S110 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 i 2 S110 2 S110^3 2 S22 2 S110^7 2 S110^9 2 S10 2 S110^13 2 S110^15 2 S110^17 2 S110^19 2 S110^21 2 S110^23 2 S110^25 2 S110^27 2 S110^29 2 S110^31 2 S110^33 2 S110^35 2 S110^37 2 S110^39 2 S110^41 2 S110^43 2 S110^45 2 S110^47 2 S110^49 2 S110^51 2 S110^53 <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> Ag 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 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 E1g * 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 -1.99674 -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 TT. ... ..TT. ....... ......TT. ........... ..........TT. E2g * 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 1.98696 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... E3g * 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 -1.97071 -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 ... ... ..... ....... ..TT..... ........... ......TT..... E4g * 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 1.94802 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....... E5g * 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 -1.91899 -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 ... ... ..... ....... ......... ........... ..TT......... E6g * 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 1.88369 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........... E7g * 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 -1.84225 -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 ... ... ..... ....... ......... ........... ............. E8g * 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 1.79480 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 ... ... ..... ....... ......... ........... ............. E9g * 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 -1.74149 -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 ... ... ..... ....... ......... ........... ............. E10g * 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 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 ... ... ..... ....... ......... ........... ............. E11g * 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 -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 ... ... ..... ....... ......... ........... ............. E12g * 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 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 ... ... ..... ....... ......... ........... ............. E13g * 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 -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 ... ... ..... ....... ......... ........... ............. E14g * 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 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 ... ... ..... ....... ......... ........... ............. E15g * 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 -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 ... ... ..... ....... ......... ........... ............. E16g * 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 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 ... ... ..... ....... ......... ........... ............. E17g * 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 -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 ... ... ..... ....... ......... ........... ............. E18g * 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 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 ... ... ..... ....... ......... ........... ............. E19g * 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 -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 ... ... ..... ....... ......... ........... ............. E20g * 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 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 ... ... ..... ....... ......... ........... ............. E21g * 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 -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 ... ... ..... ....... ......... ........... ............. E22g * 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 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 ... ... ..... ....... ......... ........... ............. E23g * 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 -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 ... ... ..... ....... ......... ........... ............. E24g * 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 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 ... ... ..... ....... ......... ........... ............. E25g * 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 -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 ... ... ..... ....... ......... ........... ............. E26g * 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 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 ... ... ..... ....... ......... ........... ............. E27g * 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 -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 ... ... ..... ....... ......... ........... ............. Au 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -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 ............. E1u * 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 1.99674 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 ... TT. ..... ....TT. ......... ........TT. ............. E2u * 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 -1.98696 -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... ............. E3u * 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 1.97071 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 ... ... ..... TT..... ......... ....TT..... ............. E4u * 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 -1.94802 -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....... ............. E5u * 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 1.91899 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 ... ... ..... ....... ......... TT......... ............. E6u * 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 -1.88369 -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 ... ... ..... ....... ......... ........... ............. E7u * 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 1.84225 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 ... ... ..... ....... ......... ........... ............. E8u * 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 -1.79480 -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 ... ... ..... ....... ......... ........... ............. E9u * 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 1.74149 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 ... ... ..... ....... ......... ........... ............. E10u * 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 -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 ... ... ..... ....... ......... ........... ............. E11u * 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 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 ... ... ..... ....... ......... ........... ............. E12u * 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 -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 ... ... ..... ....... ......... ........... ............. E13u * 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 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 ... ... ..... ....... ......... ........... ............. E14u * 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 -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 ... ... ..... ....... ......... ........... ............. E15u * 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 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 ... ... ..... ....... ......... ........... ............. E16u * 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 -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 ... ... ..... ....... ......... ........... ............. E17u * 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 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 ... ... ..... ....... ......... ........... ............. E18u * 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 -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 ... ... ..... ....... ......... ........... ............. E19u * 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 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 ... ... ..... ....... ......... ........... ............. E20u * 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 -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 ... ... ..... ....... ......... ........... ............. E21u * 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 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 ... ... ..... ....... ......... ........... ............. E22u * 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 -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 ... ... ..... ....... ......... ........... ............. E23u * 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 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 ... ... ..... ....... ......... ........... ............. E24u * 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 -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 ... ... ..... ....... ......... ........... ............. E25u * 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 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 ... ... ..... ....... ......... ........... ............. E26u * 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 -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 ... ... ..... ....... ......... ........... ............. E27u * 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 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 ... ... ..... ....... ......... ........... ............. 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) = (1+√5)/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) 1.309721467890 = 2*cos(30*π/110) = 2*cos(3*π/11) 1.221295759271 = 2*cos(32*π/110) = 2*cos(16*π/55) 1.128886437733 = 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) 0.284629676546 = 2*cos(50*π/110) = 2*cos(5*π/11) 0.171150016957 = 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 Ag R+d+g+i Rz, z2, z4, z6 E1g R+d+g+i {Rx, Ry}, {xz, yz}, {xz3, yz3}, {xz5, yz5} E2g d+g+i {x2–y2, xy}, {z2(x2–y2), xyz2}, {z4(x2–y2), xyz4} E3g g+i {xz(x2–3y2), yz(3x2–y2)}, {xz3(x2–3y2), yz3(3x2–y2)} E4g g+i {(x2–y2)2–2x2y2, xy(x2–y2)}, {z2((x2–y2)2–4x2y2), xyz2(x2–y2)} E5g i {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)} E6g i {x2(x2–3y2)2–y2(3x2–y2)2, xy(x2–3y2)(3x2–y2)} Au p+f+h z, z3, z5 E1u p+f+h {x, y}, {xz2, yz2}, {xz4, yz4} E2u f+h {z(x2–y2), xyz}, {z3(x2–y2), xyz3} E3u f+h {x(x2–3y2), y(3x2–y2)}, {xz2(x2–3y2), yz2(3x2–y2)} E4u h {z((x2–y2)2–4x2y2), xyz(x2–y2)} E5u h {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)} Notes: α The order of the S110 point group is 110, and the order of the principal axis is 110. β The S110 point group could also be named C55i, because the S110 axis is identical to a roto-inversion axis of order 55. γ This is an Abelian point group (the commutative law holds between all symmetry operations). The S110 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. δ 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. ε 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. ζ Some of the characters in the table are irrational because the order of the principal axis is neither 1,2,3,4 nor 6. These irrational values can be expressed as cosine values, or as solutions of algebraic equations with a leading coefficient of 1. All characters are algebraic integers of a degree 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.
| S108 | ||
| C110 C110v C110h D110 D110h D110d | S110 | |
| S112 |
This Character Table for the S110 point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.