Character Table for Point Group D90

Character table for the D₉₀ point group

D90 E 2 C90 2 C45 2 C30 2 C45^2 2 C18 2 C15 2 C90^7 2 C45^4 2 C10 2 C9 2 C90^11 2 C15^2 2 C90^13 2 C45^7 2 C6 2 C45^8 2 C90^17 2 C5 2 C90^19 2 C9^2 2 C30^7 2 C45^11 2 C90^23 2 C15^4 2 C18^5 2 C45^13 2 C10^3 2 C45^14 2 C90^29 2 C3 2 C90^31 2 C45^16 2 C30^11 2 C45^17 2 C18^7 2 C5^2 2 C90^37 2 C45^19 2 C30^13 2 C9^4 2 C90^41 2 C15^7 2 C90^43 2 C45^22 C2 45 C2' 45 C2" <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 ... ... ....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 ..T ..T ..... ......T ......... ..........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 ... ... ..... ....... ......... ........... ............. 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 ... ... ..... ....... ......... ........... ............. E1 2.00000 1.99513 1.98054 1.95630 1.92252 1.87939 1.82709 1.76590 1.69610 1.61803 1.53209 1.43868 1.33826 1.23132 1.11839 1.00000 0.87674 0.74921 0.61803 0.48384 0.34730 0.20906 0.06980 -0.06980 -0.20906 -0.34730 -0.48384 -0.61803 -0.74921 -0.87674 -1.00000 -1.11839 -1.23132 -1.33826 -1.43868 -1.53209 -1.61803 -1.69610 -1.76590 -1.82709 -1.87939 -1.92252 -1.95630 -1.98054 -1.99513 -2.00000 0.00000 0.00000 TT. TT. ..TT. ....TT. ......TT. ........TT. ..........TT. E2 2.00000 1.98054 1.92252 1.82709 1.69610 1.53209 1.33826 1.11839 0.87674 0.61803 0.34730 0.06980 -0.20906 -0.48384 -0.74921 -1.00000 -1.23132 -1.43868 -1.61803 -1.76590 -1.87939 -1.95630 -1.99513 -1.99513 -1.95630 -1.87939 -1.76590 -1.61803 -1.43868 -1.23132 -1.00000 -0.74921 -0.48384 -0.20906 0.06980 0.34730 0.61803 0.87674 1.11839 1.33826 1.53209 1.69610 1.82709 1.92252 1.98054 2.00000 0.00000 0.00000 ... ... TT... ..TT... ....TT... ......TT... ........TT... E3 2.00000 1.95630 1.82709 1.61803 1.33826 1.00000 0.61803 0.20906 -0.20906 -0.61803 -1.00000 -1.33826 -1.61803 -1.82709 -1.95630 -2.00000 -1.95630 -1.82709 -1.61803 -1.33826 -1.00000 -0.61803 -0.20906 0.20906 0.61803 1.00000 1.33826 1.61803 1.82709 1.95630 2.00000 1.95630 1.82709 1.61803 1.33826 1.00000 0.61803 0.20906 -0.20906 -0.61803 -1.00000 -1.33826 -1.61803 -1.82709 -1.95630 -2.00000 0.00000 0.00000 ... ... ..... TT..... ..TT..... ....TT..... ......TT..... E4 2.00000 1.92252 1.69610 1.33826 0.87674 0.34730 -0.20906 -0.74921 -1.23132 -1.61803 -1.87939 -1.99513 -1.95630 -1.76590 -1.43868 -1.00000 -0.48384 0.06980 0.61803 1.11839 1.53209 1.82709 1.98054 1.98054 1.82709 1.53209 1.11839 0.61803 0.06980 -0.48384 -1.00000 -1.43868 -1.76590 -1.95630 -1.99513 -1.87939 -1.61803 -1.23132 -0.74921 -0.20906 0.34730 0.87674 1.33826 1.69610 1.92252 2.00000 0.00000 0.00000 ... ... ..... ....... TT....... ..TT....... ....TT....... E5 2.00000 1.87939 1.53209 1.00000 0.34730 -0.34730 -1.00000 -1.53209 -1.87939 -2.00000 -1.87939 -1.53209 -1.00000 -0.34730 0.34730 1.00000 1.53209 1.87939 2.00000 1.87939 1.53209 1.00000 0.34730 -0.34730 -1.00000 -1.53209 -1.87939 -2.00000 -1.87939 -1.53209 -1.00000 -0.34730 0.34730 1.00000 1.53209 1.87939 2.00000 1.87939 1.53209 1.00000 0.34730 -0.34730 -1.00000 -1.53209 -1.87939 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... TT......... ..TT......... E6 2.00000 1.82709 1.33826 0.61803 -0.20906 -1.00000 -1.61803 -1.95630 -1.95630 -1.61803 -1.00000 -0.20906 0.61803 1.33826 1.82709 2.00000 1.82709 1.33826 0.61803 -0.20906 -1.00000 -1.61803 -1.95630 -1.95630 -1.61803 -1.00000 -0.20906 0.61803 1.33826 1.82709 2.00000 1.82709 1.33826 0.61803 -0.20906 -1.00000 -1.61803 -1.95630 -1.95630 -1.61803 -1.00000 -0.20906 0.61803 1.33826 1.82709 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... TT........... E7 2.00000 1.76590 1.11839 0.20906 -0.74921 -1.53209 -1.95630 -1.92252 -1.43868 -0.61803 0.34730 1.23132 1.82709 1.99513 1.69610 1.00000 0.06980 -0.87674 -1.61803 -1.98054 -1.87939 -1.33826 -0.48384 0.48384 1.33826 1.87939 1.98054 1.61803 0.87674 -0.06980 -1.00000 -1.69610 -1.99513 -1.82709 -1.23132 -0.34730 0.61803 1.43868 1.92252 1.95630 1.53209 0.74921 -0.20906 -1.11839 -1.76590 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E8 2.00000 1.69610 0.87674 -0.20906 -1.23132 -1.87939 -1.95630 -1.43868 -0.48384 0.61803 1.53209 1.98054 1.82709 1.11839 0.06980 -1.00000 -1.76590 -1.99513 -1.61803 -0.74921 0.34730 1.33826 1.92252 1.92252 1.33826 0.34730 -0.74921 -1.61803 -1.99513 -1.76590 -1.00000 0.06980 1.11839 1.82709 1.98054 1.53209 0.61803 -0.48384 -1.43868 -1.95630 -1.87939 -1.23132 -0.20906 0.87674 1.69610 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E9 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 1.61803 0.61803 -0.61803 -1.61803 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E10 2.00000 1.53209 0.34730 -1.00000 -1.87939 -1.87939 -1.00000 0.34730 1.53209 2.00000 1.53209 0.34730 -1.00000 -1.87939 -1.87939 -1.00000 0.34730 1.53209 2.00000 1.53209 0.34730 -1.00000 -1.87939 -1.87939 -1.00000 0.34730 1.53209 2.00000 1.53209 0.34730 -1.00000 -1.87939 -1.87939 -1.00000 0.34730 1.53209 2.00000 1.53209 0.34730 -1.00000 -1.87939 -1.87939 -1.00000 0.34730 1.53209 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E11 2.00000 1.43868 0.06980 -1.33826 -1.99513 -1.53209 -0.20906 1.23132 1.98054 1.61803 0.34730 -1.11839 -1.95630 -1.69610 -0.48384 1.00000 1.92252 1.76590 0.61803 -0.87674 -1.87939 -1.82709 -0.74921 0.74921 1.82709 1.87939 0.87674 -0.61803 -1.76590 -1.92252 -1.00000 0.48384 1.69610 1.95630 1.11839 -0.34730 -1.61803 -1.98054 -1.23132 0.20906 1.53209 1.99513 1.33826 -0.06980 -1.43868 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E12 2.00000 1.33826 -0.20906 -1.61803 -1.95630 -1.00000 0.61803 1.82709 1.82709 0.61803 -1.00000 -1.95630 -1.61803 -0.20906 1.33826 2.00000 1.33826 -0.20906 -1.61803 -1.95630 -1.00000 0.61803 1.82709 1.82709 0.61803 -1.00000 -1.95630 -1.61803 -0.20906 1.33826 2.00000 1.33826 -0.20906 -1.61803 -1.95630 -1.00000 0.61803 1.82709 1.82709 0.61803 -1.00000 -1.95630 -1.61803 -0.20906 1.33826 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E13 2.00000 1.23132 -0.48384 -1.82709 -1.76590 -0.34730 1.33826 1.99513 1.11839 -0.61803 -1.87939 -1.69610 -0.20906 1.43868 1.98054 1.00000 -0.74921 -1.92252 -1.61803 -0.06980 1.53209 1.95630 0.87674 -0.87674 -1.95630 -1.53209 0.06980 1.61803 1.92252 0.74921 -1.00000 -1.98054 -1.43868 0.20906 1.69610 1.87939 0.61803 -1.11839 -1.99513 -1.33826 0.34730 1.76590 1.82709 0.48384 -1.23132 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E14 2.00000 1.11839 -0.74921 -1.95630 -1.43868 0.34730 1.82709 1.69610 0.06980 -1.61803 -1.87939 -0.48384 1.33826 1.98054 0.87674 -1.00000 -1.99513 -1.23132 0.61803 1.92252 1.53209 -0.20906 -1.76590 -1.76590 -0.20906 1.53209 1.92252 0.61803 -1.23132 -1.99513 -1.00000 0.87674 1.98054 1.33826 -0.48384 -1.87939 -1.61803 0.06980 1.69610 1.82709 0.34730 -1.43868 -1.95630 -0.74921 1.11839 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E15 2.00000 1.00000 -1.00000 -2.00000 -1.00000 1.00000 2.00000 1.00000 -1.00000 -2.00000 -1.00000 1.00000 2.00000 1.00000 -1.00000 -2.00000 -1.00000 1.00000 2.00000 1.00000 -1.00000 -2.00000 -1.00000 1.00000 2.00000 1.00000 -1.00000 -2.00000 -1.00000 1.00000 2.00000 1.00000 -1.00000 -2.00000 -1.00000 1.00000 2.00000 1.00000 -1.00000 -2.00000 -1.00000 1.00000 2.00000 1.00000 -1.00000 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E16 2.00000 0.87674 -1.23132 -1.95630 -0.48384 1.53209 1.82709 0.06980 -1.76590 -1.61803 0.34730 1.92252 1.33826 -0.74921 -1.99513 -1.00000 1.11839 1.98054 0.61803 -1.43868 -1.87939 -0.20906 1.69610 1.69610 -0.20906 -1.87939 -1.43868 0.61803 1.98054 1.11839 -1.00000 -1.99513 -0.74921 1.33826 1.92252 0.34730 -1.61803 -1.76590 0.06980 1.82709 1.53209 -0.48384 -1.95630 -1.23132 0.87674 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E17 2.00000 0.74921 -1.43868 -1.82709 0.06980 1.87939 1.33826 -0.87674 -1.99513 -0.61803 1.53209 1.76590 -0.20906 -1.92252 -1.23132 1.00000 1.98054 0.48384 -1.61803 -1.69610 0.34730 1.95630 1.11839 -1.11839 -1.95630 -0.34730 1.69610 1.61803 -0.48384 -1.98054 -1.00000 1.23132 1.92252 0.20906 -1.76590 -1.53209 0.61803 1.99513 0.87674 -1.33826 -1.87939 -0.06980 1.82709 1.43868 -0.74921 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E18 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 0.61803 -1.61803 -1.61803 0.61803 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E19 2.00000 0.48384 -1.76590 -1.33826 1.11839 1.87939 -0.20906 -1.98054 -0.74921 1.61803 1.53209 -0.87674 -1.95630 -0.06980 1.92252 1.00000 -1.43868 -1.69610 0.61803 1.99513 0.34730 -1.82709 -1.23132 1.23132 1.82709 -0.34730 -1.99513 -0.61803 1.69610 1.43868 -1.00000 -1.92252 0.06980 1.95630 0.87674 -1.53209 -1.61803 0.74921 1.98054 0.20906 -1.87939 -1.11839 1.33826 1.76590 -0.48384 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E20 2.00000 0.34730 -1.87939 -1.00000 1.53209 1.53209 -1.00000 -1.87939 0.34730 2.00000 0.34730 -1.87939 -1.00000 1.53209 1.53209 -1.00000 -1.87939 0.34730 2.00000 0.34730 -1.87939 -1.00000 1.53209 1.53209 -1.00000 -1.87939 0.34730 2.00000 0.34730 -1.87939 -1.00000 1.53209 1.53209 -1.00000 -1.87939 0.34730 2.00000 0.34730 -1.87939 -1.00000 1.53209 1.53209 -1.00000 -1.87939 0.34730 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E21 2.00000 0.20906 -1.95630 -0.61803 1.82709 1.00000 -1.61803 -1.33826 1.33826 1.61803 -1.00000 -1.82709 0.61803 1.95630 -0.20906 -2.00000 -0.20906 1.95630 0.61803 -1.82709 -1.00000 1.61803 1.33826 -1.33826 -1.61803 1.00000 1.82709 -0.61803 -1.95630 0.20906 2.00000 0.20906 -1.95630 -0.61803 1.82709 1.00000 -1.61803 -1.33826 1.33826 1.61803 -1.00000 -1.82709 0.61803 1.95630 -0.20906 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E22 2.00000 0.06980 -1.99513 -0.20906 1.98054 0.34730 -1.95630 -0.48384 1.92252 0.61803 -1.87939 -0.74921 1.82709 0.87674 -1.76590 -1.00000 1.69610 1.11839 -1.61803 -1.23132 1.53209 1.33826 -1.43868 -1.43868 1.33826 1.53209 -1.23132 -1.61803 1.11839 1.69610 -1.00000 -1.76590 0.87674 1.82709 -0.74921 -1.87939 0.61803 1.92252 -0.48384 -1.95630 0.34730 1.98054 -0.20906 -1.99513 0.06980 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E23 2.00000 -0.06980 -1.99513 0.20906 1.98054 -0.34730 -1.95630 0.48384 1.92252 -0.61803 -1.87939 0.74921 1.82709 -0.87674 -1.76590 1.00000 1.69610 -1.11839 -1.61803 1.23132 1.53209 -1.33826 -1.43868 1.43868 1.33826 -1.53209 -1.23132 1.61803 1.11839 -1.69610 -1.00000 1.76590 0.87674 -1.82709 -0.74921 1.87939 0.61803 -1.92252 -0.48384 1.95630 0.34730 -1.98054 -0.20906 1.99513 0.06980 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E24 2.00000 -0.20906 -1.95630 0.61803 1.82709 -1.00000 -1.61803 1.33826 1.33826 -1.61803 -1.00000 1.82709 0.61803 -1.95630 -0.20906 2.00000 -0.20906 -1.95630 0.61803 1.82709 -1.00000 -1.61803 1.33826 1.33826 -1.61803 -1.00000 1.82709 0.61803 -1.95630 -0.20906 2.00000 -0.20906 -1.95630 0.61803 1.82709 -1.00000 -1.61803 1.33826 1.33826 -1.61803 -1.00000 1.82709 0.61803 -1.95630 -0.20906 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E25 2.00000 -0.34730 -1.87939 1.00000 1.53209 -1.53209 -1.00000 1.87939 0.34730 -2.00000 0.34730 1.87939 -1.00000 -1.53209 1.53209 1.00000 -1.87939 -0.34730 2.00000 -0.34730 -1.87939 1.00000 1.53209 -1.53209 -1.00000 1.87939 0.34730 -2.00000 0.34730 1.87939 -1.00000 -1.53209 1.53209 1.00000 -1.87939 -0.34730 2.00000 -0.34730 -1.87939 1.00000 1.53209 -1.53209 -1.00000 1.87939 0.34730 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E26 2.00000 -0.48384 -1.76590 1.33826 1.11839 -1.87939 -0.20906 1.98054 -0.74921 -1.61803 1.53209 0.87674 -1.95630 0.06980 1.92252 -1.00000 -1.43868 1.69610 0.61803 -1.99513 0.34730 1.82709 -1.23132 -1.23132 1.82709 0.34730 -1.99513 0.61803 1.69610 -1.43868 -1.00000 1.92252 0.06980 -1.95630 0.87674 1.53209 -1.61803 -0.74921 1.98054 -0.20906 -1.87939 1.11839 1.33826 -1.76590 -0.48384 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E27 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 -0.61803 -1.61803 1.61803 0.61803 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E28 2.00000 -0.74921 -1.43868 1.82709 0.06980 -1.87939 1.33826 0.87674 -1.99513 0.61803 1.53209 -1.76590 -0.20906 1.92252 -1.23132 -1.00000 1.98054 -0.48384 -1.61803 1.69610 0.34730 -1.95630 1.11839 1.11839 -1.95630 0.34730 1.69610 -1.61803 -0.48384 1.98054 -1.00000 -1.23132 1.92252 -0.20906 -1.76590 1.53209 0.61803 -1.99513 0.87674 1.33826 -1.87939 0.06980 1.82709 -1.43868 -0.74921 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E29 2.00000 -0.87674 -1.23132 1.95630 -0.48384 -1.53209 1.82709 -0.06980 -1.76590 1.61803 0.34730 -1.92252 1.33826 0.74921 -1.99513 1.00000 1.11839 -1.98054 0.61803 1.43868 -1.87939 0.20906 1.69610 -1.69610 -0.20906 1.87939 -1.43868 -0.61803 1.98054 -1.11839 -1.00000 1.99513 -0.74921 -1.33826 1.92252 -0.34730 -1.61803 1.76590 0.06980 -1.82709 1.53209 0.48384 -1.95630 1.23132 0.87674 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E30 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E31 2.00000 -1.11839 -0.74921 1.95630 -1.43868 -0.34730 1.82709 -1.69610 0.06980 1.61803 -1.87939 0.48384 1.33826 -1.98054 0.87674 1.00000 -1.99513 1.23132 0.61803 -1.92252 1.53209 0.20906 -1.76590 1.76590 -0.20906 -1.53209 1.92252 -0.61803 -1.23132 1.99513 -1.00000 -0.87674 1.98054 -1.33826 -0.48384 1.87939 -1.61803 -0.06980 1.69610 -1.82709 0.34730 1.43868 -1.95630 0.74921 1.11839 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E32 2.00000 -1.23132 -0.48384 1.82709 -1.76590 0.34730 1.33826 -1.99513 1.11839 0.61803 -1.87939 1.69610 -0.20906 -1.43868 1.98054 -1.00000 -0.74921 1.92252 -1.61803 0.06980 1.53209 -1.95630 0.87674 0.87674 -1.95630 1.53209 0.06980 -1.61803 1.92252 -0.74921 -1.00000 1.98054 -1.43868 -0.20906 1.69610 -1.87939 0.61803 1.11839 -1.99513 1.33826 0.34730 -1.76590 1.82709 -0.48384 -1.23132 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E33 2.00000 -1.33826 -0.20906 1.61803 -1.95630 1.00000 0.61803 -1.82709 1.82709 -0.61803 -1.00000 1.95630 -1.61803 0.20906 1.33826 -2.00000 1.33826 0.20906 -1.61803 1.95630 -1.00000 -0.61803 1.82709 -1.82709 0.61803 1.00000 -1.95630 1.61803 -0.20906 -1.33826 2.00000 -1.33826 -0.20906 1.61803 -1.95630 1.00000 0.61803 -1.82709 1.82709 -0.61803 -1.00000 1.95630 -1.61803 0.20906 1.33826 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E34 2.00000 -1.43868 0.06980 1.33826 -1.99513 1.53209 -0.20906 -1.23132 1.98054 -1.61803 0.34730 1.11839 -1.95630 1.69610 -0.48384 -1.00000 1.92252 -1.76590 0.61803 0.87674 -1.87939 1.82709 -0.74921 -0.74921 1.82709 -1.87939 0.87674 0.61803 -1.76590 1.92252 -1.00000 -0.48384 1.69610 -1.95630 1.11839 0.34730 -1.61803 1.98054 -1.23132 -0.20906 1.53209 -1.99513 1.33826 0.06980 -1.43868 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E35 2.00000 -1.53209 0.34730 1.00000 -1.87939 1.87939 -1.00000 -0.34730 1.53209 -2.00000 1.53209 -0.34730 -1.00000 1.87939 -1.87939 1.00000 0.34730 -1.53209 2.00000 -1.53209 0.34730 1.00000 -1.87939 1.87939 -1.00000 -0.34730 1.53209 -2.00000 1.53209 -0.34730 -1.00000 1.87939 -1.87939 1.00000 0.34730 -1.53209 2.00000 -1.53209 0.34730 1.00000 -1.87939 1.87939 -1.00000 -0.34730 1.53209 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E36 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 -1.61803 0.61803 0.61803 -1.61803 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E37 2.00000 -1.69610 0.87674 0.20906 -1.23132 1.87939 -1.95630 1.43868 -0.48384 -0.61803 1.53209 -1.98054 1.82709 -1.11839 0.06980 1.00000 -1.76590 1.99513 -1.61803 0.74921 0.34730 -1.33826 1.92252 -1.92252 1.33826 -0.34730 -0.74921 1.61803 -1.99513 1.76590 -1.00000 -0.06980 1.11839 -1.82709 1.98054 -1.53209 0.61803 0.48384 -1.43868 1.95630 -1.87939 1.23132 -0.20906 -0.87674 1.69610 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E38 2.00000 -1.76590 1.11839 -0.20906 -0.74921 1.53209 -1.95630 1.92252 -1.43868 0.61803 0.34730 -1.23132 1.82709 -1.99513 1.69610 -1.00000 0.06980 0.87674 -1.61803 1.98054 -1.87939 1.33826 -0.48384 -0.48384 1.33826 -1.87939 1.98054 -1.61803 0.87674 0.06980 -1.00000 1.69610 -1.99513 1.82709 -1.23132 0.34730 0.61803 -1.43868 1.92252 -1.95630 1.53209 -0.74921 -0.20906 1.11839 -1.76590 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E39 2.00000 -1.82709 1.33826 -0.61803 -0.20906 1.00000 -1.61803 1.95630 -1.95630 1.61803 -1.00000 0.20906 0.61803 -1.33826 1.82709 -2.00000 1.82709 -1.33826 0.61803 0.20906 -1.00000 1.61803 -1.95630 1.95630 -1.61803 1.00000 -0.20906 -0.61803 1.33826 -1.82709 2.00000 -1.82709 1.33826 -0.61803 -0.20906 1.00000 -1.61803 1.95630 -1.95630 1.61803 -1.00000 0.20906 0.61803 -1.33826 1.82709 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E40 2.00000 -1.87939 1.53209 -1.00000 0.34730 0.34730 -1.00000 1.53209 -1.87939 2.00000 -1.87939 1.53209 -1.00000 0.34730 0.34730 -1.00000 1.53209 -1.87939 2.00000 -1.87939 1.53209 -1.00000 0.34730 0.34730 -1.00000 1.53209 -1.87939 2.00000 -1.87939 1.53209 -1.00000 0.34730 0.34730 -1.00000 1.53209 -1.87939 2.00000 -1.87939 1.53209 -1.00000 0.34730 0.34730 -1.00000 1.53209 -1.87939 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E41 2.00000 -1.92252 1.69610 -1.33826 0.87674 -0.34730 -0.20906 0.74921 -1.23132 1.61803 -1.87939 1.99513 -1.95630 1.76590 -1.43868 1.00000 -0.48384 -0.06980 0.61803 -1.11839 1.53209 -1.82709 1.98054 -1.98054 1.82709 -1.53209 1.11839 -0.61803 0.06980 0.48384 -1.00000 1.43868 -1.76590 1.95630 -1.99513 1.87939 -1.61803 1.23132 -0.74921 0.20906 0.34730 -0.87674 1.33826 -1.69610 1.92252 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E42 2.00000 -1.95630 1.82709 -1.61803 1.33826 -1.00000 0.61803 -0.20906 -0.20906 0.61803 -1.00000 1.33826 -1.61803 1.82709 -1.95630 2.00000 -1.95630 1.82709 -1.61803 1.33826 -1.00000 0.61803 -0.20906 -0.20906 0.61803 -1.00000 1.33826 -1.61803 1.82709 -1.95630 2.00000 -1.95630 1.82709 -1.61803 1.33826 -1.00000 0.61803 -0.20906 -0.20906 0.61803 -1.00000 1.33826 -1.61803 1.82709 -1.95630 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E43 2.00000 -1.98054 1.92252 -1.82709 1.69610 -1.53209 1.33826 -1.11839 0.87674 -0.61803 0.34730 -0.06980 -0.20906 0.48384 -0.74921 1.00000 -1.23132 1.43868 -1.61803 1.76590 -1.87939 1.95630 -1.99513 1.99513 -1.95630 1.87939 -1.76590 1.61803 -1.43868 1.23132 -1.00000 0.74921 -0.48384 0.20906 0.06980 -0.34730 0.61803 -0.87674 1.11839 -1.33826 1.53209 -1.69610 1.82709 -1.92252 1.98054 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E44 2.00000 -1.99513 1.98054 -1.95630 1.92252 -1.87939 1.82709 -1.76590 1.69610 -1.61803 1.53209 -1.43868 1.33826 -1.23132 1.11839 -1.00000 0.87674 -0.74921 0.61803 -0.48384 0.34730 -0.20906 0.06980 0.06980 -0.20906 0.34730 -0.48384 0.61803 -0.74921 0.87674 -1.00000 1.11839 -1.23132 1.33826 -1.43868 1.53209 -1.61803 1.69610 -1.76590 1.82709 -1.87939 1.92252 -1.95630 1.98054 -1.99513 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.995128100520 = 2*cos(2*π/90) = 2*cos(π/45) 1.980536137483 = 2*cos(4*π/90) = 2*cos(2*π/45) 1.956295201468 = 2*cos(6*π/90) = 2*cos(π/15) = (√5-1+√30+6*√5)/4 1.922523391877 = 2*cos(8*π/90) = 2*cos(4*π/45) 1.879385241572 = 2*cos(10*π/90) = 2*cos(π/9) 1.827090915285 = 2*cos(12*π/90) = 2*cos(2*π/15) = (1+√5+√30-6*√5)/4 1.765895185718 = 2*cos(14*π/90) = 2*cos(7*π/45) 1.696096192313 = 2*cos(16*π/90) = 2*cos(8*π/45) 1.618033988750 = 2*cos(18*π/90) = 2*cos(π/5) = (1+√5)/2 1.532088886238 = 2*cos(20*π/90) = 2*cos(2*π/9) 1.438679600677 = 2*cos(22*π/90) = 2*cos(11*π/45) 1.338261212718 = 2*cos(24*π/90) = 2*cos(4*π/15) = (1-√5+√30+6*√5)/4 1.231322950651 = 2*cos(26*π/90) = 2*cos(13*π/45) 1.118385806941 = 2*cos(28*π/90) = 2*cos(14*π/45) 0.876742293578 = 2*cos(32*π/90) = 2*cos(16*π/45) 0.749213186832 = 2*cos(34*π/90) = 2*cos(17*π/45) 0.618033988750 = 2*cos(36*π/90) = 2*cos(2*π/5) = (√5-1)/2 0.483843791199 = 2*cos(38*π/90) = 2*cos(19*π/45) 0.347296355334 = 2*cos(40*π/90) = 2*cos(4*π/9) 0.209056926535 = 2*cos(42*π/90) = 2*cos(7*π/15) = (√30-6*√5-√5-1)/4 0.069798993405 = 2*cos(44*π/90) = 2*cos(22*π/45) Symmetry of Rotations and Cartesian products A1 d+g+i z², z⁴, z⁶ A2 R+p+f+h z, R_z, z³, z⁵ E1 R+p+d+f+g+h+i {x, y}, {R_x, R_y}, {xz, yz}, {xz², yz²}, {xz³, yz³}, {xz⁴, yz⁴}, {xz⁵, yz⁵} E2 d+f+g+h+i {x²–y², xy}, {z(x²–y²), xyz}, {z²(x²–y²), xyz²}, {z³(x²–y²), xyz³}, {z⁴(x²–y²), xyz⁴} E3 f+g+h+i {x(x²–3y²), y(3x²–y²)}, {xz(x²–3y²), yz(3x²–y²)}, {xz²(x²–3y²), yz²(3x²–y²)}, {xz³(x²–3y²), yz³(3x²–y²)} E4 g+h+i {(x²–y²)²–2x²y², xy(x²–y²)}, {z((x²–y²)²–4x²y²), xyz(x²–y²)}, {z²((x²–y²)²–4x²y²), xyz²(x²–y²)} E5 h+i {x(x²–(5+2√5)y²)(x²–(5–2√5)y²), y(y²–(5+2√5)x²)(y²–(5–2√5)x²)(y²–(5–2√5)x²)}, {xz(x²–(5+2√5)y²)(x²–(5–2√5)y²), yz(y²–(5+2√5)x²)(y²–(5–2√5)x²)(y²–(5–2√5)x²)} E6 i {x²(x²–3y²)²–y²(3x²–y²)², xy(x²–3y²)(3x²–y²)} Notes: 1 The order of the D₉₀ point group is 180, and the order of the principal axis is 90. 2 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^′ has the x axis as a member, while the y axis is a member of the C₂^″ set. 3 This point group is non-Abelian (some symmetry operations are not commutative). Therefore, the character table contains multi-membered classes and degenerate irreducible representations. 4 The point group is chiral, as it does not contain any mirroring operation. 5 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 less than half the order of the principal axis (and often much less). For this group, however, some of the characters cannot be expressed by a closed algebraic form using real numbers only.

This Character Table for the D₉₀ point group was created by Gernot Katzer.

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

	D₈₉
C₉₀ C_90v C_90h	D₉₀	D_90h D_90d S₉₀
	D₉₁

Character table for the D90 point group

Character table for the D₉₀ point group