Character Table for Point Group C55h

Character table for the C_55h point group

C55h E 2 C55 2 C55^2 2 C55^3 2 C55^4 2 C11 2 C55^6 2 C55^7 2 C55^8 2 C55^9 2 C11^2 2 C5 2 C55^12 2 C55^13 2 C55^14 2 C11^3 2 C55^16 2 C55^17 2 C55^18 2 C55^19 2 C11^4 2 C55^21 2 C5^2 2 C55^23 2 C55^24 2 C11^5 2 C55^26 2 C55^27 sh 2 S55 2 S55^3 2 S11 2 S55^7 2 S55^9 2 S5 2 S55^13 2 S11^3 2 S55^17 2 S55^19 2 S55^21 2 S55^23 2 S11^5 2 S55^27 2 S55^29 2 S55^31 2 S5^3 2 S11^7 2 S55^37 2 S55^39 2 S55^41 2 S55^43 2 S11^9 2 S55^47 2 S55^49 2 S55^51 2 S55^53 <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 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 -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 1.98696 1.94802 1.88369 1.79480 1.68251 1.54828 1.39388 1.22130 1.03279 0.83083 0.61803 0.39718 0.17115 -0.05711 -0.28463 -0.50844 -0.72562 -0.93333 -1.12889 -1.30972 -1.47348 -1.61803 -1.74149 -1.84225 -1.91899 -1.97071 -1.99674 2.00000 1.98696 1.88369 1.68251 1.39388 1.03279 0.61803 0.17115 -0.28463 -0.72562 -1.12889 -1.47348 -1.74149 -1.91899 -1.99674 -1.97071 -1.84225 -1.61803 -1.30972 -0.93333 -0.50844 -0.05711 0.39718 0.83083 1.22130 1.54828 1.79480 1.94802 ... TT. ..... ....TT. ......... ........TT. ............. E1" * 2.00000 1.98696 1.94802 1.88369 1.79480 1.68251 1.54828 1.39388 1.22130 1.03279 0.83083 0.61803 0.39718 0.17115 -0.05711 -0.28463 -0.50844 -0.72562 -0.93333 -1.12889 -1.30972 -1.47348 -1.61803 -1.74149 -1.84225 -1.91899 -1.97071 -1.99674 -2.00000 -1.98696 -1.88369 -1.68251 -1.39388 -1.03279 -0.61803 -0.17115 0.28463 0.72562 1.12889 1.47348 1.74149 1.91899 1.99674 1.97071 1.84225 1.61803 1.30972 0.93333 0.50844 0.05711 -0.39718 -0.83083 -1.22130 -1.54828 -1.79480 -1.94802 TT. ... ..TT. ....... ......TT. ........... ..........TT. E2' * 2.00000 1.94802 1.79480 1.54828 1.22130 0.83083 0.39718 -0.05711 -0.50844 -0.93333 -1.30972 -1.61803 -1.84225 -1.97071 -1.99674 -1.91899 -1.74149 -1.47348 -1.12889 -0.72562 -0.28463 0.17115 0.61803 1.03279 1.39388 1.68251 1.88369 1.98696 2.00000 1.94802 1.54828 0.83083 -0.05711 -0.93333 -1.61803 -1.97071 -1.91899 -1.47348 -0.72562 0.17115 1.03279 1.68251 1.98696 1.88369 1.39388 0.61803 -0.28463 -1.12889 -1.74149 -1.99674 -1.84225 -1.30972 -0.50844 0.39718 1.22130 1.79480 ... ... TT... ....... ....TT... ........... ........TT... E2" * 2.00000 1.94802 1.79480 1.54828 1.22130 0.83083 0.39718 -0.05711 -0.50844 -0.93333 -1.30972 -1.61803 -1.84225 -1.97071 -1.99674 -1.91899 -1.74149 -1.47348 -1.12889 -0.72562 -0.28463 0.17115 0.61803 1.03279 1.39388 1.68251 1.88369 1.98696 -2.00000 -1.94802 -1.54828 -0.83083 0.05711 0.93333 1.61803 1.97071 1.91899 1.47348 0.72562 -0.17115 -1.03279 -1.68251 -1.98696 -1.88369 -1.39388 -0.61803 0.28463 1.12889 1.74149 1.99674 1.84225 1.30972 0.50844 -0.39718 -1.22130 -1.79480 ... ... ..... ..TT... ......... ......TT... ............. E3' * 2.00000 1.88369 1.54828 1.03279 0.39718 -0.28463 -0.93333 -1.47348 -1.84225 -1.99674 -1.91899 -1.61803 -1.12889 -0.50844 0.17115 0.83083 1.39388 1.79480 1.98696 1.94802 1.68251 1.22130 0.61803 -0.05711 -0.72562 -1.30972 -1.74149 -1.97071 2.00000 1.88369 1.03279 -0.28463 -1.47348 -1.99674 -1.61803 -0.50844 0.83083 1.79480 1.94802 1.22130 -0.05711 -1.30972 -1.97071 -1.74149 -0.72562 0.61803 1.68251 1.98696 1.39388 0.17115 -1.12889 -1.91899 -1.84225 -0.93333 0.39718 1.54828 ... ... ..... TT..... ......... ....TT..... ............. E3" * 2.00000 1.88369 1.54828 1.03279 0.39718 -0.28463 -0.93333 -1.47348 -1.84225 -1.99674 -1.91899 -1.61803 -1.12889 -0.50844 0.17115 0.83083 1.39388 1.79480 1.98696 1.94802 1.68251 1.22130 0.61803 -0.05711 -0.72562 -1.30972 -1.74149 -1.97071 -2.00000 -1.88369 -1.03279 0.28463 1.47348 1.99674 1.61803 0.50844 -0.83083 -1.79480 -1.94802 -1.22130 0.05711 1.30972 1.97071 1.74149 0.72562 -0.61803 -1.68251 -1.98696 -1.39388 -0.17115 1.12889 1.91899 1.84225 0.93333 -0.39718 -1.54828 ... ... ..... ....... ..TT..... ........... ......TT..... E4' * 2.00000 1.79480 1.22130 0.39718 -0.50844 -1.30972 -1.84225 -1.99674 -1.74149 -1.12889 -0.28463 0.61803 1.39388 1.88369 1.98696 1.68251 1.03279 0.17115 -0.72562 -1.47348 -1.91899 -1.97071 -1.61803 -0.93333 -0.05711 0.83083 1.54828 1.94802 2.00000 1.79480 0.39718 -1.30972 -1.99674 -1.12889 0.61803 1.88369 1.68251 0.17115 -1.47348 -1.97071 -0.93333 0.83083 1.94802 1.54828 -0.05711 -1.61803 -1.91899 -0.72562 1.03279 1.98696 1.39388 -0.28463 -1.74149 -1.84225 -0.50844 1.22130 ... ... ..... ....... TT....... ........... ....TT....... E4" * 2.00000 1.79480 1.22130 0.39718 -0.50844 -1.30972 -1.84225 -1.99674 -1.74149 -1.12889 -0.28463 0.61803 1.39388 1.88369 1.98696 1.68251 1.03279 0.17115 -0.72562 -1.47348 -1.91899 -1.97071 -1.61803 -0.93333 -0.05711 0.83083 1.54828 1.94802 -2.00000 -1.79480 -0.39718 1.30972 1.99674 1.12889 -0.61803 -1.88369 -1.68251 -0.17115 1.47348 1.97071 0.93333 -0.83083 -1.94802 -1.54828 0.05711 1.61803 1.91899 0.72562 -1.03279 -1.98696 -1.39388 0.28463 1.74149 1.84225 0.50844 -1.22130 ... ... ..... ....... ......... ..TT....... ............. E5' * 2.00000 1.68251 0.83083 -0.28463 -1.30972 -1.91899 -1.91899 -1.30972 -0.28463 0.83083 1.68251 2.00000 1.68251 0.83083 -0.28463 -1.30972 -1.91899 -1.91899 -1.30972 -0.28463 0.83083 1.68251 2.00000 1.68251 0.83083 -0.28463 -1.30972 -1.91899 2.00000 1.68251 -0.28463 -1.91899 -1.30972 0.83083 2.00000 0.83083 -1.30972 -1.91899 -0.28463 1.68251 1.68251 -0.28463 -1.91899 -1.30972 0.83083 2.00000 0.83083 -1.30972 -1.91899 -0.28463 1.68251 1.68251 -0.28463 -1.91899 -1.30972 0.83083 ... ... ..... ....... ......... TT......... ............. E5" * 2.00000 1.68251 0.83083 -0.28463 -1.30972 -1.91899 -1.91899 -1.30972 -0.28463 0.83083 1.68251 2.00000 1.68251 0.83083 -0.28463 -1.30972 -1.91899 -1.91899 -1.30972 -0.28463 0.83083 1.68251 2.00000 1.68251 0.83083 -0.28463 -1.30972 -1.91899 -2.00000 -1.68251 0.28463 1.91899 1.30972 -0.83083 -2.00000 -0.83083 1.30972 1.91899 0.28463 -1.68251 -1.68251 0.28463 1.91899 1.30972 -0.83083 -2.00000 -0.83083 1.30972 1.91899 0.28463 -1.68251 -1.68251 0.28463 1.91899 1.30972 -0.83083 ... ... ..... ....... ......... ........... ..TT......... E6' * 2.00000 1.54828 0.39718 -0.93333 -1.84225 -1.91899 -1.12889 0.17115 1.39388 1.98696 1.68251 0.61803 -0.72562 -1.74149 -1.97071 -1.30972 -0.05711 1.22130 1.94802 1.79480 0.83083 -0.50844 -1.61803 -1.99674 -1.47348 -0.28463 1.03279 1.88369 2.00000 1.54828 -0.93333 -1.91899 0.17115 1.98696 0.61803 -1.74149 -1.30972 1.22130 1.79480 -0.50844 -1.99674 -0.28463 1.88369 1.03279 -1.47348 -1.61803 0.83083 1.94802 -0.05711 -1.97071 -0.72562 1.68251 1.39388 -1.12889 -1.84225 0.39718 ... ... ..... ....... ......... ........... TT........... E6" * 2.00000 1.54828 0.39718 -0.93333 -1.84225 -1.91899 -1.12889 0.17115 1.39388 1.98696 1.68251 0.61803 -0.72562 -1.74149 -1.97071 -1.30972 -0.05711 1.22130 1.94802 1.79480 0.83083 -0.50844 -1.61803 -1.99674 -1.47348 -0.28463 1.03279 1.88369 -2.00000 -1.54828 0.93333 1.91899 -0.17115 -1.98696 -0.61803 1.74149 1.30972 -1.22130 -1.79480 0.50844 1.99674 0.28463 -1.88369 -1.03279 1.47348 1.61803 -0.83083 -1.94802 0.05711 1.97071 0.72562 -1.68251 -1.39388 1.12889 1.84225 -0.39718 ... ... ..... ....... ......... ........... ............. E7' * 2.00000 1.39388 -0.05711 -1.47348 -1.99674 -1.30972 0.17115 1.54828 1.98696 1.22130 -0.28463 -1.61803 -1.97071 -1.12889 0.39718 1.68251 1.94802 1.03279 -0.50844 -1.74149 -1.91899 -0.93333 0.61803 1.79480 1.88369 0.83083 -0.72562 -1.84225 2.00000 1.39388 -1.47348 -1.30972 1.54828 1.22130 -1.61803 -1.12889 1.68251 1.03279 -1.74149 -0.93333 1.79480 0.83083 -1.84225 -0.72562 1.88369 0.61803 -1.91899 -0.50844 1.94802 0.39718 -1.97071 -0.28463 1.98696 0.17115 -1.99674 -0.05711 ... ... ..... ....... ......... ........... ............. E7" * 2.00000 1.39388 -0.05711 -1.47348 -1.99674 -1.30972 0.17115 1.54828 1.98696 1.22130 -0.28463 -1.61803 -1.97071 -1.12889 0.39718 1.68251 1.94802 1.03279 -0.50844 -1.74149 -1.91899 -0.93333 0.61803 1.79480 1.88369 0.83083 -0.72562 -1.84225 -2.00000 -1.39388 1.47348 1.30972 -1.54828 -1.22130 1.61803 1.12889 -1.68251 -1.03279 1.74149 0.93333 -1.79480 -0.83083 1.84225 0.72562 -1.88369 -0.61803 1.91899 0.50844 -1.94802 -0.39718 1.97071 0.28463 -1.98696 -0.17115 1.99674 0.05711 ... ... ..... ....... ......... ........... ............. E8' * 2.00000 1.22130 -0.50844 -1.84225 -1.74149 -0.28463 1.39388 1.98696 1.03279 -0.72562 -1.91899 -1.61803 -0.05711 1.54828 1.94802 0.83083 -0.93333 -1.97071 -1.47348 0.17115 1.68251 1.88369 0.61803 -1.12889 -1.99674 -1.30972 0.39718 1.79480 2.00000 1.22130 -1.84225 -0.28463 1.98696 -0.72562 -1.61803 1.54828 0.83083 -1.97071 0.17115 1.88369 -1.12889 -1.30972 1.79480 0.39718 -1.99674 0.61803 1.68251 -1.47348 -0.93333 1.94802 -0.05711 -1.91899 1.03279 1.39388 -1.74149 -0.50844 ... ... ..... ....... ......... ........... ............. E8" * 2.00000 1.22130 -0.50844 -1.84225 -1.74149 -0.28463 1.39388 1.98696 1.03279 -0.72562 -1.91899 -1.61803 -0.05711 1.54828 1.94802 0.83083 -0.93333 -1.97071 -1.47348 0.17115 1.68251 1.88369 0.61803 -1.12889 -1.99674 -1.30972 0.39718 1.79480 -2.00000 -1.22130 1.84225 0.28463 -1.98696 0.72562 1.61803 -1.54828 -0.83083 1.97071 -0.17115 -1.88369 1.12889 1.30972 -1.79480 -0.39718 1.99674 -0.61803 -1.68251 1.47348 0.93333 -1.94802 0.05711 1.91899 -1.03279 -1.39388 1.74149 0.50844 ... ... ..... ....... ......... ........... ............. E9' * 2.00000 1.03279 -0.93333 -1.99674 -1.12889 0.83083 1.98696 1.22130 -0.72562 -1.97071 -1.30972 0.61803 1.94802 1.39388 -0.50844 -1.91899 -1.47348 0.39718 1.88369 1.54828 -0.28463 -1.84225 -1.61803 0.17115 1.79480 1.68251 -0.05711 -1.74149 2.00000 1.03279 -1.99674 0.83083 1.22130 -1.97071 0.61803 1.39388 -1.91899 0.39718 1.54828 -1.84225 0.17115 1.68251 -1.74149 -0.05711 1.79480 -1.61803 -0.28463 1.88369 -1.47348 -0.50844 1.94802 -1.30972 -0.72562 1.98696 -1.12889 -0.93333 ... ... ..... ....... ......... ........... ............. E9" * 2.00000 1.03279 -0.93333 -1.99674 -1.12889 0.83083 1.98696 1.22130 -0.72562 -1.97071 -1.30972 0.61803 1.94802 1.39388 -0.50844 -1.91899 -1.47348 0.39718 1.88369 1.54828 -0.28463 -1.84225 -1.61803 0.17115 1.79480 1.68251 -0.05711 -1.74149 -2.00000 -1.03279 1.99674 -0.83083 -1.22130 1.97071 -0.61803 -1.39388 1.91899 -0.39718 -1.54828 1.84225 -0.17115 -1.68251 1.74149 0.05711 -1.79480 1.61803 0.28463 -1.88369 1.47348 0.50844 -1.94802 1.30972 0.72562 -1.98696 1.12889 0.93333 ... ... ..... ....... ......... ........... ............. E10' * 2.00000 0.83083 -1.30972 -1.91899 -0.28463 1.68251 1.68251 -0.28463 -1.91899 -1.30972 0.83083 2.00000 0.83083 -1.30972 -1.91899 -0.28463 1.68251 1.68251 -0.28463 -1.91899 -1.30972 0.83083 2.00000 0.83083 -1.30972 -1.91899 -0.28463 1.68251 2.00000 0.83083 -1.91899 1.68251 -0.28463 -1.30972 2.00000 -1.30972 -0.28463 1.68251 -1.91899 0.83083 0.83083 -1.91899 1.68251 -0.28463 -1.30972 2.00000 -1.30972 -0.28463 1.68251 -1.91899 0.83083 0.83083 -1.91899 1.68251 -0.28463 -1.30972 ... ... ..... ....... ......... ........... ............. E10" * 2.00000 0.83083 -1.30972 -1.91899 -0.28463 1.68251 1.68251 -0.28463 -1.91899 -1.30972 0.83083 2.00000 0.83083 -1.30972 -1.91899 -0.28463 1.68251 1.68251 -0.28463 -1.91899 -1.30972 0.83083 2.00000 0.83083 -1.30972 -1.91899 -0.28463 1.68251 -2.00000 -0.83083 1.91899 -1.68251 0.28463 1.30972 -2.00000 1.30972 0.28463 -1.68251 1.91899 -0.83083 -0.83083 1.91899 -1.68251 0.28463 1.30972 -2.00000 1.30972 0.28463 -1.68251 1.91899 -0.83083 -0.83083 1.91899 -1.68251 0.28463 1.30972 ... ... ..... ....... ......... ........... ............. E11' * 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 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 2.00000 -1.61803 0.61803 0.61803 -1.61803 ... ... ..... ....... ......... ........... ............. E11" * 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 -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 -2.00000 1.61803 -0.61803 -0.61803 1.61803 ... ... ..... ....... ......... ........... ............. E12' * 2.00000 0.39718 -1.84225 -1.12889 1.39388 1.68251 -0.72562 -1.97071 -0.05711 1.94802 0.83083 -1.61803 -1.47348 1.03279 1.88369 -0.28463 -1.99674 -0.50844 1.79480 1.22130 -1.30972 -1.74149 0.61803 1.98696 0.17115 -1.91899 -0.93333 1.54828 2.00000 0.39718 -1.12889 1.68251 -1.97071 1.94802 -1.61803 1.03279 -0.28463 -0.50844 1.22130 -1.74149 1.98696 -1.91899 1.54828 -0.93333 0.17115 0.61803 -1.30972 1.79480 -1.99674 1.88369 -1.47348 0.83083 -0.05711 -0.72562 1.39388 -1.84225 ... ... ..... ....... ......... ........... ............. E12" * 2.00000 0.39718 -1.84225 -1.12889 1.39388 1.68251 -0.72562 -1.97071 -0.05711 1.94802 0.83083 -1.61803 -1.47348 1.03279 1.88369 -0.28463 -1.99674 -0.50844 1.79480 1.22130 -1.30972 -1.74149 0.61803 1.98696 0.17115 -1.91899 -0.93333 1.54828 -2.00000 -0.39718 1.12889 -1.68251 1.97071 -1.94802 1.61803 -1.03279 0.28463 0.50844 -1.22130 1.74149 -1.98696 1.91899 -1.54828 0.93333 -0.17115 -0.61803 1.30972 -1.79480 1.99674 -1.88369 1.47348 -0.83083 0.05711 0.72562 -1.39388 1.84225 ... ... ..... ....... ......... ........... ............. E13' * 2.00000 0.17115 -1.97071 -0.50844 1.88369 0.83083 -1.74149 -1.12889 1.54828 1.39388 -1.30972 -1.61803 1.03279 1.79480 -0.72562 -1.91899 0.39718 1.98696 -0.05711 -1.99674 -0.28463 1.94802 0.61803 -1.84225 -0.93333 1.68251 1.22130 -1.47348 2.00000 0.17115 -0.50844 0.83083 -1.12889 1.39388 -1.61803 1.79480 -1.91899 1.98696 -1.99674 1.94802 -1.84225 1.68251 -1.47348 1.22130 -0.93333 0.61803 -0.28463 -0.05711 0.39718 -0.72562 1.03279 -1.30972 1.54828 -1.74149 1.88369 -1.97071 ... ... ..... ....... ......... ........... ............. E13" * 2.00000 0.17115 -1.97071 -0.50844 1.88369 0.83083 -1.74149 -1.12889 1.54828 1.39388 -1.30972 -1.61803 1.03279 1.79480 -0.72562 -1.91899 0.39718 1.98696 -0.05711 -1.99674 -0.28463 1.94802 0.61803 -1.84225 -0.93333 1.68251 1.22130 -1.47348 -2.00000 -0.17115 0.50844 -0.83083 1.12889 -1.39388 1.61803 -1.79480 1.91899 -1.98696 1.99674 -1.94802 1.84225 -1.68251 1.47348 -1.22130 0.93333 -0.61803 0.28463 0.05711 -0.39718 0.72562 -1.03279 1.30972 -1.54828 1.74149 -1.88369 1.97071 ... ... ..... ....... ......... ........... ............. E14' * 2.00000 -0.05711 -1.99674 0.17115 1.98696 -0.28463 -1.97071 0.39718 1.94802 -0.50844 -1.91899 0.61803 1.88369 -0.72562 -1.84225 0.83083 1.79480 -0.93333 -1.74149 1.03279 1.68251 -1.12889 -1.61803 1.22130 1.54828 -1.30972 -1.47348 1.39388 2.00000 -0.05711 0.17115 -0.28463 0.39718 -0.50844 0.61803 -0.72562 0.83083 -0.93333 1.03279 -1.12889 1.22130 -1.30972 1.39388 -1.47348 1.54828 -1.61803 1.68251 -1.74149 1.79480 -1.84225 1.88369 -1.91899 1.94802 -1.97071 1.98696 -1.99674 ... ... ..... ....... ......... ........... ............. E14" * 2.00000 -0.05711 -1.99674 0.17115 1.98696 -0.28463 -1.97071 0.39718 1.94802 -0.50844 -1.91899 0.61803 1.88369 -0.72562 -1.84225 0.83083 1.79480 -0.93333 -1.74149 1.03279 1.68251 -1.12889 -1.61803 1.22130 1.54828 -1.30972 -1.47348 1.39388 -2.00000 0.05711 -0.17115 0.28463 -0.39718 0.50844 -0.61803 0.72562 -0.83083 0.93333 -1.03279 1.12889 -1.22130 1.30972 -1.39388 1.47348 -1.54828 1.61803 -1.68251 1.74149 -1.79480 1.84225 -1.88369 1.91899 -1.94802 1.97071 -1.98696 1.99674 ... ... ..... ....... ......... ........... ............. E15' * 2.00000 -0.28463 -1.91899 0.83083 1.68251 -1.30972 -1.30972 1.68251 0.83083 -1.91899 -0.28463 2.00000 -0.28463 -1.91899 0.83083 1.68251 -1.30972 -1.30972 1.68251 0.83083 -1.91899 -0.28463 2.00000 -0.28463 -1.91899 0.83083 1.68251 -1.30972 2.00000 -0.28463 0.83083 -1.30972 1.68251 -1.91899 2.00000 -1.91899 1.68251 -1.30972 0.83083 -0.28463 -0.28463 0.83083 -1.30972 1.68251 -1.91899 2.00000 -1.91899 1.68251 -1.30972 0.83083 -0.28463 -0.28463 0.83083 -1.30972 1.68251 -1.91899 ... ... ..... ....... ......... ........... ............. E15" * 2.00000 -0.28463 -1.91899 0.83083 1.68251 -1.30972 -1.30972 1.68251 0.83083 -1.91899 -0.28463 2.00000 -0.28463 -1.91899 0.83083 1.68251 -1.30972 -1.30972 1.68251 0.83083 -1.91899 -0.28463 2.00000 -0.28463 -1.91899 0.83083 1.68251 -1.30972 -2.00000 0.28463 -0.83083 1.30972 -1.68251 1.91899 -2.00000 1.91899 -1.68251 1.30972 -0.83083 0.28463 0.28463 -0.83083 1.30972 -1.68251 1.91899 -2.00000 1.91899 -1.68251 1.30972 -0.83083 0.28463 0.28463 -0.83083 1.30972 -1.68251 1.91899 ... ... ..... ....... ......... ........... ............. E16' * 2.00000 -0.50844 -1.74149 1.39388 1.03279 -1.91899 -0.05711 1.94802 -0.93333 -1.47348 1.68251 0.61803 -1.99674 0.39718 1.79480 -1.30972 -1.12889 1.88369 0.17115 -1.97071 0.83083 1.54828 -1.61803 -0.72562 1.98696 -0.28463 -1.84225 1.22130 2.00000 -0.50844 1.39388 -1.91899 1.94802 -1.47348 0.61803 0.39718 -1.30972 1.88369 -1.97071 1.54828 -0.72562 -0.28463 1.22130 -1.84225 1.98696 -1.61803 0.83083 0.17115 -1.12889 1.79480 -1.99674 1.68251 -0.93333 -0.05711 1.03279 -1.74149 ... ... ..... ....... ......... ........... ............. E16" * 2.00000 -0.50844 -1.74149 1.39388 1.03279 -1.91899 -0.05711 1.94802 -0.93333 -1.47348 1.68251 0.61803 -1.99674 0.39718 1.79480 -1.30972 -1.12889 1.88369 0.17115 -1.97071 0.83083 1.54828 -1.61803 -0.72562 1.98696 -0.28463 -1.84225 1.22130 -2.00000 0.50844 -1.39388 1.91899 -1.94802 1.47348 -0.61803 -0.39718 1.30972 -1.88369 1.97071 -1.54828 0.72562 0.28463 -1.22130 1.84225 -1.98696 1.61803 -0.83083 -0.17115 1.12889 -1.79480 1.99674 -1.68251 0.93333 0.05711 -1.03279 1.74149 ... ... ..... ....... ......... ........... ............. E17' * 2.00000 -0.72562 -1.47348 1.79480 0.17115 -1.91899 1.22130 1.03279 -1.97071 0.39718 1.68251 -1.61803 -0.50844 1.98696 -0.93333 -1.30972 1.88369 -0.05711 -1.84225 1.39388 0.83083 -1.99674 0.61803 1.54828 -1.74149 -0.28463 1.94802 -1.12889 2.00000 -0.72562 1.79480 -1.91899 1.03279 0.39718 -1.61803 1.98696 -1.30972 -0.05711 1.39388 -1.99674 1.54828 -0.28463 -1.12889 1.94802 -1.74149 0.61803 0.83083 -1.84225 1.88369 -0.93333 -0.50844 1.68251 -1.97071 1.22130 0.17115 -1.47348 ... ... ..... ....... ......... ........... ............. E17" * 2.00000 -0.72562 -1.47348 1.79480 0.17115 -1.91899 1.22130 1.03279 -1.97071 0.39718 1.68251 -1.61803 -0.50844 1.98696 -0.93333 -1.30972 1.88369 -0.05711 -1.84225 1.39388 0.83083 -1.99674 0.61803 1.54828 -1.74149 -0.28463 1.94802 -1.12889 -2.00000 0.72562 -1.79480 1.91899 -1.03279 -0.39718 1.61803 -1.98696 1.30972 0.05711 -1.39388 1.99674 -1.54828 0.28463 1.12889 -1.94802 1.74149 -0.61803 -0.83083 1.84225 -1.88369 0.93333 0.50844 -1.68251 1.97071 -1.22130 -0.17115 1.47348 ... ... ..... ....... ......... ........... ............. E18' * 2.00000 -0.93333 -1.12889 1.98696 -0.72562 -1.30972 1.94802 -0.50844 -1.47348 1.88369 -0.28463 -1.61803 1.79480 -0.05711 -1.74149 1.68251 0.17115 -1.84225 1.54828 0.39718 -1.91899 1.39388 0.61803 -1.97071 1.22130 0.83083 -1.99674 1.03279 2.00000 -0.93333 1.98696 -1.30972 -0.50844 1.88369 -1.61803 -0.05711 1.68251 -1.84225 0.39718 1.39388 -1.97071 0.83083 1.03279 -1.99674 1.22130 0.61803 -1.91899 1.54828 0.17115 -1.74149 1.79480 -0.28463 -1.47348 1.94802 -0.72562 -1.12889 ... ... ..... ....... ......... ........... ............. E18" * 2.00000 -0.93333 -1.12889 1.98696 -0.72562 -1.30972 1.94802 -0.50844 -1.47348 1.88369 -0.28463 -1.61803 1.79480 -0.05711 -1.74149 1.68251 0.17115 -1.84225 1.54828 0.39718 -1.91899 1.39388 0.61803 -1.97071 1.22130 0.83083 -1.99674 1.03279 -2.00000 0.93333 -1.98696 1.30972 0.50844 -1.88369 1.61803 0.05711 -1.68251 1.84225 -0.39718 -1.39388 1.97071 -0.83083 -1.03279 1.99674 -1.22130 -0.61803 1.91899 -1.54828 -0.17115 1.74149 -1.79480 0.28463 1.47348 -1.94802 0.72562 1.12889 ... ... ..... ....... ......... ........... ............. E19' * 2.00000 -1.12889 -0.72562 1.94802 -1.47348 -0.28463 1.79480 -1.74149 0.17115 1.54828 -1.91899 0.61803 1.22130 -1.99674 1.03279 0.83083 -1.97071 1.39388 0.39718 -1.84225 1.68251 -0.05711 -1.61803 1.88369 -0.50844 -1.30972 1.98696 -0.93333 2.00000 -1.12889 1.94802 -0.28463 -1.74149 1.54828 0.61803 -1.99674 0.83083 1.39388 -1.84225 -0.05711 1.88369 -1.30972 -0.93333 1.98696 -0.50844 -1.61803 1.68251 0.39718 -1.97071 1.03279 1.22130 -1.91899 0.17115 1.79480 -1.47348 -0.72562 ... ... ..... ....... ......... ........... ............. E19" * 2.00000 -1.12889 -0.72562 1.94802 -1.47348 -0.28463 1.79480 -1.74149 0.17115 1.54828 -1.91899 0.61803 1.22130 -1.99674 1.03279 0.83083 -1.97071 1.39388 0.39718 -1.84225 1.68251 -0.05711 -1.61803 1.88369 -0.50844 -1.30972 1.98696 -0.93333 -2.00000 1.12889 -1.94802 0.28463 1.74149 -1.54828 -0.61803 1.99674 -0.83083 -1.39388 1.84225 0.05711 -1.88369 1.30972 0.93333 -1.98696 0.50844 1.61803 -1.68251 -0.39718 1.97071 -1.03279 -1.22130 1.91899 -0.17115 -1.79480 1.47348 0.72562 ... ... ..... ....... ......... ........... ............. E20' * 2.00000 -1.30972 -0.28463 1.68251 -1.91899 0.83083 0.83083 -1.91899 1.68251 -0.28463 -1.30972 2.00000 -1.30972 -0.28463 1.68251 -1.91899 0.83083 0.83083 -1.91899 1.68251 -0.28463 -1.30972 2.00000 -1.30972 -0.28463 1.68251 -1.91899 0.83083 2.00000 -1.30972 1.68251 0.83083 -1.91899 -0.28463 2.00000 -0.28463 -1.91899 0.83083 1.68251 -1.30972 -1.30972 1.68251 0.83083 -1.91899 -0.28463 2.00000 -0.28463 -1.91899 0.83083 1.68251 -1.30972 -1.30972 1.68251 0.83083 -1.91899 -0.28463 ... ... ..... ....... ......... ........... ............. E20" * 2.00000 -1.30972 -0.28463 1.68251 -1.91899 0.83083 0.83083 -1.91899 1.68251 -0.28463 -1.30972 2.00000 -1.30972 -0.28463 1.68251 -1.91899 0.83083 0.83083 -1.91899 1.68251 -0.28463 -1.30972 2.00000 -1.30972 -0.28463 1.68251 -1.91899 0.83083 -2.00000 1.30972 -1.68251 -0.83083 1.91899 0.28463 -2.00000 0.28463 1.91899 -0.83083 -1.68251 1.30972 1.30972 -1.68251 -0.83083 1.91899 0.28463 -2.00000 0.28463 1.91899 -0.83083 -1.68251 1.30972 1.30972 -1.68251 -0.83083 1.91899 0.28463 ... ... ..... ....... ......... ........... ............. E21' * 2.00000 -1.47348 0.17115 1.22130 -1.97071 1.68251 -0.50844 -0.93333 1.88369 -1.84225 0.83083 0.61803 -1.74149 1.94802 -1.12889 -0.28463 1.54828 -1.99674 1.39388 -0.05711 -1.30972 1.98696 -1.61803 0.39718 1.03279 -1.91899 1.79480 -0.72562 2.00000 -1.47348 1.22130 1.68251 -0.93333 -1.84225 0.61803 1.94802 -0.28463 -1.99674 -0.05711 1.98696 0.39718 -1.91899 -0.72562 1.79480 1.03279 -1.61803 -1.30972 1.39388 1.54828 -1.12889 -1.74149 0.83083 1.88369 -0.50844 -1.97071 0.17115 ... ... ..... ....... ......... ........... ............. E21" * 2.00000 -1.47348 0.17115 1.22130 -1.97071 1.68251 -0.50844 -0.93333 1.88369 -1.84225 0.83083 0.61803 -1.74149 1.94802 -1.12889 -0.28463 1.54828 -1.99674 1.39388 -0.05711 -1.30972 1.98696 -1.61803 0.39718 1.03279 -1.91899 1.79480 -0.72562 -2.00000 1.47348 -1.22130 -1.68251 0.93333 1.84225 -0.61803 -1.94802 0.28463 1.99674 0.05711 -1.98696 -0.39718 1.91899 0.72562 -1.79480 -1.03279 1.61803 1.30972 -1.39388 -1.54828 1.12889 1.74149 -0.83083 -1.88369 0.50844 1.97071 -0.17115 ... ... ..... ....... ......... ........... ............. E22' * 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 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 2.00000 0.61803 -1.61803 -1.61803 0.61803 ... ... ..... ....... ......... ........... ............. E22" * 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 -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 -2.00000 -0.61803 1.61803 1.61803 -0.61803 ... ... ..... ....... ......... ........... ............. E23' * 2.00000 -1.74149 1.03279 -0.05711 -0.93333 1.68251 -1.99674 1.79480 -1.12889 0.17115 0.83083 -1.61803 1.98696 -1.84225 1.22130 -0.28463 -0.72562 1.54828 -1.97071 1.88369 -1.30972 0.39718 0.61803 -1.47348 1.94802 -1.91899 1.39388 -0.50844 2.00000 -1.74149 -0.05711 1.68251 1.79480 0.17115 -1.61803 -1.84225 -0.28463 1.54828 1.88369 0.39718 -1.47348 -1.91899 -0.50844 1.39388 1.94802 0.61803 -1.30972 -1.97071 -0.72562 1.22130 1.98696 0.83083 -1.12889 -1.99674 -0.93333 1.03279 ... ... ..... ....... ......... ........... ............. E23" * 2.00000 -1.74149 1.03279 -0.05711 -0.93333 1.68251 -1.99674 1.79480 -1.12889 0.17115 0.83083 -1.61803 1.98696 -1.84225 1.22130 -0.28463 -0.72562 1.54828 -1.97071 1.88369 -1.30972 0.39718 0.61803 -1.47348 1.94802 -1.91899 1.39388 -0.50844 -2.00000 1.74149 0.05711 -1.68251 -1.79480 -0.17115 1.61803 1.84225 0.28463 -1.54828 -1.88369 -0.39718 1.47348 1.91899 0.50844 -1.39388 -1.94802 -0.61803 1.30972 1.97071 0.72562 -1.22130 -1.98696 -0.83083 1.12889 1.99674 0.93333 -1.03279 ... ... ..... ....... ......... ........... ............. E24' * 2.00000 -1.84225 1.39388 -0.72562 -0.05711 0.83083 -1.47348 1.88369 -1.99674 1.79480 -1.30972 0.61803 0.17115 -0.93333 1.54828 -1.91899 1.98696 -1.74149 1.22130 -0.50844 -0.28463 1.03279 -1.61803 1.94802 -1.97071 1.68251 -1.12889 0.39718 2.00000 -1.84225 -0.72562 0.83083 1.88369 1.79480 0.61803 -0.93333 -1.91899 -1.74149 -0.50844 1.03279 1.94802 1.68251 0.39718 -1.12889 -1.97071 -1.61803 -0.28463 1.22130 1.98696 1.54828 0.17115 -1.30972 -1.99674 -1.47348 -0.05711 1.39388 ... ... ..... ....... ......... ........... ............. E24" * 2.00000 -1.84225 1.39388 -0.72562 -0.05711 0.83083 -1.47348 1.88369 -1.99674 1.79480 -1.30972 0.61803 0.17115 -0.93333 1.54828 -1.91899 1.98696 -1.74149 1.22130 -0.50844 -0.28463 1.03279 -1.61803 1.94802 -1.97071 1.68251 -1.12889 0.39718 -2.00000 1.84225 0.72562 -0.83083 -1.88369 -1.79480 -0.61803 0.93333 1.91899 1.74149 0.50844 -1.03279 -1.94802 -1.68251 -0.39718 1.12889 1.97071 1.61803 0.28463 -1.22130 -1.98696 -1.54828 -0.17115 1.30972 1.99674 1.47348 0.05711 -1.39388 ... ... ..... ....... ......... ........... ............. E25' * 2.00000 -1.91899 1.68251 -1.30972 0.83083 -0.28463 -0.28463 0.83083 -1.30972 1.68251 -1.91899 2.00000 -1.91899 1.68251 -1.30972 0.83083 -0.28463 -0.28463 0.83083 -1.30972 1.68251 -1.91899 2.00000 -1.91899 1.68251 -1.30972 0.83083 -0.28463 2.00000 -1.91899 -1.30972 -0.28463 0.83083 1.68251 2.00000 1.68251 0.83083 -0.28463 -1.30972 -1.91899 -1.91899 -1.30972 -0.28463 0.83083 1.68251 2.00000 1.68251 0.83083 -0.28463 -1.30972 -1.91899 -1.91899 -1.30972 -0.28463 0.83083 1.68251 ... ... ..... ....... ......... ........... ............. E25" * 2.00000 -1.91899 1.68251 -1.30972 0.83083 -0.28463 -0.28463 0.83083 -1.30972 1.68251 -1.91899 2.00000 -1.91899 1.68251 -1.30972 0.83083 -0.28463 -0.28463 0.83083 -1.30972 1.68251 -1.91899 2.00000 -1.91899 1.68251 -1.30972 0.83083 -0.28463 -2.00000 1.91899 1.30972 0.28463 -0.83083 -1.68251 -2.00000 -1.68251 -0.83083 0.28463 1.30972 1.91899 1.91899 1.30972 0.28463 -0.83083 -1.68251 -2.00000 -1.68251 -0.83083 0.28463 1.30972 1.91899 1.91899 1.30972 0.28463 -0.83083 -1.68251 ... ... ..... ....... ......... ........... ............. E26' * 2.00000 -1.97071 1.88369 -1.74149 1.54828 -1.30972 1.03279 -0.72562 0.39718 -0.05711 -0.28463 0.61803 -0.93333 1.22130 -1.47348 1.68251 -1.84225 1.94802 -1.99674 1.98696 -1.91899 1.79480 -1.61803 1.39388 -1.12889 0.83083 -0.50844 0.17115 2.00000 -1.97071 -1.74149 -1.30972 -0.72562 -0.05711 0.61803 1.22130 1.68251 1.94802 1.98696 1.79480 1.39388 0.83083 0.17115 -0.50844 -1.12889 -1.61803 -1.91899 -1.99674 -1.84225 -1.47348 -0.93333 -0.28463 0.39718 1.03279 1.54828 1.88369 ... ... ..... ....... ......... ........... ............. E26" * 2.00000 -1.97071 1.88369 -1.74149 1.54828 -1.30972 1.03279 -0.72562 0.39718 -0.05711 -0.28463 0.61803 -0.93333 1.22130 -1.47348 1.68251 -1.84225 1.94802 -1.99674 1.98696 -1.91899 1.79480 -1.61803 1.39388 -1.12889 0.83083 -0.50844 0.17115 -2.00000 1.97071 1.74149 1.30972 0.72562 0.05711 -0.61803 -1.22130 -1.68251 -1.94802 -1.98696 -1.79480 -1.39388 -0.83083 -0.17115 0.50844 1.12889 1.61803 1.91899 1.99674 1.84225 1.47348 0.93333 0.28463 -0.39718 -1.03279 -1.54828 -1.88369 ... ... ..... ....... ......... ........... ............. E27' * 2.00000 -1.99674 1.98696 -1.97071 1.94802 -1.91899 1.88369 -1.84225 1.79480 -1.74149 1.68251 -1.61803 1.54828 -1.47348 1.39388 -1.30972 1.22130 -1.12889 1.03279 -0.93333 0.83083 -0.72562 0.61803 -0.50844 0.39718 -0.28463 0.17115 -0.05711 2.00000 -1.99674 -1.97071 -1.91899 -1.84225 -1.74149 -1.61803 -1.47348 -1.30972 -1.12889 -0.93333 -0.72562 -0.50844 -0.28463 -0.05711 0.17115 0.39718 0.61803 0.83083 1.03279 1.22130 1.39388 1.54828 1.68251 1.79480 1.88369 1.94802 1.98696 ... ... ..... ....... ......... ........... ............. E27" * 2.00000 -1.99674 1.98696 -1.97071 1.94802 -1.91899 1.88369 -1.84225 1.79480 -1.74149 1.68251 -1.61803 1.54828 -1.47348 1.39388 -1.30972 1.22130 -1.12889 1.03279 -0.93333 0.83083 -0.72562 0.61803 -0.50844 0.39718 -0.28463 0.17115 -0.05711 -2.00000 1.99674 1.97071 1.91899 1.84225 1.74149 1.61803 1.47348 1.30972 1.12889 0.93333 0.72562 0.50844 0.28463 0.05711 -0.17115 -0.39718 -0.61803 -0.83083 -1.03279 -1.22130 -1.39388 -1.54828 -1.68251 -1.79480 -1.88369 -1.94802 -1.98696 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.996738207852 = 2*cos(2*π/110) = 2*cos(π/55) 1.986963470697 = 2*cos(4*π/110) = 2*cos(2*π/55) 1.970707671695 = 2*cos(6*π/110) = 2*cos(3*π/55) 1.948023833885 = 2*cos(8*π/110) = 2*cos(4*π/55) 1.918985947229 = 2*cos(10*π/110) = 2*cos(π/11) 1.883688727279 = 2*cos(12*π/110) = 2*cos(6*π/55) 1.842247306230 = 2*cos(14*π/110) = 2*cos(7*π/55) 1.794796857383 = 2*cos(16*π/110) = 2*cos(8*π/55) 1.741492154240 = 2*cos(18*π/110) = 2*cos(9*π/55) 1.682507065662 = 2*cos(20*π/110) = 2*cos(2*π/11) 1.618033988750 = 2*cos(22*π/110) = 2*cos(π/5) = (√5+1)/2 1.548283221278 = 2*cos(24*π/110) = 2*cos(12*π/55) 1.473482275753 = 2*cos(26*π/110) = 2*cos(13*π/55) 1.393875137311 = 2*cos(28*π/110) = 2*cos(14*π/55) 1.309721467891 = 2*cos(30*π/110) = 2*cos(3*π/11) 1.221295759271 = 2*cos(32*π/110) = 2*cos(16*π/55) 1.128886437734 = 2*cos(34*π/110) = 2*cos(17*π/55) 1.032794923278 = 2*cos(36*π/110) = 2*cos(18*π/55) 0.933334646451 = 2*cos(38*π/110) = 2*cos(19*π/55) 0.830830026004 = 2*cos(40*π/110) = 2*cos(4*π/11) 0.725615410701 = 2*cos(42*π/110) = 2*cos(21*π/55) 0.618033988750 = 2*cos(44*π/110) = 2*cos(2*π/5) = (√5−1)/2 0.508436668387 = 2*cos(46*π/110) = 2*cos(23*π/55) 0.397180933291 = 2*cos(48*π/110) = 2*cos(24*π/55) 0.284629676547 = 2*cos(50*π/110) = 2*cos(5*π/11) 0.171150016958 = 2*cos(52*π/110) = 2*cos(26*π/55) 0.057112101587 = 2*cos(54*π/110) = 2*cos(27*π/55) 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_55h point group is 110, and the order of the principal axis (S₅₅) is 110. The group has 56 irreducible representations. β The C_55h point group could also be named S₅₅, as it contains the S₅₅ axis as its only symmetry element. Another rare designation is C_110i because the S₅₅ axis is identical to a roto-inversion axis of order 110. γ The C_55h point group is isomorphic to C₁₁₀ and S₁₁₀. δ The C_55h point group is generated by one single symmetry element, S₅₅. Therefore, it is a cyclic group. The canonical choice, however, is to use redundant generators: C₅₅ and σ_h. ε The lowest nonvanishing multipole moment in C_55h is 4 (quadrupole moment). ζ This is an Abelian point group (the commutative law holds between all symmetry operations). The C_55h 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 108 cases have been combined into 54 two-dimensional representations that are no longer irreducible but have real-valued characters. Accordingly, 54 pairs of left and right rotations have been combined into one two-membered pseudo-class each. θ The 54 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. 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 C_55h point group was created by Gernot Katzer.

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

	C_53h
	C_54h
C₅₅ C_55v	C_55h	D₅₅ D_55h D_55d
	C_56h
	C_57h

Character table for the C55h point group

Character table for the C_55h point group