Character Table for Point Group C45h

Character table for the C_45h point group

C45h E 2 C45 2 C45^2 2 C15 2 C45^4 2 C9 2 C15^2 2 C45^7 2 C45^8 2 C5 2 C9^2 2 C45^11 2 C15^4 2 C45^13 2 C45^14 2 C3 2 C45^16 2 C45^17 2 C5^2 2 C45^19 2 C9^4 2 C15^7 2 C45^22 sh 2 S45 2 S15 2 S9 2 S45^7 2 S5 2 S45^11 2 S45^13 2 S3 2 S45^17 2 S45^19 2 S15^7 2 S45^23 2 S9^5 2 S5^3 2 S45^29 2 S45^31 2 S15^11 2 S9^7 2 S45^37 2 S15^13 2 S45^41 2 S45^43 <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> A' 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 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 A" 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -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 1.98054 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 2.00000 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 ... TT. ..... ....TT. ......... ........TT. ............. E1" * 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 -2.00000 -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 TT. ... ..TT. ....... ......TT. ........... ..........TT. E2' * 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 2.00000 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 ... ... TT... ....... ....TT... ........... ........TT... E2" * 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 -2.00000 -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 ... ... ..... ..TT... ......... ......TT... ............. E3' * 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 2.00000 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 ... ... ..... TT..... ......... ....TT..... ............. E3" * 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 -2.00000 -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 ... ... ..... ....... ..TT..... ........... ......TT..... E4' * 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 2.00000 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 ... ... ..... ....... TT....... ........... ....TT....... E4" * 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 -2.00000 -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 ... ... ..... ....... ......... ..TT....... ............. E5' * 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 2.00000 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 ... ... ..... ....... ......... TT......... ............. E5" * 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 -2.00000 -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 ... ... ..... ....... ......... ........... ..TT......... E6' * 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 2.00000 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 ... ... ..... ....... ......... ........... TT........... E6" * 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 -2.00000 -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 ... ... ..... ....... ......... ........... ............. E7' * 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 2.00000 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 ... ... ..... ....... ......... ........... ............. E7" * 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 -2.00000 -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 ... ... ..... ....... ......... ........... ............. E8' * 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 2.00000 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 ... ... ..... ....... ......... ........... ............. E8" * 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 -2.00000 -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 ... ... ..... ....... ......... ........... ............. E9' * 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 2.00000 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 ... ... ..... ....... ......... ........... ............. E9" * 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 -2.00000 -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 ... ... ..... ....... ......... ........... ............. E10' * 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 2.00000 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 ... ... ..... ....... ......... ........... ............. E10" * 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 -2.00000 -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 ... ... ..... ....... ......... ........... ............. E11' * 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 2.00000 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 ... ... ..... ....... ......... ........... ............. E11" * 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 -2.00000 -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 ... ... ..... ....... ......... ........... ............. E12' * 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 2.00000 -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 ... ... ..... ....... ......... ........... ............. E12" * 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 -2.00000 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 ... ... ..... ....... ......... ........... ............. E13' * 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 2.00000 -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 ... ... ..... ....... ......... ........... ............. E13" * 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 -2.00000 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 ... ... ..... ....... ......... ........... ............. E14' * 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 2.00000 -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 ... ... ..... ....... ......... ........... ............. E14" * 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 -2.00000 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 ... ... ..... ....... ......... ........... ............. 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 2.00000 -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 ... ... ..... ....... ......... ........... ............. 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 -2.00000 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 ... ... ..... ....... ......... ........... ............. E16' * 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 2.00000 -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 ... ... ..... ....... ......... ........... ............. E16" * 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 -2.00000 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 ... ... ..... ....... ......... ........... ............. E17' * 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 2.00000 -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 ... ... ..... ....... ......... ........... ............. E17" * 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 -2.00000 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 ... ... ..... ....... ......... ........... ............. E18' * 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 2.00000 -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 ... ... ..... ....... ......... ........... ............. E18" * 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 -2.00000 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 ... ... ..... ....... ......... ........... ............. E19' * 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 2.00000 -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 ... ... ..... ....... ......... ........... ............. E19" * 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 -2.00000 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 ... ... ..... ....... ......... ........... ............. E20' * 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 2.00000 -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 ... ... ..... ....... ......... ........... ............. E20" * 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 -2.00000 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 ... ... ..... ....... ......... ........... ............. E21' * 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 2.00000 -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 ... ... ..... ....... ......... ........... ............. E21" * 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 -2.00000 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 ... ... ..... ....... ......... ........... ............. E22' * 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 2.00000 -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 ... ... ..... ....... ......... ........... ............. E22" * 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 -2.00000 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 ... ... ..... ....... ......... ........... ............. 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) = (√5+1)/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 A' R+d+g+i+k+m R_z, z², z⁴, z⁶ A" p+f+h+j+l z, z³, z⁵ E1' p+f+h+j+l {x, y}, {xz², yz²}, {xz⁴, yz⁴} E1" R+d+g+i+k+m {R_x, R_y}, {xz, yz}, {xz³, yz³}, {xz⁵, yz⁵} E2' d+g+i+k+m {x²−y², xy}, {z²(x²−y²), xyz²}, {z⁴(x²−y²), xyz⁴} E2" f+h+j+l {z(x²−y²), xyz}, {z³(x²−y²), xyz³} E3' f+h+j+l {x(x²−3y²), y(3x²−y²)}, {xz²(x²−3y²), yz²(3x²−y²)} E3" g+i+k+m {xz(x²−3y²), yz(3x²−y²)}, {xz³(x²−3y²), yz³(3x²−y²)} E4' g+i+k+m {(x²−y²)²−4x²y², xy(x²−y²)}, {z²((x²−y²)²−4x²y²), xyz²(x²−y²)} E4" h+j+l {z((x²−y²)²−4x²y²), xyz(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)} E5" 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)} E6' i+k+m {x²(x²−3y²)²−y²(3x²−y²)², xy(x²−3y²)(3x²−y²)} E6" j+l E7' j+l E7" k+m E8' k+m E8" l E9' l E9" m E10' m Notes: α The order of the C_45h point group is 90, and the order of the principal axis (S₄₅) is 90. The group has 46 irreducible representations. β The C_45h point group could also be named S₄₅, as it contains the S₄₅ axis as its only symmetry element. Another rare designation is C_90i because the S₄₅ axis is identical to a roto-inversion axis of order 90. γ The C_45h point group is isomorphic to C₉₀ and S₉₀. δ The C_45h point group is generated by one single symmetry element, S₄₅. Therefore, it is a cyclic group. The canonical choice, however, is to use redundant generators: C₄₅ and σ_h. ε The lowest nonvanishing multipole moment in C_45h is 4 (quadrupole moment). ζ This is an Abelian point group (the commutative law holds between all symmetry operations). The C_45h group is Abelian because it contains only one symmetry element, all the powers of which necessarily commute (sufficient condition). In Abelian groups, all symmetry operations form a class of their own, and all irreducible representations are one-dimensional. η Because the group is Abelian and the maximum order of rotation is >2, some irreducible representations have complex characters. These 88 cases have been combined into 44 two-dimensional representations that are no longer irreducible but have real-valued characters. Accordingly, 44 pairs of left and right rotations have been combined into one two-membered pseudo-class each. θ The 44 reducible “E” representations almost behave like true irreducible representations. Their norm, however, is twice the group order. Therefore, they have been marked with an asterisk in the table. This is essential when trying to decompose a reducible representation into “irreducible” ones using the familiar projection formula. ι 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. κ The point group corresponds to a polygon inconstructible by the classical means of ruler and compass. Yet it becomes constructible if angle trisection is allowed, e.g., with neusis construction or origami. This is because the order of the principal axis is given by a product of any number of different Pierpont primes (...,5,7,13,17,19,37,73,97,109,163,...) times arbitrary powers of two and three. While some characters of this group can be expressed using real integers and square roots alone, others need complex numbers and third roots.

This Character Table for the C_45h point group was created by Gernot Katzer.

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

	C_43h
	C_44h
C₄₅ C_45v	C_45h	D₄₅ D_45h D_45d
	C_46h
	C_47h

Character table for the C45h point group

Character table for the C_45h point group