ....... ......... ........... ............. E15u 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 -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 ... ... ..... ....... ......... ........... ............. E16u 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 -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 ... ... ..... ....... ......... ........... ............. E17u 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 -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 ... ... ..... ....... ......... ........... ............. E18u 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 -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 ... ... ..... ....... ......... ........... ............. E19u 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 -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 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.975376681190 = 2*cos(2*π/40) = 2*cos(π/20) = (√2+√10+2*(√5−√5))/4 1.902113032590 = 2*cos(4*π/40) = 2*cos(π/10) = (√10+2*√5)/2 1.782013048377 = 2*cos(6*π/40) = 2*cos(3*π/20) = (√10−√2+2*√5+√5)/4 1.618033988750 = 2*cos(8*π/40) = 2*cos(π/5) = (√5+1)/2 1.414213562373 = 2*cos(10*π/40) = 2*cos(π/4) = √2 1.175570504585 = 2*cos(12*π/40) = 2*cos(3*π/10) = (√10−2*√5)/2 0.907980999479 = 2*cos(14*π/40) = 2*cos(7*π/20) = (√2−√10+2*√5+√5)/4 0.618033988750 = 2*cos(16*π/40) = 2*cos(2*π/5) = (√5−1)/2 0.312868930080 = 2*cos(18*π/40) = 2*cos(9*π/20) = (√2+√10−2*(√5−√5))/4 Symmetry of Rotations and Cartesian products A1g d+g+i+k+m z², z⁴, z⁶ A2g R R_z E1g R+d+g+i+k+m {R_x, R_y}, {xz, yz}, {xz³, yz³}, {xz⁵, yz⁵} E2g d+g+i+k+m {x²−y², xy}, {z²(x²−y²), xyz²}, {z⁴(x²−y²), xyz⁴} E3g g+i+k+m {xz(x²−3y²), yz(3x²−y²)}, {xz³(x²−3y²), yz³(3x²−y²)} E4g g+i+k+m {(x²−y²)²−4x²y², xy(x²−y²)}, {z²((x²−y²)²−4x²y²), xyz²(x²−y²)} E5g 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)} E6g i+k+m {x²(x²−3y²)²−y²(3x²−y²)², xy(x²−3y²)(3x²−y²)} E7g k+m E8g k+m E9g m E10g m A2u p+f+h+j+l z, z³, z⁵ E1u p+f+h+j+l {x, y}, {xz², yz²}, {xz⁴, yz⁴} E2u f+h+j+l {z(x²−y²), xyz}, {z³(x²−y²), xyz³} E3u f+h+j+l {x(x²−3y²), y(3x²−y²)}, {xz²(x²−3y²), yz²(3x²−y²)} E4u h+j+l {z((x²−y²)²−4x²y²), xyz(x²−y²)} E5u 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)} E6u j+l E7u j+l E8u l E9u l Notes: α The order of the D_40h point group is 160, and the order of the principal axis (C₄₀) is 40. The group has 46 irreducible representations. β The D_40h point group is generated by three symmetry elements that are canonically chosen C₄₀, C₂^′ and i. Other choices include σ_h instead of i, or any of C₂^″, σ_v or σ_d instead of C₂^′. Also, some ternary combinations of C₂^′, C₂^″, σ_v and σ_d act as generators. Lastly, the S₄₀ can be chosen, together with i or σ_h and any one of C₂^′, C₂^″, σ_v or σ_d. γ There are two different sets of twofold symmetry axes perpendicular to the principal axis (z axis in standard orientation). By convention, the set denoted as C₂^′ contains both the x and y axes. δ There are two different sets of symmetry planes containing the principal axis (z axis in standard orientation). By convention, the set denoted as σ_v contains both the xz and the yz planes. ε The lowest nonvanishing multipole moment in D_40h is 4 (quadrupole moment). ζ This point group is non-Abelian (some symmetry operations are not commutative). Therefore, the character table contains multi-membered classes and degenerate irreducible representations. η 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. θ 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. ι 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_40h point group was created by Gernot Katzer.

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

	D_38h
	D_39h
C₄₀ C_40v C_40h D₄₀	D_40h	D_40d S₄₀
	D_41h
	D_42h

Character table for the D40h point group

Character table for the D_40h point group