Character Table for Point Group D116d

Character table for the D_116d point group

D116d E 2 S232 2 C116 2 S232^3 2 C58 2 S232^5 2 C116^3 2 S232^7 2 C29 2 S232^9 2 C116^5 2 S232^11 2 C58^3 2 S232^13 2 C116^7 2 S232^15 2 C29^2 2 S232^17 2 C116^9 2 S232^19 2 C58^5 2 S232^21 2 C116^11 2 S232^23 2 C29^3 2 S232^25 2 C116^13 2 S232^27 2 C58^7 2 S8 2 C116^15 2 S232^31 2 C29^4 2 S232^33 2 C116^17 2 S232^35 2 C58^9 2 S232^37 2 C116^19 2 S232^39 2 C29^5 2 S232^41 2 C116^21 2 S232^43 2 C58^11 2 S232^45 2 C116^23 2 S232^47 2 C29^6 2 S232^49 2 C116^25 2 S232^51 2 C58^13 2 S232^53 2 C116^27 2 S232^55 2 C29^7 2 S232^57 2 C4 2 S232^59 2 C58^15 2 S232^61 2 C116^31 2 S232^63 2 C29^8 2 S232^65 2 C116^33 2 S232^67 2 C58^17 2 S232^69 2 C116^35 2 S232^71 2 C29^9 2 S232^73 2 C116^37 2 S232^75 2 C58^19 2 S232^77 2 C116^39 2 S232^79 2 C29^10 2 S232^81 2 C116^41 2 S232^83 2 C58^21 2 S232^85 2 C116^43 2 S8^3 2 C29^11 2 S232^89 2 C116^45 2 S232^91 2 C58^23 2 S232^93 2 C116^47 2 S232^95 2 C29^12 2 S232^97 2 C116^49 2 S232^99 2 C58^25 2 S232^101 2 C116^51 2 S232^103 2 C29^13 2 S232^105 2 C116^53 2 S232^107 2 C58^27 2 S232^109 2 C116^55 2 S232^111 2 C29^14 2 S232^113 2 C116^57 2 S232^115 C2 116 C2' 116 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 ... ... ....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 ..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 ... ... ..... ....... ......... ........... ............. 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 ... ..T ..... ......T ......... ..........T ............. E1 2.00000 1.99927 1.99707 1.99340 1.98828 1.98169 1.97365 1.96417 1.95324 1.94088 1.92710 1.91190 1.89531 1.87732 1.85795 1.83723 1.81515 1.79174 1.76702 1.74101 1.71371 1.68516 1.65538 1.62438 1.59219 1.55883 1.52432 1.48870 1.45199 1.41421 1.37540 1.33558 1.29477 1.25302 1.21035 1.16679 1.12237 1.07714 1.03111 0.98432 0.93682 0.88862 0.83978 0.79032 0.74028 0.68969 0.63860 0.58705 0.53506 0.48268 0.42994 0.37689 0.32356 0.27000 0.21624 0.16232 0.10828 0.05416 0.00000 -0.05416 -0.10828 -0.16232 -0.21624 -0.27000 -0.32356 -0.37689 -0.42994 -0.48268 -0.53506 -0.58705 -0.63860 -0.68969 -0.74028 -0.79032 -0.83978 -0.88862 -0.93682 -0.98432 -1.03111 -1.07714 -1.12237 -1.16679 -1.21035 -1.25302 -1.29477 -1.33558 -1.37540 -1.41421 -1.45199 -1.48870 -1.52432 -1.55883 -1.59219 -1.62438 -1.65538 -1.68516 -1.71371 -1.74101 -1.76702 -1.79174 -1.81515 -1.83723 -1.85795 -1.87732 -1.89531 -1.91190 -1.92710 -1.94088 -1.95324 -1.96417 -1.97365 -1.98169 -1.98828 -1.99340 -1.99707 -1.99927 -2.00000 0.00000 0.00000 ... TT. ..... ....TT. ......... ........TT. ............. E2 2.00000 1.99707 1.98828 1.97365 1.95324 1.92710 1.89531 1.85795 1.81515 1.76702 1.71371 1.65538 1.59219 1.52432 1.45199 1.37540 1.29477 1.21035 1.12237 1.03111 0.93682 0.83978 0.74028 0.63860 0.53506 0.42994 0.32356 0.21624 0.10828 0.00000 -0.10828 -0.21624 -0.32356 -0.42994 -0.53506 -0.63860 -0.74028 -0.83978 -0.93682 -1.03111 -1.12237 -1.21035 -1.29477 -1.37540 -1.45199 -1.52432 -1.59219 -1.65538 -1.71371 -1.76702 -1.81515 -1.85795 -1.89531 -1.92710 -1.95324 -1.97365 -1.98828 -1.99707 -2.00000 -1.99707 -1.98828 -1.97365 -1.95324 -1.92710 -1.89531 -1.85795 -1.81515 -1.76702 -1.71371 -1.65538 -1.59219 -1.52432 -1.45199 -1.37540 -1.29477 -1.21035 -1.12237 -1.03111 -0.93682 -0.83978 -0.74028 -0.63860 -0.53506 -0.42994 -0.32356 -0.21624 -0.10828 0.00000 0.10828 0.21624 0.32356 0.42994 0.53506 0.63860 0.74028 0.83978 0.93682 1.03111 1.12237 1.21035 1.29477 1.37540 1.45199 1.52432 1.59219 1.65538 1.71371 1.76702 1.81515 1.85795 1.89531 1.92710 1.95324 1.97365 1.98828 1.99707 2.00000 0.00000 0.00000 ... ... TT... ....... ....TT... ........... ........TT... E3 2.00000 1.99340 1.97365 1.94088 1.89531 1.83723 1.76702 1.68516 1.59219 1.48870 1.37540 1.25302 1.12237 0.98432 0.83978 0.68969 0.53506 0.37689 0.21624 0.05416 -0.10828 -0.27000 -0.42994 -0.58705 -0.74028 -0.88862 -1.03111 -1.16679 -1.29477 -1.41421 -1.52432 -1.62438 -1.71371 -1.79174 -1.85795 -1.91190 -1.95324 -1.98169 -1.99707 -1.99927 -1.98828 -1.96417 -1.92710 -1.87732 -1.81515 -1.74101 -1.65538 -1.55883 -1.45199 -1.33558 -1.21035 -1.07714 -0.93682 -0.79032 -0.63860 -0.48268 -0.32356 -0.16232 0.00000 0.16232 0.32356 0.48268 0.63860 0.79032 0.93682 1.07714 1.21035 1.33558 1.45199 1.55883 1.65538 1.74101 1.81515 1.87732 1.92710 1.96417 1.98828 1.99927 1.99707 1.98169 1.95324 1.91190 1.85795 1.79174 1.71371 1.62438 1.52432 1.41421 1.29477 1.16679 1.03111 0.88862 0.74028 0.58705 0.42994 0.27000 0.10828 -0.05416 -0.21624 -0.37689 -0.53506 -0.68969 -0.83978 -0.98432 -1.12237 -1.25302 -1.37540 -1.48870 -1.59219 -1.68516 -1.76702 -1.83723 -1.89531 -1.94088 -1.97365 -1.99340 -2.00000 0.00000 0.00000 ... ... ..... TT..... ......... ....TT..... ............. E4 2.00000 1.98828 1.95324 1.89531 1.81515 1.71371 1.59219 1.45199 1.29477 1.12237 0.93682 0.74028 0.53506 0.32356 0.10828 -0.10828 -0.32356 -0.53506 -0.74028 -0.93682 -1.12237 -1.29477 -1.45199 -1.59219 -1.71371 -1.81515 -1.89531 -1.95324 -1.98828 -2.00000 -1.98828 -1.95324 -1.89531 -1.81515 -1.71371 -1.59219 -1.45199 -1.29477 -1.12237 -0.93682 -0.74028 -0.53506 -0.32356 -0.10828 0.10828 0.32356 0.53506 0.74028 0.93682 1.12237 1.29477 1.45199 1.59219 1.71371 1.81515 1.89531 1.95324 1.98828 2.00000 1.98828 1.95324 1.89531 1.81515 1.71371 1.59219 1.45199 1.29477 1.12237 0.93682 0.74028 0.53506 0.32356 0.10828 -0.10828 -0.32356 -0.53506 -0.74028 -0.93682 -1.12237 -1.29477 -1.45199 -1.59219 -1.71371 -1.81515 -1.89531 -1.95324 -1.98828 -2.00000 -1.98828 -1.95324 -1.89531 -1.81515 -1.71371 -1.59219 -1.45199 -1.29477 -1.12237 -0.93682 -0.74028 -0.53506 -0.32356 -0.10828 0.10828 0.32356 0.53506 0.74028 0.93682 1.12237 1.29477 1.45199 1.59219 1.71371 1.81515 1.89531 1.95324 1.98828 2.00000 0.00000 0.00000 ... ... ..... ....... TT....... ........... ....TT....... E5 2.00000 1.98169 1.92710 1.83723 1.71371 1.55883 1.37540 1.16679 0.93682 0.68969 0.42994 0.16232 -0.10828 -0.37689 -0.63860 -0.88862 -1.12237 -1.33558 -1.52432 -1.68516 -1.81515 -1.91190 -1.97365 -1.99927 -1.98828 -1.94088 -1.85795 -1.74101 -1.59219 -1.41421 -1.21035 -0.98432 -0.74028 -0.48268 -0.21624 0.05416 0.32356 0.58705 0.83978 1.07714 1.29477 1.48870 1.65538 1.79174 1.89531 1.96417 1.99707 1.99340 1.95324 1.87732 1.76702 1.62438 1.45199 1.25302 1.03111 0.79032 0.53506 0.27000 0.00000 -0.27000 -0.53506 -0.79032 -1.03111 -1.25302 -1.45199 -1.62438 -1.76702 -1.87732 -1.95324 -1.99340 -1.99707 -1.96417 -1.89531 -1.79174 -1.65538 -1.48870 -1.29477 -1.07714 -0.83978 -0.58705 -0.32356 -0.05416 0.21624 0.48268 0.74028 0.98432 1.21035 1.41421 1.59219 1.74101 1.85795 1.94088 1.98828 1.99927 1.97365 1.91190 1.81515 1.68516 1.52432 1.33558 1.12237 0.88862 0.63860 0.37689 0.10828 -0.16232 -0.42994 -0.68969 -0.93682 -1.16679 -1.37540 -1.55883 -1.71371 -1.83723 -1.92710 -1.98169 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... TT......... ............. E6 2.00000 1.97365 1.89531 1.76702 1.59219 1.37540 1.12237 0.83978 0.53506 0.21624 -0.10828 -0.42994 -0.74028 -1.03111 -1.29477 -1.52432 -1.71371 -1.85795 -1.95324 -1.99707 -1.98828 -1.92710 -1.81515 -1.65538 -1.45199 -1.21035 -0.93682 -0.63860 -0.32356 0.00000 0.32356 0.63860 0.93682 1.21035 1.45199 1.65538 1.81515 1.92710 1.98828 1.99707 1.95324 1.85795 1.71371 1.52432 1.29477 1.03111 0.74028 0.42994 0.10828 -0.21624 -0.53506 -0.83978 -1.12237 -1.37540 -1.59219 -1.76702 -1.89531 -1.97365 -2.00000 -1.97365 -1.89531 -1.76702 -1.59219 -1.37540 -1.12237 -0.83978 -0.53506 -0.21624 0.10828 0.42994 0.74028 1.03111 1.29477 1.52432 1.71371 1.85795 1.95324 1.99707 1.98828 1.92710 1.81515 1.65538 1.45199 1.21035 0.93682 0.63860 0.32356 0.00000 -0.32356 -0.63860 -0.93682 -1.21035 -1.45199 -1.65538 -1.81515 -1.92710 -1.98828 -1.99707 -1.95324 -1.85795 -1.71371 -1.52432 -1.29477 -1.03111 -0.74028 -0.42994 -0.10828 0.21624 0.53506 0.83978 1.12237 1.37540 1.59219 1.76702 1.89531 1.97365 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... TT........... E7 2.00000 1.96417 1.85795 1.68516 1.45199 1.16679 0.83978 0.48268 0.10828 -0.27000 -0.63860 -0.98432 -1.29477 -1.55883 -1.76702 -1.91190 -1.98828 -1.99340 -1.92710 -1.79174 -1.59219 -1.33558 -1.03111 -0.68969 -0.32356 0.05416 0.42994 0.79032 1.12237 1.41421 1.65538 1.83723 1.95324 1.99927 1.97365 1.87732 1.71371 1.48870 1.21035 0.88862 0.53506 0.16232 -0.21624 -0.58705 -0.93682 -1.25302 -1.52432 -1.74101 -1.89531 -1.98169 -1.99707 -1.94088 -1.81515 -1.62438 -1.37540 -1.07714 -0.74028 -0.37689 0.00000 0.37689 0.74028 1.07714 1.37540 1.62438 1.81515 1.94088 1.99707 1.98169 1.89531 1.74101 1.52432 1.25302 0.93682 0.58705 0.21624 -0.16232 -0.53506 -0.88862 -1.21035 -1.48870 -1.71371 -1.87732 -1.97365 -1.99927 -1.95324 -1.83723 -1.65538 -1.41421 -1.12237 -0.79032 -0.42994 -0.05416 0.32356 0.68969 1.03111 1.33558 1.59219 1.79174 1.92710 1.99340 1.98828 1.91190 1.76702 1.55883 1.29477 0.98432 0.63860 0.27000 -0.10828 -0.48268 -0.83978 -1.16679 -1.45199 -1.68516 -1.85795 -1.96417 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E8 2.00000 1.95324 1.81515 1.59219 1.29477 0.93682 0.53506 0.10828 -0.32356 -0.74028 -1.12237 -1.45199 -1.71371 -1.89531 -1.98828 -1.98828 -1.89531 -1.71371 -1.45199 -1.12237 -0.74028 -0.32356 0.10828 0.53506 0.93682 1.29477 1.59219 1.81515 1.95324 2.00000 1.95324 1.81515 1.59219 1.29477 0.93682 0.53506 0.10828 -0.32356 -0.74028 -1.12237 -1.45199 -1.71371 -1.89531 -1.98828 -1.98828 -1.89531 -1.71371 -1.45199 -1.12237 -0.74028 -0.32356 0.10828 0.53506 0.93682 1.29477 1.59219 1.81515 1.95324 2.00000 1.95324 1.81515 1.59219 1.29477 0.93682 0.53506 0.10828 -0.32356 -0.74028 -1.12237 -1.45199 -1.71371 -1.89531 -1.98828 -1.98828 -1.89531 -1.71371 -1.45199 -1.12237 -0.74028 -0.32356 0.10828 0.53506 0.93682 1.29477 1.59219 1.81515 1.95324 2.00000 1.95324 1.81515 1.59219 1.29477 0.93682 0.53506 0.10828 -0.32356 -0.74028 -1.12237 -1.45199 -1.71371 -1.89531 -1.98828 -1.98828 -1.89531 -1.71371 -1.45199 -1.12237 -0.74028 -0.32356 0.10828 0.53506 0.93682 1.29477 1.59219 1.81515 1.95324 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. 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E114 2.00000 -1.99707 1.98828 -1.97365 1.95324 -1.92710 1.89531 -1.85795 1.81515 -1.76702 1.71371 -1.65538 1.59219 -1.52432 1.45199 -1.37540 1.29477 -1.21035 1.12237 -1.03111 0.93682 -0.83978 0.74028 -0.63860 0.53506 -0.42994 0.32356 -0.21624 0.10828 0.00000 -0.10828 0.21624 -0.32356 0.42994 -0.53506 0.63860 -0.74028 0.83978 -0.93682 1.03111 -1.12237 1.21035 -1.29477 1.37540 -1.45199 1.52432 -1.59219 1.65538 -1.71371 1.76702 -1.81515 1.85795 -1.89531 1.92710 -1.95324 1.97365 -1.98828 1.99707 -2.00000 1.99707 -1.98828 1.97365 -1.95324 1.92710 -1.89531 1.85795 -1.81515 1.76702 -1.71371 1.65538 -1.59219 1.52432 -1.45199 1.37540 -1.29477 1.21035 -1.12237 1.03111 -0.93682 0.83978 -0.74028 0.63860 -0.53506 0.42994 -0.32356 0.21624 -0.10828 0.00000 0.10828 -0.21624 0.32356 -0.42994 0.53506 -0.63860 0.74028 -0.83978 0.93682 -1.03111 1.12237 -1.21035 1.29477 -1.37540 1.45199 -1.52432 1.59219 -1.65538 1.71371 -1.76702 1.81515 -1.85795 1.89531 -1.92710 1.95324 -1.97365 1.98828 -1.99707 2.00000 0.00000 0.00000 ... ... ..... ..TT... ......... ......TT... ............. E115 2.00000 -1.99927 1.99707 -1.99340 1.98828 -1.98169 1.97365 -1.96417 1.95324 -1.94088 1.92710 -1.91190 1.89531 -1.87732 1.85795 -1.83723 1.81515 -1.79174 1.76702 -1.74101 1.71371 -1.68516 1.65538 -1.62438 1.59219 -1.55883 1.52432 -1.48870 1.45199 -1.41421 1.37540 -1.33558 1.29477 -1.25302 1.21035 -1.16679 1.12237 -1.07714 1.03111 -0.98432 0.93682 -0.88862 0.83978 -0.79032 0.74028 -0.68969 0.63860 -0.58705 0.53506 -0.48268 0.42994 -0.37689 0.32356 -0.27000 0.21624 -0.16232 0.10828 -0.05416 0.00000 0.05416 -0.10828 0.16232 -0.21624 0.27000 -0.32356 0.37689 -0.42994 0.48268 -0.53506 0.58705 -0.63860 0.68969 -0.74028 0.79032 -0.83978 0.88862 -0.93682 0.98432 -1.03111 1.07714 -1.12237 1.16679 -1.21035 1.25302 -1.29477 1.33558 -1.37540 1.41421 -1.45199 1.48870 -1.52432 1.55883 -1.59219 1.62438 -1.65538 1.68516 -1.71371 1.74101 -1.76702 1.79174 -1.81515 1.83723 -1.85795 1.87732 -1.89531 1.91190 -1.92710 1.94088 -1.95324 1.96417 -1.97365 1.98169 -1.98828 1.99340 -1.99707 1.99927 -2.00000 0.00000 0.00000 TT. ... ..TT. ....... ......TT. ........... ..........TT. Irrational character values: 1.999266572447 = 2*cos(2*π/232) = 2*cos(π/116) 1.997066827702 = 2*cos(4*π/232) = 2*cos(π/58) 1.993402379120 = 2*cos(6*π/232) = 2*cos(3*π/116) 1.988275914309 = 2*cos(8*π/232) = 2*cos(π/29) 1.981691193158 = 2*cos(10*π/232) = 2*cos(5*π/116) 1.973653045083 = 2*cos(12*π/232) = 2*cos(3*π/58) 1.964167365484 = 2*cos(14*π/232) = 2*cos(7*π/116) 1.953241111420 = 2*cos(16*π/232) = 2*cos(2*π/29) 1.940882296506 = 2*cos(18*π/232) = 2*cos(9*π/116) 1.927099985038 = 2*cos(20*π/232) = 2*cos(5*π/58) 1.911904285343 = 2*cos(22*π/232) = 2*cos(11*π/116) 1.895306342366 = 2*cos(24*π/232) = 2*cos(3*π/29) 1.877318329494 = 2*cos(26*π/232) = 2*cos(13*π/116) 1.857953439634 = 2*cos(28*π/232) = 2*cos(7*π/58) 1.837225875527 = 2*cos(30*π/232) = 2*cos(15*π/116) 1.815150839342 = 2*cos(32*π/232) = 2*cos(4*π/29) 1.791744521517 = 2*cos(34*π/232) = 2*cos(17*π/116) 1.767024088892 = 2*cos(36*π/232) = 2*cos(9*π/58) 1.741007672112 = 2*cos(38*π/232) = 2*cos(19*π/116) 1.713714352335 = 2*cos(40*π/232) = 2*cos(5*π/29) 1.685164147233 = 2*cos(42*π/232) = 2*cos(21*π/116) 1.655377996314 = 2*cos(44*π/232) = 2*cos(11*π/58) 1.624377745560 = 2*cos(46*π/232) = 2*cos(23*π/116) 1.592186131411 = 2*cos(48*π/232) = 2*cos(6*π/29) 1.558826764083 = 2*cos(50*π/232) = 2*cos(25*π/116) 1.524324110255 = 2*cos(52*π/232) = 2*cos(13*π/58) 1.488703475125 = 2*cos(54*π/232) = 2*cos(27*π/116) 1.451990983846 = 2*cos(56*π/232) = 2*cos(7*π/29) 1.414213562373 = 2*cos(58*π/232) = 2*cos(π/4) = √2 1.375398917707 = 2*cos(60*π/232) = 2*cos(15*π/58) 1.335575517577 = 2*cos(62*π/232) = 2*cos(31*π/116) 1.294772569564 = 2*cos(64*π/232) = 2*cos(8*π/29) 1.253019999672 = 2*cos(66*π/232) = 2*cos(33*π/116) 1.210348430388 = 2*cos(68*π/232) = 2*cos(17*π/58) 1.166789158215 = 2*cos(70*π/232) = 2*cos(35*π/116) 1.122374130725 = 2*cos(72*π/232) = 2*cos(9*π/29) 1.077135923122 = 2*cos(74*π/232) = 2*cos(37*π/116) 1.031107714354 = 2*cos(76*π/232) = 2*cos(19*π/58) 0.984323262778 = 2*cos(78*π/232) = 2*cos(39*π/116) 0.936816881400 = 2*cos(80*π/232) = 2*cos(10*π/29) 0.888623412708 = 2*cos(82*π/232) = 2*cos(41*π/116) 0.839778203121 = 2*cos(84*π/232) = 2*cos(21*π/58) 0.790317077060 = 2*cos(86*π/232) = 2*cos(43*π/116) 0.740276310680 = 2*cos(88*π/232) = 2*cos(11*π/29) 0.689692605256 = 2*cos(90*π/232) = 2*cos(45*π/116) 0.638603060272 = 2*cos(92*π/232) = 2*cos(23*π/58) 0.587045146208 = 2*cos(94*π/232) = 2*cos(47*π/116) 0.535056677058 = 2*cos(96*π/232) = 2*cos(12*π/29) 0.482675782599 = 2*cos(98*π/232) = 2*cos(49*π/116) 0.429940880422 = 2*cos(100*π/232) = 2*cos(25*π/58) 0.376890647757 = 2*cos(102*π/232) = 2*cos(51*π/116) 0.323563993106 = 2*cos(104*π/232) = 2*cos(13*π/29) 0.270000027707 = 2*cos(106*π/232) = 2*cos(53*π/116) 0.216238036848 = 2*cos(108*π/232) = 2*cos(27*π/58) 0.162317451055 = 2*cos(110*π/232) = 2*cos(55*π/116) 0.108277817171 = 2*cos(112*π/232) = 2*cos(14*π/29) 0.054158769352 = 2*cos(114*π/232) = 2*cos(57*π/116) 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 E107 m E108 l E109 k+m E110 j+l E111 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)} E112 h+j+l {z((x²−y²)²−4x²y²), xyz(x²−y²)} E113 g+i+k+m {xz(x²−3y²), yz(3x²−y²)}, {xz³(x²−3y²), yz³(3x²−y²)} E114 f+h+j+l {z(x²−y²), xyz}, {z³(x²−y²), xyz³} E115 R+d+g+i+k+m {R_x, R_y}, {xz, yz}, {xz³, yz³}, {xz⁵, yz⁵} Notes: α The order of the D_116d point group is 464, and the order of the principal axis (S₂₃₂) is 232. The group has 119 irreducible representations. β The D_116d point group is isomorphic to C_232v and D₂₃₂. γ The D_116d point group is generated by two symmetry elements, S₂₃₂ 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 lowest nonvanishing multipole moment in D_116d is 4 (quadrupole moment). ε This point group is non-Abelian (some symmetry operations are not commutative). Therefore, the character table contains multi-membered classes and degenerate irreducible representations. ζ 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. For this group, however, some of the irrational characters cannot be expressed by a closed algebraic form using real numbers only.

This Character Table for the D_116d point group was created by Gernot Katzer.

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

	D_114d
	D_115d
C₁₁₆ C_116v C_116h D₁₁₆ D_116h	D_116d	S₁₁₆
	D_117d
	D_118d

Character table for the D116d point group

Character table for the D_116d point group