Character Table for Point Group D45h

Character table for the D_45h point group

D45h 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 45 C2' 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 45 sv <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 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 ... ... ..... ....... ......... ........... ............. 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 ... ..... ....... ......... ........... ............. 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 ............. 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 0.00000 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 0.00000 ... 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 0.00000 -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 0.00000 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 0.00000 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 0.00000 ... ... 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 0.00000 -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 0.00000 ... ... ..... ..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 0.00000 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 0.00000 ... ... ..... 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 0.00000 -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 0.00000 ... ... ..... ....... ..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 0.00000 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 0.00000 ... ... ..... ....... 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 0.00000 -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 0.00000 ... ... ..... ....... ......... ..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 0.00000 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 0.00000 ... ... ..... ....... ......... 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 0.00000 -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 0.00000 ... ... ..... ....... ......... ........... ..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 0.00000 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 0.00000 ... ... ..... ....... ......... ........... 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 0.00000 -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 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 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 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 -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 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 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 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 -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 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 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 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 -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 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 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 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 -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 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 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 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 -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 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 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 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 -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 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 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 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 -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 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 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 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 -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 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 0.00000 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 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 0.00000 -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 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 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 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 -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 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 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 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 -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 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 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 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 -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 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 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 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 -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 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 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 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 -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 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 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 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 -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 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 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 0.00000 ... ... ..... ....... ......... ........... ............. 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 0.00000 -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 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) = (√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 A1' d+g+i+k+m z², z⁴, z⁶ A2' R R_z A2" 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 D_45h point group is 180, and the order of the principal axis (S₄₅) is 90. The group has 48 irreducible representations. β The D_45h point group is isomorphic to D_45d, C_90v and D₉₀. γ The D_45h point group is generated by two symmetry elements, S₄₅ and either a perpendicular C₂^′ or a vertical σ_v. Also, the group may be generated from two σ_v planes, or a σ_v and a C₂^′ (some pairs will yield smaller groups, though; choosing a minimum angle is safe). The canonical choice, however, is to use redundant generators: C₄₅, C₂^′ and σ_h. δ The group contains one set of C₂^′ symmetry axes perpendicular to the principal (z) axis. The x axis (but not the y axis) is a member of that set. Similarly, the single set of symmetry planes denoted σ_d contains the xz plane but not the yz plane. ε The lowest nonvanishing multipole moment in D_45h 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 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 D_45h point group was created by Gernot Katzer.

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

	D_43h
	D_44h
C₄₅ C_45v C_45h D₄₅	D_45h	D_45d
	D_46h
	D_47h

Character table for the D45h point group

Character table for the D_45h point group