Character Table for Point Group C68h

Character table for the C_68h point group

C68h E 2 C68 2 C34 2 C68^3 2 C17 2 C68^5 2 C34^3 2 C68^7 2 C17^2 2 C68^9 2 C34^5 2 C68^11 2 C17^3 2 C68^13 2 C34^7 2 C68^15 2 C17^4 2 C4 2 C34^9 2 C68^19 2 C17^5 2 C68^21 2 C34^11 2 C68^23 2 C17^6 2 C68^25 2 C34^13 2 C68^27 2 C17^7 2 C68^29 2 C34^15 2 C68^31 2 C17^8 2 C68^33 C2 i 2 S68 2 S34 2 S68^3 2 S17 2 S68^5 2 S34^3 2 S68^7 2 S17^2 2 S68^9 2 S34^5 2 S68^11 2 S17^3 2 S68^13 2 S34^7 2 S68^15 2 S17^4 2 S4 2 S34^9 2 S68^19 2 S17^5 2 S68^21 2 S34^11 2 S68^23 2 S17^6 2 S68^25 2 S34^13 2 S68^27 2 S17^7 2 S68^29 2 S34^15 2 S68^31 2 S17^8 2 S68^33 sh <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 1.00000 1.00000 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 Bg 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 ... ... ..... ....... ......... ........... ............. E1g * 2.00000 1.99147 1.96595 1.92365 1.86494 1.79033 1.70043 1.59603 1.47802 1.34739 1.20527 1.05286 0.89148 0.72248 0.54733 0.36750 0.18454 0.00000 -0.18454 -0.36750 -0.54733 -0.72248 -0.89148 -1.05286 -1.20527 -1.34739 -1.47802 -1.59603 -1.70043 -1.79033 -1.86494 -1.92365 -1.96595 -1.99147 -2.00000 2.00000 -1.99147 -1.96595 -1.92365 -1.86494 -1.79033 -1.70043 -1.59603 -1.47802 -1.34739 -1.20527 -1.05286 -0.89148 -0.72248 -0.54733 -0.36750 -0.18454 0.00000 0.18454 0.36750 0.54733 0.72248 0.89148 1.05286 1.20527 1.34739 1.47802 1.59603 1.70043 1.79033 1.86494 1.92365 1.96595 1.99147 -2.00000 TT. ... ..TT. ....... ......TT. ........... ..........TT. E2g * 2.00000 1.96595 1.86494 1.70043 1.47802 1.20527 0.89148 0.54733 0.18454 -0.18454 -0.54733 -0.89148 -1.20527 -1.47802 -1.70043 -1.86494 -1.96595 -2.00000 -1.96595 -1.86494 -1.70043 -1.47802 -1.20527 -0.89148 -0.54733 -0.18454 0.18454 0.54733 0.89148 1.20527 1.47802 1.70043 1.86494 1.96595 2.00000 2.00000 1.96595 1.86494 1.70043 1.47802 1.20527 0.89148 0.54733 0.18454 -0.18454 -0.54733 -0.89148 -1.20527 -1.47802 -1.70043 -1.86494 -1.96595 -2.00000 -1.96595 -1.86494 -1.70043 -1.47802 -1.20527 -0.89148 -0.54733 -0.18454 0.18454 0.54733 0.89148 1.20527 1.47802 1.70043 1.86494 1.96595 2.00000 ... ... TT... ....... ....TT... ........... ........TT... E3g * 2.00000 1.92365 1.70043 1.34739 0.89148 0.36750 -0.18454 -0.72248 -1.20527 -1.59603 -1.86494 -1.99147 -1.96595 -1.79033 -1.47802 -1.05286 -0.54733 0.00000 0.54733 1.05286 1.47802 1.79033 1.96595 1.99147 1.86494 1.59603 1.20527 0.72248 0.18454 -0.36750 -0.89148 -1.34739 -1.70043 -1.92365 -2.00000 2.00000 -1.92365 -1.70043 -1.34739 -0.89148 -0.36750 0.18454 0.72248 1.20527 1.59603 1.86494 1.99147 1.96595 1.79033 1.47802 1.05286 0.54733 0.00000 -0.54733 -1.05286 -1.47802 -1.79033 -1.96595 -1.99147 -1.86494 -1.59603 -1.20527 -0.72248 -0.18454 0.36750 0.89148 1.34739 1.70043 1.92365 -2.00000 ... ... ..... ....... ..TT..... ........... ......TT..... E4g * 2.00000 1.86494 1.47802 0.89148 0.18454 -0.54733 -1.20527 -1.70043 -1.96595 -1.96595 -1.70043 -1.20527 -0.54733 0.18454 0.89148 1.47802 1.86494 2.00000 1.86494 1.47802 0.89148 0.18454 -0.54733 -1.20527 -1.70043 -1.96595 -1.96595 -1.70043 -1.20527 -0.54733 0.18454 0.89148 1.47802 1.86494 2.00000 2.00000 1.86494 1.47802 0.89148 0.18454 -0.54733 -1.20527 -1.70043 -1.96595 -1.96595 -1.70043 -1.20527 -0.54733 0.18454 0.89148 1.47802 1.86494 2.00000 1.86494 1.47802 0.89148 0.18454 -0.54733 -1.20527 -1.70043 -1.96595 -1.96595 -1.70043 -1.20527 -0.54733 0.18454 0.89148 1.47802 1.86494 2.00000 ... ... ..... ....... TT....... ........... ....TT....... E5g * 2.00000 1.79033 1.20527 0.36750 -0.54733 -1.34739 -1.86494 -1.99147 -1.70043 -1.05286 -0.18454 0.72248 1.47802 1.92365 1.96595 1.59603 0.89148 0.00000 -0.89148 -1.59603 -1.96595 -1.92365 -1.47802 -0.72248 0.18454 1.05286 1.70043 1.99147 1.86494 1.34739 0.54733 -0.36750 -1.20527 -1.79033 -2.00000 2.00000 -1.79033 -1.20527 -0.36750 0.54733 1.34739 1.86494 1.99147 1.70043 1.05286 0.18454 -0.72248 -1.47802 -1.92365 -1.96595 -1.59603 -0.89148 0.00000 0.89148 1.59603 1.96595 1.92365 1.47802 0.72248 -0.18454 -1.05286 -1.70043 -1.99147 -1.86494 -1.34739 -0.54733 0.36750 1.20527 1.79033 -2.00000 ... ... ..... ....... ......... ........... ..TT......... E6g * 2.00000 1.70043 0.89148 -0.18454 -1.20527 -1.86494 -1.96595 -1.47802 -0.54733 0.54733 1.47802 1.96595 1.86494 1.20527 0.18454 -0.89148 -1.70043 -2.00000 -1.70043 -0.89148 0.18454 1.20527 1.86494 1.96595 1.47802 0.54733 -0.54733 -1.47802 -1.96595 -1.86494 -1.20527 -0.18454 0.89148 1.70043 2.00000 2.00000 1.70043 0.89148 -0.18454 -1.20527 -1.86494 -1.96595 -1.47802 -0.54733 0.54733 1.47802 1.96595 1.86494 1.20527 0.18454 -0.89148 -1.70043 -2.00000 -1.70043 -0.89148 0.18454 1.20527 1.86494 1.96595 1.47802 0.54733 -0.54733 -1.47802 -1.96595 -1.86494 -1.20527 -0.18454 0.89148 1.70043 2.00000 ... ... ..... ....... ......... ........... TT........... E7g * 2.00000 1.59603 0.54733 -0.72248 -1.70043 -1.99147 -1.47802 -0.36750 0.89148 1.79033 1.96595 1.34739 0.18454 -1.05286 -1.86494 -1.92365 -1.20527 0.00000 1.20527 1.92365 1.86494 1.05286 -0.18454 -1.34739 -1.96595 -1.79033 -0.89148 0.36750 1.47802 1.99147 1.70043 0.72248 -0.54733 -1.59603 -2.00000 2.00000 -1.59603 -0.54733 0.72248 1.70043 1.99147 1.47802 0.36750 -0.89148 -1.79033 -1.96595 -1.34739 -0.18454 1.05286 1.86494 1.92365 1.20527 0.00000 -1.20527 -1.92365 -1.86494 -1.05286 0.18454 1.34739 1.96595 1.79033 0.89148 -0.36750 -1.47802 -1.99147 -1.70043 -0.72248 0.54733 1.59603 -2.00000 ... ... ..... ....... ......... ........... ............. E8g * 2.00000 1.47802 0.18454 -1.20527 -1.96595 -1.70043 -0.54733 0.89148 1.86494 1.86494 0.89148 -0.54733 -1.70043 -1.96595 -1.20527 0.18454 1.47802 2.00000 1.47802 0.18454 -1.20527 -1.96595 -1.70043 -0.54733 0.89148 1.86494 1.86494 0.89148 -0.54733 -1.70043 -1.96595 -1.20527 0.18454 1.47802 2.00000 2.00000 1.47802 0.18454 -1.20527 -1.96595 -1.70043 -0.54733 0.89148 1.86494 1.86494 0.89148 -0.54733 -1.70043 -1.96595 -1.20527 0.18454 1.47802 2.00000 1.47802 0.18454 -1.20527 -1.96595 -1.70043 -0.54733 0.89148 1.86494 1.86494 0.89148 -0.54733 -1.70043 -1.96595 -1.20527 0.18454 1.47802 2.00000 ... ... ..... ....... ......... ........... ............. E9g * 2.00000 1.34739 -0.18454 -1.59603 -1.96595 -1.05286 0.54733 1.79033 1.86494 0.72248 -0.89148 -1.92365 -1.70043 -0.36750 1.20527 1.99147 1.47802 0.00000 -1.47802 -1.99147 -1.20527 0.36750 1.70043 1.92365 0.89148 -0.72248 -1.86494 -1.79033 -0.54733 1.05286 1.96595 1.59603 0.18454 -1.34739 -2.00000 2.00000 -1.34739 0.18454 1.59603 1.96595 1.05286 -0.54733 -1.79033 -1.86494 -0.72248 0.89148 1.92365 1.70043 0.36750 -1.20527 -1.99147 -1.47802 0.00000 1.47802 1.99147 1.20527 -0.36750 -1.70043 -1.92365 -0.89148 0.72248 1.86494 1.79033 0.54733 -1.05286 -1.96595 -1.59603 -0.18454 1.34739 -2.00000 ... ... ..... ....... ......... ........... ............. E10g * 2.00000 1.20527 -0.54733 -1.86494 -1.70043 -0.18454 1.47802 1.96595 0.89148 -0.89148 -1.96595 -1.47802 0.18454 1.70043 1.86494 0.54733 -1.20527 -2.00000 -1.20527 0.54733 1.86494 1.70043 0.18454 -1.47802 -1.96595 -0.89148 0.89148 1.96595 1.47802 -0.18454 -1.70043 -1.86494 -0.54733 1.20527 2.00000 2.00000 1.20527 -0.54733 -1.86494 -1.70043 -0.18454 1.47802 1.96595 0.89148 -0.89148 -1.96595 -1.47802 0.18454 1.70043 1.86494 0.54733 -1.20527 -2.00000 -1.20527 0.54733 1.86494 1.70043 0.18454 -1.47802 -1.96595 -0.89148 0.89148 1.96595 1.47802 -0.18454 -1.70043 -1.86494 -0.54733 1.20527 2.00000 ... ... ..... ....... ......... ........... ............. E11g * 2.00000 1.05286 -0.89148 -1.99147 -1.20527 0.72248 1.96595 1.34739 -0.54733 -1.92365 -1.47802 0.36750 1.86494 1.59603 -0.18454 -1.79033 -1.70043 0.00000 1.70043 1.79033 0.18454 -1.59603 -1.86494 -0.36750 1.47802 1.92365 0.54733 -1.34739 -1.96595 -0.72248 1.20527 1.99147 0.89148 -1.05286 -2.00000 2.00000 -1.05286 0.89148 1.99147 1.20527 -0.72248 -1.96595 -1.34739 0.54733 1.92365 1.47802 -0.36750 -1.86494 -1.59603 0.18454 1.79033 1.70043 0.00000 -1.70043 -1.79033 -0.18454 1.59603 1.86494 0.36750 -1.47802 -1.92365 -0.54733 1.34739 1.96595 0.72248 -1.20527 -1.99147 -0.89148 1.05286 -2.00000 ... ... ..... ....... ......... ........... ............. E12g * 2.00000 0.89148 -1.20527 -1.96595 -0.54733 1.47802 1.86494 0.18454 -1.70043 -1.70043 0.18454 1.86494 1.47802 -0.54733 -1.96595 -1.20527 0.89148 2.00000 0.89148 -1.20527 -1.96595 -0.54733 1.47802 1.86494 0.18454 -1.70043 -1.70043 0.18454 1.86494 1.47802 -0.54733 -1.96595 -1.20527 0.89148 2.00000 2.00000 0.89148 -1.20527 -1.96595 -0.54733 1.47802 1.86494 0.18454 -1.70043 -1.70043 0.18454 1.86494 1.47802 -0.54733 -1.96595 -1.20527 0.89148 2.00000 0.89148 -1.20527 -1.96595 -0.54733 1.47802 1.86494 0.18454 -1.70043 -1.70043 0.18454 1.86494 1.47802 -0.54733 -1.96595 -1.20527 0.89148 2.00000 ... ... ..... ....... ......... ........... ............. E13g * 2.00000 0.72248 -1.47802 -1.79033 0.18454 1.92365 1.20527 -1.05286 -1.96595 -0.36750 1.70043 1.59603 -0.54733 -1.99147 -0.89148 1.34739 1.86494 0.00000 -1.86494 -1.34739 0.89148 1.99147 0.54733 -1.59603 -1.70043 0.36750 1.96595 1.05286 -1.20527 -1.92365 -0.18454 1.79033 1.47802 -0.72248 -2.00000 2.00000 -0.72248 1.47802 1.79033 -0.18454 -1.92365 -1.20527 1.05286 1.96595 0.36750 -1.70043 -1.59603 0.54733 1.99147 0.89148 -1.34739 -1.86494 0.00000 1.86494 1.34739 -0.89148 -1.99147 -0.54733 1.59603 1.70043 -0.36750 -1.96595 -1.05286 1.20527 1.92365 0.18454 -1.79033 -1.47802 0.72248 -2.00000 ... ... ..... ....... ......... ........... ............. E14g * 2.00000 0.54733 -1.70043 -1.47802 0.89148 1.96595 0.18454 -1.86494 -1.20527 1.20527 1.86494 -0.18454 -1.96595 -0.89148 1.47802 1.70043 -0.54733 -2.00000 -0.54733 1.70043 1.47802 -0.89148 -1.96595 -0.18454 1.86494 1.20527 -1.20527 -1.86494 0.18454 1.96595 0.89148 -1.47802 -1.70043 0.54733 2.00000 2.00000 0.54733 -1.70043 -1.47802 0.89148 1.96595 0.18454 -1.86494 -1.20527 1.20527 1.86494 -0.18454 -1.96595 -0.89148 1.47802 1.70043 -0.54733 -2.00000 -0.54733 1.70043 1.47802 -0.89148 -1.96595 -0.18454 1.86494 1.20527 -1.20527 -1.86494 0.18454 1.96595 0.89148 -1.47802 -1.70043 0.54733 2.00000 ... ... ..... ....... ......... ........... ............. E15g * 2.00000 0.36750 -1.86494 -1.05286 1.47802 1.59603 -0.89148 -1.92365 0.18454 1.99147 0.54733 -1.79033 -1.20527 1.34739 1.70043 -0.72248 -1.96595 0.00000 1.96595 0.72248 -1.70043 -1.34739 1.20527 1.79033 -0.54733 -1.99147 -0.18454 1.92365 0.89148 -1.59603 -1.47802 1.05286 1.86494 -0.36750 -2.00000 2.00000 -0.36750 1.86494 1.05286 -1.47802 -1.59603 0.89148 1.92365 -0.18454 -1.99147 -0.54733 1.79033 1.20527 -1.34739 -1.70043 0.72248 1.96595 0.00000 -1.96595 -0.72248 1.70043 1.34739 -1.20527 -1.79033 0.54733 1.99147 0.18454 -1.92365 -0.89148 1.59603 1.47802 -1.05286 -1.86494 0.36750 -2.00000 ... ... ..... ....... ......... ........... ............. E16g * 2.00000 0.18454 -1.96595 -0.54733 1.86494 0.89148 -1.70043 -1.20527 1.47802 1.47802 -1.20527 -1.70043 0.89148 1.86494 -0.54733 -1.96595 0.18454 2.00000 0.18454 -1.96595 -0.54733 1.86494 0.89148 -1.70043 -1.20527 1.47802 1.47802 -1.20527 -1.70043 0.89148 1.86494 -0.54733 -1.96595 0.18454 2.00000 2.00000 0.18454 -1.96595 -0.54733 1.86494 0.89148 -1.70043 -1.20527 1.47802 1.47802 -1.20527 -1.70043 0.89148 1.86494 -0.54733 -1.96595 0.18454 2.00000 0.18454 -1.96595 -0.54733 1.86494 0.89148 -1.70043 -1.20527 1.47802 1.47802 -1.20527 -1.70043 0.89148 1.86494 -0.54733 -1.96595 0.18454 2.00000 ... ... ..... ....... ......... ........... ............. E17g * 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 -2.00000 ... ... ..... ....... ......... ........... ............. E18g * 2.00000 -0.18454 -1.96595 0.54733 1.86494 -0.89148 -1.70043 1.20527 1.47802 -1.47802 -1.20527 1.70043 0.89148 -1.86494 -0.54733 1.96595 0.18454 -2.00000 0.18454 1.96595 -0.54733 -1.86494 0.89148 1.70043 -1.20527 -1.47802 1.47802 1.20527 -1.70043 -0.89148 1.86494 0.54733 -1.96595 -0.18454 2.00000 2.00000 -0.18454 -1.96595 0.54733 1.86494 -0.89148 -1.70043 1.20527 1.47802 -1.47802 -1.20527 1.70043 0.89148 -1.86494 -0.54733 1.96595 0.18454 -2.00000 0.18454 1.96595 -0.54733 -1.86494 0.89148 1.70043 -1.20527 -1.47802 1.47802 1.20527 -1.70043 -0.89148 1.86494 0.54733 -1.96595 -0.18454 2.00000 ... ... ..... ....... ......... ........... ............. E19g * 2.00000 -0.36750 -1.86494 1.05286 1.47802 -1.59603 -0.89148 1.92365 0.18454 -1.99147 0.54733 1.79033 -1.20527 -1.34739 1.70043 0.72248 -1.96595 0.00000 1.96595 -0.72248 -1.70043 1.34739 1.20527 -1.79033 -0.54733 1.99147 -0.18454 -1.92365 0.89148 1.59603 -1.47802 -1.05286 1.86494 0.36750 -2.00000 2.00000 0.36750 1.86494 -1.05286 -1.47802 1.59603 0.89148 -1.92365 -0.18454 1.99147 -0.54733 -1.79033 1.20527 1.34739 -1.70043 -0.72248 1.96595 0.00000 -1.96595 0.72248 1.70043 -1.34739 -1.20527 1.79033 0.54733 -1.99147 0.18454 1.92365 -0.89148 -1.59603 1.47802 1.05286 -1.86494 -0.36750 -2.00000 ... ... ..... ....... ......... ........... ............. E20g * 2.00000 -0.54733 -1.70043 1.47802 0.89148 -1.96595 0.18454 1.86494 -1.20527 -1.20527 1.86494 0.18454 -1.96595 0.89148 1.47802 -1.70043 -0.54733 2.00000 -0.54733 -1.70043 1.47802 0.89148 -1.96595 0.18454 1.86494 -1.20527 -1.20527 1.86494 0.18454 -1.96595 0.89148 1.47802 -1.70043 -0.54733 2.00000 2.00000 -0.54733 -1.70043 1.47802 0.89148 -1.96595 0.18454 1.86494 -1.20527 -1.20527 1.86494 0.18454 -1.96595 0.89148 1.47802 -1.70043 -0.54733 2.00000 -0.54733 -1.70043 1.47802 0.89148 -1.96595 0.18454 1.86494 -1.20527 -1.20527 1.86494 0.18454 -1.96595 0.89148 1.47802 -1.70043 -0.54733 2.00000 ... ... ..... ....... ......... ........... ............. E21g * 2.00000 -0.72248 -1.47802 1.79033 0.18454 -1.92365 1.20527 1.05286 -1.96595 0.36750 1.70043 -1.59603 -0.54733 1.99147 -0.89148 -1.34739 1.86494 0.00000 -1.86494 1.34739 0.89148 -1.99147 0.54733 1.59603 -1.70043 -0.36750 1.96595 -1.05286 -1.20527 1.92365 -0.18454 -1.79033 1.47802 0.72248 -2.00000 2.00000 0.72248 1.47802 -1.79033 -0.18454 1.92365 -1.20527 -1.05286 1.96595 -0.36750 -1.70043 1.59603 0.54733 -1.99147 0.89148 1.34739 -1.86494 0.00000 1.86494 -1.34739 -0.89148 1.99147 -0.54733 -1.59603 1.70043 0.36750 -1.96595 1.05286 1.20527 -1.92365 0.18454 1.79033 -1.47802 -0.72248 -2.00000 ... ... ..... ....... ......... ........... ............. E22g * 2.00000 -0.89148 -1.20527 1.96595 -0.54733 -1.47802 1.86494 -0.18454 -1.70043 1.70043 0.18454 -1.86494 1.47802 0.54733 -1.96595 1.20527 0.89148 -2.00000 0.89148 1.20527 -1.96595 0.54733 1.47802 -1.86494 0.18454 1.70043 -1.70043 -0.18454 1.86494 -1.47802 -0.54733 1.96595 -1.20527 -0.89148 2.00000 2.00000 -0.89148 -1.20527 1.96595 -0.54733 -1.47802 1.86494 -0.18454 -1.70043 1.70043 0.18454 -1.86494 1.47802 0.54733 -1.96595 1.20527 0.89148 -2.00000 0.89148 1.20527 -1.96595 0.54733 1.47802 -1.86494 0.18454 1.70043 -1.70043 -0.18454 1.86494 -1.47802 -0.54733 1.96595 -1.20527 -0.89148 2.00000 ... ... ..... ....... ......... ........... ............. E23g * 2.00000 -1.05286 -0.89148 1.99147 -1.20527 -0.72248 1.96595 -1.34739 -0.54733 1.92365 -1.47802 -0.36750 1.86494 -1.59603 -0.18454 1.79033 -1.70043 0.00000 1.70043 -1.79033 0.18454 1.59603 -1.86494 0.36750 1.47802 -1.92365 0.54733 1.34739 -1.96595 0.72248 1.20527 -1.99147 0.89148 1.05286 -2.00000 2.00000 1.05286 0.89148 -1.99147 1.20527 0.72248 -1.96595 1.34739 0.54733 -1.92365 1.47802 0.36750 -1.86494 1.59603 0.18454 -1.79033 1.70043 0.00000 -1.70043 1.79033 -0.18454 -1.59603 1.86494 -0.36750 -1.47802 1.92365 -0.54733 -1.34739 1.96595 -0.72248 -1.20527 1.99147 -0.89148 -1.05286 -2.00000 ... ... ..... ....... ......... ........... ............. E24g * 2.00000 -1.20527 -0.54733 1.86494 -1.70043 0.18454 1.47802 -1.96595 0.89148 0.89148 -1.96595 1.47802 0.18454 -1.70043 1.86494 -0.54733 -1.20527 2.00000 -1.20527 -0.54733 1.86494 -1.70043 0.18454 1.47802 -1.96595 0.89148 0.89148 -1.96595 1.47802 0.18454 -1.70043 1.86494 -0.54733 -1.20527 2.00000 2.00000 -1.20527 -0.54733 1.86494 -1.70043 0.18454 1.47802 -1.96595 0.89148 0.89148 -1.96595 1.47802 0.18454 -1.70043 1.86494 -0.54733 -1.20527 2.00000 -1.20527 -0.54733 1.86494 -1.70043 0.18454 1.47802 -1.96595 0.89148 0.89148 -1.96595 1.47802 0.18454 -1.70043 1.86494 -0.54733 -1.20527 2.00000 ... ... ..... ....... ......... ........... ............. E25g * 2.00000 -1.34739 -0.18454 1.59603 -1.96595 1.05286 0.54733 -1.79033 1.86494 -0.72248 -0.89148 1.92365 -1.70043 0.36750 1.20527 -1.99147 1.47802 0.00000 -1.47802 1.99147 -1.20527 -0.36750 1.70043 -1.92365 0.89148 0.72248 -1.86494 1.79033 -0.54733 -1.05286 1.96595 -1.59603 0.18454 1.34739 -2.00000 2.00000 1.34739 0.18454 -1.59603 1.96595 -1.05286 -0.54733 1.79033 -1.86494 0.72248 0.89148 -1.92365 1.70043 -0.36750 -1.20527 1.99147 -1.47802 0.00000 1.47802 -1.99147 1.20527 0.36750 -1.70043 1.92365 -0.89148 -0.72248 1.86494 -1.79033 0.54733 1.05286 -1.96595 1.59603 -0.18454 -1.34739 -2.00000 ... ... ..... ....... ......... ........... ............. E26g * 2.00000 -1.47802 0.18454 1.20527 -1.96595 1.70043 -0.54733 -0.89148 1.86494 -1.86494 0.89148 0.54733 -1.70043 1.96595 -1.20527 -0.18454 1.47802 -2.00000 1.47802 -0.18454 -1.20527 1.96595 -1.70043 0.54733 0.89148 -1.86494 1.86494 -0.89148 -0.54733 1.70043 -1.96595 1.20527 0.18454 -1.47802 2.00000 2.00000 -1.47802 0.18454 1.20527 -1.96595 1.70043 -0.54733 -0.89148 1.86494 -1.86494 0.89148 0.54733 -1.70043 1.96595 -1.20527 -0.18454 1.47802 -2.00000 1.47802 -0.18454 -1.20527 1.96595 -1.70043 0.54733 0.89148 -1.86494 1.86494 -0.89148 -0.54733 1.70043 -1.96595 1.20527 0.18454 -1.47802 2.00000 ... ... ..... ....... ......... ........... ............. E27g * 2.00000 -1.59603 0.54733 0.72248 -1.70043 1.99147 -1.47802 0.36750 0.89148 -1.79033 1.96595 -1.34739 0.18454 1.05286 -1.86494 1.92365 -1.20527 0.00000 1.20527 -1.92365 1.86494 -1.05286 -0.18454 1.34739 -1.96595 1.79033 -0.89148 -0.36750 1.47802 -1.99147 1.70043 -0.72248 -0.54733 1.59603 -2.00000 2.00000 1.59603 -0.54733 -0.72248 1.70043 -1.99147 1.47802 -0.36750 -0.89148 1.79033 -1.96595 1.34739 -0.18454 -1.05286 1.86494 -1.92365 1.20527 0.00000 -1.20527 1.92365 -1.86494 1.05286 0.18454 -1.34739 1.96595 -1.79033 0.89148 0.36750 -1.47802 1.99147 -1.70043 0.72248 0.54733 -1.59603 -2.00000 ... ... ..... ....... ......... ........... ............. E28g * 2.00000 -1.70043 0.89148 0.18454 -1.20527 1.86494 -1.96595 1.47802 -0.54733 -0.54733 1.47802 -1.96595 1.86494 -1.20527 0.18454 0.89148 -1.70043 2.00000 -1.70043 0.89148 0.18454 -1.20527 1.86494 -1.96595 1.47802 -0.54733 -0.54733 1.47802 -1.96595 1.86494 -1.20527 0.18454 0.89148 -1.70043 2.00000 2.00000 -1.70043 0.89148 0.18454 -1.20527 1.86494 -1.96595 1.47802 -0.54733 -0.54733 1.47802 -1.96595 1.86494 -1.20527 0.18454 0.89148 -1.70043 2.00000 -1.70043 0.89148 0.18454 -1.20527 1.86494 -1.96595 1.47802 -0.54733 -0.54733 1.47802 -1.96595 1.86494 -1.20527 0.18454 0.89148 -1.70043 2.00000 ... ... ..... ....... ......... ........... ............. E29g * 2.00000 -1.79033 1.20527 -0.36750 -0.54733 1.34739 -1.86494 1.99147 -1.70043 1.05286 -0.18454 -0.72248 1.47802 -1.92365 1.96595 -1.59603 0.89148 0.00000 -0.89148 1.59603 -1.96595 1.92365 -1.47802 0.72248 0.18454 -1.05286 1.70043 -1.99147 1.86494 -1.34739 0.54733 0.36750 -1.20527 1.79033 -2.00000 2.00000 1.79033 -1.20527 0.36750 0.54733 -1.34739 1.86494 -1.99147 1.70043 -1.05286 0.18454 0.72248 -1.47802 1.92365 -1.96595 1.59603 -0.89148 0.00000 0.89148 -1.59603 1.96595 -1.92365 1.47802 -0.72248 -0.18454 1.05286 -1.70043 1.99147 -1.86494 1.34739 -0.54733 -0.36750 1.20527 -1.79033 -2.00000 ... ... ..... ....... ......... ........... ............. E30g * 2.00000 -1.86494 1.47802 -0.89148 0.18454 0.54733 -1.20527 1.70043 -1.96595 1.96595 -1.70043 1.20527 -0.54733 -0.18454 0.89148 -1.47802 1.86494 -2.00000 1.86494 -1.47802 0.89148 -0.18454 -0.54733 1.20527 -1.70043 1.96595 -1.96595 1.70043 -1.20527 0.54733 0.18454 -0.89148 1.47802 -1.86494 2.00000 2.00000 -1.86494 1.47802 -0.89148 0.18454 0.54733 -1.20527 1.70043 -1.96595 1.96595 -1.70043 1.20527 -0.54733 -0.18454 0.89148 -1.47802 1.86494 -2.00000 1.86494 -1.47802 0.89148 -0.18454 -0.54733 1.20527 -1.70043 1.96595 -1.96595 1.70043 -1.20527 0.54733 0.18454 -0.89148 1.47802 -1.86494 2.00000 ... ... ..... ....... ......... ........... ............. E31g * 2.00000 -1.92365 1.70043 -1.34739 0.89148 -0.36750 -0.18454 0.72248 -1.20527 1.59603 -1.86494 1.99147 -1.96595 1.79033 -1.47802 1.05286 -0.54733 0.00000 0.54733 -1.05286 1.47802 -1.79033 1.96595 -1.99147 1.86494 -1.59603 1.20527 -0.72248 0.18454 0.36750 -0.89148 1.34739 -1.70043 1.92365 -2.00000 2.00000 1.92365 -1.70043 1.34739 -0.89148 0.36750 0.18454 -0.72248 1.20527 -1.59603 1.86494 -1.99147 1.96595 -1.79033 1.47802 -1.05286 0.54733 0.00000 -0.54733 1.05286 -1.47802 1.79033 -1.96595 1.99147 -1.86494 1.59603 -1.20527 0.72248 -0.18454 -0.36750 0.89148 -1.34739 1.70043 -1.92365 -2.00000 ... ... ..... ....... ......... ........... ............. E32g * 2.00000 -1.96595 1.86494 -1.70043 1.47802 -1.20527 0.89148 -0.54733 0.18454 0.18454 -0.54733 0.89148 -1.20527 1.47802 -1.70043 1.86494 -1.96595 2.00000 -1.96595 1.86494 -1.70043 1.47802 -1.20527 0.89148 -0.54733 0.18454 0.18454 -0.54733 0.89148 -1.20527 1.47802 -1.70043 1.86494 -1.96595 2.00000 2.00000 -1.96595 1.86494 -1.70043 1.47802 -1.20527 0.89148 -0.54733 0.18454 0.18454 -0.54733 0.89148 -1.20527 1.47802 -1.70043 1.86494 -1.96595 2.00000 -1.96595 1.86494 -1.70043 1.47802 -1.20527 0.89148 -0.54733 0.18454 0.18454 -0.54733 0.89148 -1.20527 1.47802 -1.70043 1.86494 -1.96595 2.00000 ... ... ..... ....... ......... ........... ............. E33g * 2.00000 -1.99147 1.96595 -1.92365 1.86494 -1.79033 1.70043 -1.59603 1.47802 -1.34739 1.20527 -1.05286 0.89148 -0.72248 0.54733 -0.36750 0.18454 0.00000 -0.18454 0.36750 -0.54733 0.72248 -0.89148 1.05286 -1.20527 1.34739 -1.47802 1.59603 -1.70043 1.79033 -1.86494 1.92365 -1.96595 1.99147 -2.00000 2.00000 1.99147 -1.96595 1.92365 -1.86494 1.79033 -1.70043 1.59603 -1.47802 1.34739 -1.20527 1.05286 -0.89148 0.72248 -0.54733 0.36750 -0.18454 0.00000 0.18454 -0.36750 0.54733 -0.72248 0.89148 -1.05286 1.20527 -1.34739 1.47802 -1.59603 1.70043 -1.79033 1.86494 -1.92365 1.96595 -1.99147 -2.00000 ... ... ..... ....... ......... ........... ............. 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 -1.00000 -1.00000 -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 ............. Bu 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 ... ... ..... ....... ......... ........... ............. E1u * 2.00000 1.99147 1.96595 1.92365 1.86494 1.79033 1.70043 1.59603 1.47802 1.34739 1.20527 1.05286 0.89148 0.72248 0.54733 0.36750 0.18454 0.00000 -0.18454 -0.36750 -0.54733 -0.72248 -0.89148 -1.05286 -1.20527 -1.34739 -1.47802 -1.59603 -1.70043 -1.79033 -1.86494 -1.92365 -1.96595 -1.99147 -2.00000 -2.00000 1.99147 1.96595 1.92365 1.86494 1.79033 1.70043 1.59603 1.47802 1.34739 1.20527 1.05286 0.89148 0.72248 0.54733 0.36750 0.18454 0.00000 -0.18454 -0.36750 -0.54733 -0.72248 -0.89148 -1.05286 -1.20527 -1.34739 -1.47802 -1.59603 -1.70043 -1.79033 -1.86494 -1.92365 -1.96595 -1.99147 2.00000 ... TT. ..... ....TT. ......... ........TT. ............. E2u * 2.00000 1.96595 1.86494 1.70043 1.47802 1.20527 0.89148 0.54733 0.18454 -0.18454 -0.54733 -0.89148 -1.20527 -1.47802 -1.70043 -1.86494 -1.96595 -2.00000 -1.96595 -1.86494 -1.70043 -1.47802 -1.20527 -0.89148 -0.54733 -0.18454 0.18454 0.54733 0.89148 1.20527 1.47802 1.70043 1.86494 1.96595 2.00000 -2.00000 -1.96595 -1.86494 -1.70043 -1.47802 -1.20527 -0.89148 -0.54733 -0.18454 0.18454 0.54733 0.89148 1.20527 1.47802 1.70043 1.86494 1.96595 2.00000 1.96595 1.86494 1.70043 1.47802 1.20527 0.89148 0.54733 0.18454 -0.18454 -0.54733 -0.89148 -1.20527 -1.47802 -1.70043 -1.86494 -1.96595 -2.00000 ... ... ..... ..TT... ......... ......TT... ............. E3u * 2.00000 1.92365 1.70043 1.34739 0.89148 0.36750 -0.18454 -0.72248 -1.20527 -1.59603 -1.86494 -1.99147 -1.96595 -1.79033 -1.47802 -1.05286 -0.54733 0.00000 0.54733 1.05286 1.47802 1.79033 1.96595 1.99147 1.86494 1.59603 1.20527 0.72248 0.18454 -0.36750 -0.89148 -1.34739 -1.70043 -1.92365 -2.00000 -2.00000 1.92365 1.70043 1.34739 0.89148 0.36750 -0.18454 -0.72248 -1.20527 -1.59603 -1.86494 -1.99147 -1.96595 -1.79033 -1.47802 -1.05286 -0.54733 0.00000 0.54733 1.05286 1.47802 1.79033 1.96595 1.99147 1.86494 1.59603 1.20527 0.72248 0.18454 -0.36750 -0.89148 -1.34739 -1.70043 -1.92365 2.00000 ... ... ..... TT..... ......... ....TT..... ............. E4u * 2.00000 1.86494 1.47802 0.89148 0.18454 -0.54733 -1.20527 -1.70043 -1.96595 -1.96595 -1.70043 -1.20527 -0.54733 0.18454 0.89148 1.47802 1.86494 2.00000 1.86494 1.47802 0.89148 0.18454 -0.54733 -1.20527 -1.70043 -1.96595 -1.96595 -1.70043 -1.20527 -0.54733 0.18454 0.89148 1.47802 1.86494 2.00000 -2.00000 -1.86494 -1.47802 -0.89148 -0.18454 0.54733 1.20527 1.70043 1.96595 1.96595 1.70043 1.20527 0.54733 -0.18454 -0.89148 -1.47802 -1.86494 -2.00000 -1.86494 -1.47802 -0.89148 -0.18454 0.54733 1.20527 1.70043 1.96595 1.96595 1.70043 1.20527 0.54733 -0.18454 -0.89148 -1.47802 -1.86494 -2.00000 ... ... ..... ....... ......... ..TT....... ............. E5u * 2.00000 1.79033 1.20527 0.36750 -0.54733 -1.34739 -1.86494 -1.99147 -1.70043 -1.05286 -0.18454 0.72248 1.47802 1.92365 1.96595 1.59603 0.89148 0.00000 -0.89148 -1.59603 -1.96595 -1.92365 -1.47802 -0.72248 0.18454 1.05286 1.70043 1.99147 1.86494 1.34739 0.54733 -0.36750 -1.20527 -1.79033 -2.00000 -2.00000 1.79033 1.20527 0.36750 -0.54733 -1.34739 -1.86494 -1.99147 -1.70043 -1.05286 -0.18454 0.72248 1.47802 1.92365 1.96595 1.59603 0.89148 0.00000 -0.89148 -1.59603 -1.96595 -1.92365 -1.47802 -0.72248 0.18454 1.05286 1.70043 1.99147 1.86494 1.34739 0.54733 -0.36750 -1.20527 -1.79033 2.00000 ... ... ..... ....... ......... TT......... ............. E6u * 2.00000 1.70043 0.89148 -0.18454 -1.20527 -1.86494 -1.96595 -1.47802 -0.54733 0.54733 1.47802 1.96595 1.86494 1.20527 0.18454 -0.89148 -1.70043 -2.00000 -1.70043 -0.89148 0.18454 1.20527 1.86494 1.96595 1.47802 0.54733 -0.54733 -1.47802 -1.96595 -1.86494 -1.20527 -0.18454 0.89148 1.70043 2.00000 -2.00000 -1.70043 -0.89148 0.18454 1.20527 1.86494 1.96595 1.47802 0.54733 -0.54733 -1.47802 -1.96595 -1.86494 -1.20527 -0.18454 0.89148 1.70043 2.00000 1.70043 0.89148 -0.18454 -1.20527 -1.86494 -1.96595 -1.47802 -0.54733 0.54733 1.47802 1.96595 1.86494 1.20527 0.18454 -0.89148 -1.70043 -2.00000 ... ... ..... ....... ......... ........... ............. E7u * 2.00000 1.59603 0.54733 -0.72248 -1.70043 -1.99147 -1.47802 -0.36750 0.89148 1.79033 1.96595 1.34739 0.18454 -1.05286 -1.86494 -1.92365 -1.20527 0.00000 1.20527 1.92365 1.86494 1.05286 -0.18454 -1.34739 -1.96595 -1.79033 -0.89148 0.36750 1.47802 1.99147 1.70043 0.72248 -0.54733 -1.59603 -2.00000 -2.00000 1.59603 0.54733 -0.72248 -1.70043 -1.99147 -1.47802 -0.36750 0.89148 1.79033 1.96595 1.34739 0.18454 -1.05286 -1.86494 -1.92365 -1.20527 0.00000 1.20527 1.92365 1.86494 1.05286 -0.18454 -1.34739 -1.96595 -1.79033 -0.89148 0.36750 1.47802 1.99147 1.70043 0.72248 -0.54733 -1.59603 2.00000 ... ... ..... ....... ......... ........... ............. E8u * 2.00000 1.47802 0.18454 -1.20527 -1.96595 -1.70043 -0.54733 0.89148 1.86494 1.86494 0.89148 -0.54733 -1.70043 -1.96595 -1.20527 0.18454 1.47802 2.00000 1.47802 0.18454 -1.20527 -1.96595 -1.70043 -0.54733 0.89148 1.86494 1.86494 0.89148 -0.54733 -1.70043 -1.96595 -1.20527 0.18454 1.47802 2.00000 -2.00000 -1.47802 -0.18454 1.20527 1.96595 1.70043 0.54733 -0.89148 -1.86494 -1.86494 -0.89148 0.54733 1.70043 1.96595 1.20527 -0.18454 -1.47802 -2.00000 -1.47802 -0.18454 1.20527 1.96595 1.70043 0.54733 -0.89148 -1.86494 -1.86494 -0.89148 0.54733 1.70043 1.96595 1.20527 -0.18454 -1.47802 -2.00000 ... ... ..... ....... ......... ........... ............. E9u * 2.00000 1.34739 -0.18454 -1.59603 -1.96595 -1.05286 0.54733 1.79033 1.86494 0.72248 -0.89148 -1.92365 -1.70043 -0.36750 1.20527 1.99147 1.47802 0.00000 -1.47802 -1.99147 -1.20527 0.36750 1.70043 1.92365 0.89148 -0.72248 -1.86494 -1.79033 -0.54733 1.05286 1.96595 1.59603 0.18454 -1.34739 -2.00000 -2.00000 1.34739 -0.18454 -1.59603 -1.96595 -1.05286 0.54733 1.79033 1.86494 0.72248 -0.89148 -1.92365 -1.70043 -0.36750 1.20527 1.99147 1.47802 0.00000 -1.47802 -1.99147 -1.20527 0.36750 1.70043 1.92365 0.89148 -0.72248 -1.86494 -1.79033 -0.54733 1.05286 1.96595 1.59603 0.18454 -1.34739 2.00000 ... ... ..... ....... ......... ........... ............. E10u * 2.00000 1.20527 -0.54733 -1.86494 -1.70043 -0.18454 1.47802 1.96595 0.89148 -0.89148 -1.96595 -1.47802 0.18454 1.70043 1.86494 0.54733 -1.20527 -2.00000 -1.20527 0.54733 1.86494 1.70043 0.18454 -1.47802 -1.96595 -0.89148 0.89148 1.96595 1.47802 -0.18454 -1.70043 -1.86494 -0.54733 1.20527 2.00000 -2.00000 -1.20527 0.54733 1.86494 1.70043 0.18454 -1.47802 -1.96595 -0.89148 0.89148 1.96595 1.47802 -0.18454 -1.70043 -1.86494 -0.54733 1.20527 2.00000 1.20527 -0.54733 -1.86494 -1.70043 -0.18454 1.47802 1.96595 0.89148 -0.89148 -1.96595 -1.47802 0.18454 1.70043 1.86494 0.54733 -1.20527 -2.00000 ... ... ..... ....... ......... ........... ............. E11u * 2.00000 1.05286 -0.89148 -1.99147 -1.20527 0.72248 1.96595 1.34739 -0.54733 -1.92365 -1.47802 0.36750 1.86494 1.59603 -0.18454 -1.79033 -1.70043 0.00000 1.70043 1.79033 0.18454 -1.59603 -1.86494 -0.36750 1.47802 1.92365 0.54733 -1.34739 -1.96595 -0.72248 1.20527 1.99147 0.89148 -1.05286 -2.00000 -2.00000 1.05286 -0.89148 -1.99147 -1.20527 0.72248 1.96595 1.34739 -0.54733 -1.92365 -1.47802 0.36750 1.86494 1.59603 -0.18454 -1.79033 -1.70043 0.00000 1.70043 1.79033 0.18454 -1.59603 -1.86494 -0.36750 1.47802 1.92365 0.54733 -1.34739 -1.96595 -0.72248 1.20527 1.99147 0.89148 -1.05286 2.00000 ... ... ..... ....... ......... ........... ............. E12u * 2.00000 0.89148 -1.20527 -1.96595 -0.54733 1.47802 1.86494 0.18454 -1.70043 -1.70043 0.18454 1.86494 1.47802 -0.54733 -1.96595 -1.20527 0.89148 2.00000 0.89148 -1.20527 -1.96595 -0.54733 1.47802 1.86494 0.18454 -1.70043 -1.70043 0.18454 1.86494 1.47802 -0.54733 -1.96595 -1.20527 0.89148 2.00000 -2.00000 -0.89148 1.20527 1.96595 0.54733 -1.47802 -1.86494 -0.18454 1.70043 1.70043 -0.18454 -1.86494 -1.47802 0.54733 1.96595 1.20527 -0.89148 -2.00000 -0.89148 1.20527 1.96595 0.54733 -1.47802 -1.86494 -0.18454 1.70043 1.70043 -0.18454 -1.86494 -1.47802 0.54733 1.96595 1.20527 -0.89148 -2.00000 ... ... ..... ....... ......... ........... ............. E13u * 2.00000 0.72248 -1.47802 -1.79033 0.18454 1.92365 1.20527 -1.05286 -1.96595 -0.36750 1.70043 1.59603 -0.54733 -1.99147 -0.89148 1.34739 1.86494 0.00000 -1.86494 -1.34739 0.89148 1.99147 0.54733 -1.59603 -1.70043 0.36750 1.96595 1.05286 -1.20527 -1.92365 -0.18454 1.79033 1.47802 -0.72248 -2.00000 -2.00000 0.72248 -1.47802 -1.79033 0.18454 1.92365 1.20527 -1.05286 -1.96595 -0.36750 1.70043 1.59603 -0.54733 -1.99147 -0.89148 1.34739 1.86494 0.00000 -1.86494 -1.34739 0.89148 1.99147 0.54733 -1.59603 -1.70043 0.36750 1.96595 1.05286 -1.20527 -1.92365 -0.18454 1.79033 1.47802 -0.72248 2.00000 ... ... ..... ....... ......... ........... ............. E14u * 2.00000 0.54733 -1.70043 -1.47802 0.89148 1.96595 0.18454 -1.86494 -1.20527 1.20527 1.86494 -0.18454 -1.96595 -0.89148 1.47802 1.70043 -0.54733 -2.00000 -0.54733 1.70043 1.47802 -0.89148 -1.96595 -0.18454 1.86494 1.20527 -1.20527 -1.86494 0.18454 1.96595 0.89148 -1.47802 -1.70043 0.54733 2.00000 -2.00000 -0.54733 1.70043 1.47802 -0.89148 -1.96595 -0.18454 1.86494 1.20527 -1.20527 -1.86494 0.18454 1.96595 0.89148 -1.47802 -1.70043 0.54733 2.00000 0.54733 -1.70043 -1.47802 0.89148 1.96595 0.18454 -1.86494 -1.20527 1.20527 1.86494 -0.18454 -1.96595 -0.89148 1.47802 1.70043 -0.54733 -2.00000 ... ... ..... ....... ......... ........... ............. E15u * 2.00000 0.36750 -1.86494 -1.05286 1.47802 1.59603 -0.89148 -1.92365 0.18454 1.99147 0.54733 -1.79033 -1.20527 1.34739 1.70043 -0.72248 -1.96595 0.00000 1.96595 0.72248 -1.70043 -1.34739 1.20527 1.79033 -0.54733 -1.99147 -0.18454 1.92365 0.89148 -1.59603 -1.47802 1.05286 1.86494 -0.36750 -2.00000 -2.00000 0.36750 -1.86494 -1.05286 1.47802 1.59603 -0.89148 -1.92365 0.18454 1.99147 0.54733 -1.79033 -1.20527 1.34739 1.70043 -0.72248 -1.96595 0.00000 1.96595 0.72248 -1.70043 -1.34739 1.20527 1.79033 -0.54733 -1.99147 -0.18454 1.92365 0.89148 -1.59603 -1.47802 1.05286 1.86494 -0.36750 2.00000 ... ... ..... ....... ......... ........... ............. E16u * 2.00000 0.18454 -1.96595 -0.54733 1.86494 0.89148 -1.70043 -1.20527 1.47802 1.47802 -1.20527 -1.70043 0.89148 1.86494 -0.54733 -1.96595 0.18454 2.00000 0.18454 -1.96595 -0.54733 1.86494 0.89148 -1.70043 -1.20527 1.47802 1.47802 -1.20527 -1.70043 0.89148 1.86494 -0.54733 -1.96595 0.18454 2.00000 -2.00000 -0.18454 1.96595 0.54733 -1.86494 -0.89148 1.70043 1.20527 -1.47802 -1.47802 1.20527 1.70043 -0.89148 -1.86494 0.54733 1.96595 -0.18454 -2.00000 -0.18454 1.96595 0.54733 -1.86494 -0.89148 1.70043 1.20527 -1.47802 -1.47802 1.20527 1.70043 -0.89148 -1.86494 0.54733 1.96595 -0.18454 -2.00000 ... ... ..... ....... ......... ........... ............. E17u * 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 -2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 2.00000 ... ... ..... ....... ......... ........... ............. E18u * 2.00000 -0.18454 -1.96595 0.54733 1.86494 -0.89148 -1.70043 1.20527 1.47802 -1.47802 -1.20527 1.70043 0.89148 -1.86494 -0.54733 1.96595 0.18454 -2.00000 0.18454 1.96595 -0.54733 -1.86494 0.89148 1.70043 -1.20527 -1.47802 1.47802 1.20527 -1.70043 -0.89148 1.86494 0.54733 -1.96595 -0.18454 2.00000 -2.00000 0.18454 1.96595 -0.54733 -1.86494 0.89148 1.70043 -1.20527 -1.47802 1.47802 1.20527 -1.70043 -0.89148 1.86494 0.54733 -1.96595 -0.18454 2.00000 -0.18454 -1.96595 0.54733 1.86494 -0.89148 -1.70043 1.20527 1.47802 -1.47802 -1.20527 1.70043 0.89148 -1.86494 -0.54733 1.96595 0.18454 -2.00000 ... ... ..... ....... ......... ........... ............. E19u * 2.00000 -0.36750 -1.86494 1.05286 1.47802 -1.59603 -0.89148 1.92365 0.18454 -1.99147 0.54733 1.79033 -1.20527 -1.34739 1.70043 0.72248 -1.96595 0.00000 1.96595 -0.72248 -1.70043 1.34739 1.20527 -1.79033 -0.54733 1.99147 -0.18454 -1.92365 0.89148 1.59603 -1.47802 -1.05286 1.86494 0.36750 -2.00000 -2.00000 -0.36750 -1.86494 1.05286 1.47802 -1.59603 -0.89148 1.92365 0.18454 -1.99147 0.54733 1.79033 -1.20527 -1.34739 1.70043 0.72248 -1.96595 0.00000 1.96595 -0.72248 -1.70043 1.34739 1.20527 -1.79033 -0.54733 1.99147 -0.18454 -1.92365 0.89148 1.59603 -1.47802 -1.05286 1.86494 0.36750 2.00000 ... ... ..... ....... ......... ........... ............. E20u * 2.00000 -0.54733 -1.70043 1.47802 0.89148 -1.96595 0.18454 1.86494 -1.20527 -1.20527 1.86494 0.18454 -1.96595 0.89148 1.47802 -1.70043 -0.54733 2.00000 -0.54733 -1.70043 1.47802 0.89148 -1.96595 0.18454 1.86494 -1.20527 -1.20527 1.86494 0.18454 -1.96595 0.89148 1.47802 -1.70043 -0.54733 2.00000 -2.00000 0.54733 1.70043 -1.47802 -0.89148 1.96595 -0.18454 -1.86494 1.20527 1.20527 -1.86494 -0.18454 1.96595 -0.89148 -1.47802 1.70043 0.54733 -2.00000 0.54733 1.70043 -1.47802 -0.89148 1.96595 -0.18454 -1.86494 1.20527 1.20527 -1.86494 -0.18454 1.96595 -0.89148 -1.47802 1.70043 0.54733 -2.00000 ... ... ..... ....... ......... ........... ............. E21u * 2.00000 -0.72248 -1.47802 1.79033 0.18454 -1.92365 1.20527 1.05286 -1.96595 0.36750 1.70043 -1.59603 -0.54733 1.99147 -0.89148 -1.34739 1.86494 0.00000 -1.86494 1.34739 0.89148 -1.99147 0.54733 1.59603 -1.70043 -0.36750 1.96595 -1.05286 -1.20527 1.92365 -0.18454 -1.79033 1.47802 0.72248 -2.00000 -2.00000 -0.72248 -1.47802 1.79033 0.18454 -1.92365 1.20527 1.05286 -1.96595 0.36750 1.70043 -1.59603 -0.54733 1.99147 -0.89148 -1.34739 1.86494 0.00000 -1.86494 1.34739 0.89148 -1.99147 0.54733 1.59603 -1.70043 -0.36750 1.96595 -1.05286 -1.20527 1.92365 -0.18454 -1.79033 1.47802 0.72248 2.00000 ... ... ..... ....... ......... ........... ............. E22u * 2.00000 -0.89148 -1.20527 1.96595 -0.54733 -1.47802 1.86494 -0.18454 -1.70043 1.70043 0.18454 -1.86494 1.47802 0.54733 -1.96595 1.20527 0.89148 -2.00000 0.89148 1.20527 -1.96595 0.54733 1.47802 -1.86494 0.18454 1.70043 -1.70043 -0.18454 1.86494 -1.47802 -0.54733 1.96595 -1.20527 -0.89148 2.00000 -2.00000 0.89148 1.20527 -1.96595 0.54733 1.47802 -1.86494 0.18454 1.70043 -1.70043 -0.18454 1.86494 -1.47802 -0.54733 1.96595 -1.20527 -0.89148 2.00000 -0.89148 -1.20527 1.96595 -0.54733 -1.47802 1.86494 -0.18454 -1.70043 1.70043 0.18454 -1.86494 1.47802 0.54733 -1.96595 1.20527 0.89148 -2.00000 ... ... ..... ....... ......... ........... ............. E23u * 2.00000 -1.05286 -0.89148 1.99147 -1.20527 -0.72248 1.96595 -1.34739 -0.54733 1.92365 -1.47802 -0.36750 1.86494 -1.59603 -0.18454 1.79033 -1.70043 0.00000 1.70043 -1.79033 0.18454 1.59603 -1.86494 0.36750 1.47802 -1.92365 0.54733 1.34739 -1.96595 0.72248 1.20527 -1.99147 0.89148 1.05286 -2.00000 -2.00000 -1.05286 -0.89148 1.99147 -1.20527 -0.72248 1.96595 -1.34739 -0.54733 1.92365 -1.47802 -0.36750 1.86494 -1.59603 -0.18454 1.79033 -1.70043 0.00000 1.70043 -1.79033 0.18454 1.59603 -1.86494 0.36750 1.47802 -1.92365 0.54733 1.34739 -1.96595 0.72248 1.20527 -1.99147 0.89148 1.05286 2.00000 ... ... ..... ....... ......... ........... ............. E24u * 2.00000 -1.20527 -0.54733 1.86494 -1.70043 0.18454 1.47802 -1.96595 0.89148 0.89148 -1.96595 1.47802 0.18454 -1.70043 1.86494 -0.54733 -1.20527 2.00000 -1.20527 -0.54733 1.86494 -1.70043 0.18454 1.47802 -1.96595 0.89148 0.89148 -1.96595 1.47802 0.18454 -1.70043 1.86494 -0.54733 -1.20527 2.00000 -2.00000 1.20527 0.54733 -1.86494 1.70043 -0.18454 -1.47802 1.96595 -0.89148 -0.89148 1.96595 -1.47802 -0.18454 1.70043 -1.86494 0.54733 1.20527 -2.00000 1.20527 0.54733 -1.86494 1.70043 -0.18454 -1.47802 1.96595 -0.89148 -0.89148 1.96595 -1.47802 -0.18454 1.70043 -1.86494 0.54733 1.20527 -2.00000 ... ... ..... ....... ......... ........... ............. E25u * 2.00000 -1.34739 -0.18454 1.59603 -1.96595 1.05286 0.54733 -1.79033 1.86494 -0.72248 -0.89148 1.92365 -1.70043 0.36750 1.20527 -1.99147 1.47802 0.00000 -1.47802 1.99147 -1.20527 -0.36750 1.70043 -1.92365 0.89148 0.72248 -1.86494 1.79033 -0.54733 -1.05286 1.96595 -1.59603 0.18454 1.34739 -2.00000 -2.00000 -1.34739 -0.18454 1.59603 -1.96595 1.05286 0.54733 -1.79033 1.86494 -0.72248 -0.89148 1.92365 -1.70043 0.36750 1.20527 -1.99147 1.47802 0.00000 -1.47802 1.99147 -1.20527 -0.36750 1.70043 -1.92365 0.89148 0.72248 -1.86494 1.79033 -0.54733 -1.05286 1.96595 -1.59603 0.18454 1.34739 2.00000 ... ... ..... ....... ......... ........... ............. E26u * 2.00000 -1.47802 0.18454 1.20527 -1.96595 1.70043 -0.54733 -0.89148 1.86494 -1.86494 0.89148 0.54733 -1.70043 1.96595 -1.20527 -0.18454 1.47802 -2.00000 1.47802 -0.18454 -1.20527 1.96595 -1.70043 0.54733 0.89148 -1.86494 1.86494 -0.89148 -0.54733 1.70043 -1.96595 1.20527 0.18454 -1.47802 2.00000 -2.00000 1.47802 -0.18454 -1.20527 1.96595 -1.70043 0.54733 0.89148 -1.86494 1.86494 -0.89148 -0.54733 1.70043 -1.96595 1.20527 0.18454 -1.47802 2.00000 -1.47802 0.18454 1.20527 -1.96595 1.70043 -0.54733 -0.89148 1.86494 -1.86494 0.89148 0.54733 -1.70043 1.96595 -1.20527 -0.18454 1.47802 -2.00000 ... ... ..... ....... ......... ........... ............. E27u * 2.00000 -1.59603 0.54733 0.72248 -1.70043 1.99147 -1.47802 0.36750 0.89148 -1.79033 1.96595 -1.34739 0.18454 1.05286 -1.86494 1.92365 -1.20527 0.00000 1.20527 -1.92365 1.86494 -1.05286 -0.18454 1.34739 -1.96595 1.79033 -0.89148 -0.36750 1.47802 -1.99147 1.70043 -0.72248 -0.54733 1.59603 -2.00000 -2.00000 -1.59603 0.54733 0.72248 -1.70043 1.99147 -1.47802 0.36750 0.89148 -1.79033 1.96595 -1.34739 0.18454 1.05286 -1.86494 1.92365 -1.20527 0.00000 1.20527 -1.92365 1.86494 -1.05286 -0.18454 1.34739 -1.96595 1.79033 -0.89148 -0.36750 1.47802 -1.99147 1.70043 -0.72248 -0.54733 1.59603 2.00000 ... ... ..... ....... ......... ........... ............. E28u * 2.00000 -1.70043 0.89148 0.18454 -1.20527 1.86494 -1.96595 1.47802 -0.54733 -0.54733 1.47802 -1.96595 1.86494 -1.20527 0.18454 0.89148 -1.70043 2.00000 -1.70043 0.89148 0.18454 -1.20527 1.86494 -1.96595 1.47802 -0.54733 -0.54733 1.47802 -1.96595 1.86494 -1.20527 0.18454 0.89148 -1.70043 2.00000 -2.00000 1.70043 -0.89148 -0.18454 1.20527 -1.86494 1.96595 -1.47802 0.54733 0.54733 -1.47802 1.96595 -1.86494 1.20527 -0.18454 -0.89148 1.70043 -2.00000 1.70043 -0.89148 -0.18454 1.20527 -1.86494 1.96595 -1.47802 0.54733 0.54733 -1.47802 1.96595 -1.86494 1.20527 -0.18454 -0.89148 1.70043 -2.00000 ... ... ..... ....... ......... ........... ............. E29u * 2.00000 -1.79033 1.20527 -0.36750 -0.54733 1.34739 -1.86494 1.99147 -1.70043 1.05286 -0.18454 -0.72248 1.47802 -1.92365 1.96595 -1.59603 0.89148 0.00000 -0.89148 1.59603 -1.96595 1.92365 -1.47802 0.72248 0.18454 -1.05286 1.70043 -1.99147 1.86494 -1.34739 0.54733 0.36750 -1.20527 1.79033 -2.00000 -2.00000 -1.79033 1.20527 -0.36750 -0.54733 1.34739 -1.86494 1.99147 -1.70043 1.05286 -0.18454 -0.72248 1.47802 -1.92365 1.96595 -1.59603 0.89148 0.00000 -0.89148 1.59603 -1.96595 1.92365 -1.47802 0.72248 0.18454 -1.05286 1.70043 -1.99147 1.86494 -1.34739 0.54733 0.36750 -1.20527 1.79033 2.00000 ... ... ..... ....... ......... ........... ............. E30u * 2.00000 -1.86494 1.47802 -0.89148 0.18454 0.54733 -1.20527 1.70043 -1.96595 1.96595 -1.70043 1.20527 -0.54733 -0.18454 0.89148 -1.47802 1.86494 -2.00000 1.86494 -1.47802 0.89148 -0.18454 -0.54733 1.20527 -1.70043 1.96595 -1.96595 1.70043 -1.20527 0.54733 0.18454 -0.89148 1.47802 -1.86494 2.00000 -2.00000 1.86494 -1.47802 0.89148 -0.18454 -0.54733 1.20527 -1.70043 1.96595 -1.96595 1.70043 -1.20527 0.54733 0.18454 -0.89148 1.47802 -1.86494 2.00000 -1.86494 1.47802 -0.89148 0.18454 0.54733 -1.20527 1.70043 -1.96595 1.96595 -1.70043 1.20527 -0.54733 -0.18454 0.89148 -1.47802 1.86494 -2.00000 ... ... ..... ....... ......... ........... ............. E31u * 2.00000 -1.92365 1.70043 -1.34739 0.89148 -0.36750 -0.18454 0.72248 -1.20527 1.59603 -1.86494 1.99147 -1.96595 1.79033 -1.47802 1.05286 -0.54733 0.00000 0.54733 -1.05286 1.47802 -1.79033 1.96595 -1.99147 1.86494 -1.59603 1.20527 -0.72248 0.18454 0.36750 -0.89148 1.34739 -1.70043 1.92365 -2.00000 -2.00000 -1.92365 1.70043 -1.34739 0.89148 -0.36750 -0.18454 0.72248 -1.20527 1.59603 -1.86494 1.99147 -1.96595 1.79033 -1.47802 1.05286 -0.54733 0.00000 0.54733 -1.05286 1.47802 -1.79033 1.96595 -1.99147 1.86494 -1.59603 1.20527 -0.72248 0.18454 0.36750 -0.89148 1.34739 -1.70043 1.92365 2.00000 ... ... ..... ....... ......... ........... ............. E32u * 2.00000 -1.96595 1.86494 -1.70043 1.47802 -1.20527 0.89148 -0.54733 0.18454 0.18454 -0.54733 0.89148 -1.20527 1.47802 -1.70043 1.86494 -1.96595 2.00000 -1.96595 1.86494 -1.70043 1.47802 -1.20527 0.89148 -0.54733 0.18454 0.18454 -0.54733 0.89148 -1.20527 1.47802 -1.70043 1.86494 -1.96595 2.00000 -2.00000 1.96595 -1.86494 1.70043 -1.47802 1.20527 -0.89148 0.54733 -0.18454 -0.18454 0.54733 -0.89148 1.20527 -1.47802 1.70043 -1.86494 1.96595 -2.00000 1.96595 -1.86494 1.70043 -1.47802 1.20527 -0.89148 0.54733 -0.18454 -0.18454 0.54733 -0.89148 1.20527 -1.47802 1.70043 -1.86494 1.96595 -2.00000 ... ... ..... ....... ......... ........... ............. E33u * 2.00000 -1.99147 1.96595 -1.92365 1.86494 -1.79033 1.70043 -1.59603 1.47802 -1.34739 1.20527 -1.05286 0.89148 -0.72248 0.54733 -0.36750 0.18454 0.00000 -0.18454 0.36750 -0.54733 0.72248 -0.89148 1.05286 -1.20527 1.34739 -1.47802 1.59603 -1.70043 1.79033 -1.86494 1.92365 -1.96595 1.99147 -2.00000 -2.00000 -1.99147 1.96595 -1.92365 1.86494 -1.79033 1.70043 -1.59603 1.47802 -1.34739 1.20527 -1.05286 0.89148 -0.72248 0.54733 -0.36750 0.18454 0.00000 -0.18454 0.36750 -0.54733 0.72248 -0.89148 1.05286 -1.20527 1.34739 -1.47802 1.59603 -1.70043 1.79033 -1.86494 1.92365 -1.96595 1.99147 2.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.991468352590 = 2*cos(2*π/68) = 2*cos(π/34) = √2*√17−√17+√34−2*√17+2*√17+3*√17+√34−2*√17+2*√34+2*√17/4 1.965946199368 = 2*cos(4*π/68) = 2*cos(π/17) = (1−√17+√34−2*√17+2*√17+3*√17+√34−2*√17+2*√34+2*√17)/8 1.923651286346 = 2*cos(6*π/68) = 2*cos(3*π/34) = √2*√17+√17+√34+2*√17+2*√17−3*√17+√34+2*√17−2*√34−2*√17/4 1.864944458809 = 2*cos(8*π/68) = 2*cos(2*π/17) = (−1+√17+√34−2*√17+2*√17+3*√17−√34−2*√17−2*√34+2*√17)/8 1.790326582710 = 2*cos(10*π/68) = 2*cos(5*π/34) = √2*√17+√17+√34+2*√17−2*√17−3*√17+√34+2*√17−2*√34−2*√17/4 1.700434271459 = 2*cos(12*π/68) = 2*cos(3*π/17) = (1+√17+√34+2*√17+2*√17−3*√17+√34+2*√17−2*√34−2*√17)/8 1.596034454560 = 2*cos(14*π/68) = 2*cos(7*π/34) = √2*√17+√17−√34+2*√17+2*√17−3*√17−√34+2*√17+2*√34−2*√17/4 1.478017834441 = 2*cos(16*π/68) = 2*cos(4*π/17) = (−1+√17−√34−2*√17+2*√17+3*√17+√34−2*√17+2*√34+2*√17)/8 1.347391287293 = 2*cos(18*π/68) = 2*cos(9*π/34) = √2*√17−√17−√34−2*√17+2*√17+3*√17−√34−2*√17−2*√34+2*√17/4 1.205269272759 = 2*cos(20*π/68) = 2*cos(5*π/17) = (1+√17+√34+2*√17−2*√17−3*√17+√34+2*√17−2*√34−2*√17)/8 1.052864325755 = 2*cos(22*π/68) = 2*cos(11*π/34) = √2*√17+√17−√34+2*√17−2*√17−3*√17−√34+2*√17+2*√34−2*√17/4 0.891476711553 = 2*cos(24*π/68) = 2*cos(6*π/17) = (−1−√17+√34+2*√17+2*√17−3*√17−√34+2*√17+2*√34−2*√17)/8 0.722483332374 = 2*cos(26*π/68) = 2*cos(13*π/34) = √2*√17−√17+√34−2*√17−2*√17+3*√17+√34−2*√17+2*√34+2*√17/4 0.547325980144 = 2*cos(28*π/68) = 2*cos(7*π/17) = (1+√17−√34+2*√17+2*√17−3*√17−√34+2*√17+2*√34−2*√17)/8 0.367499035633 = 2*cos(30*π/68) = 2*cos(15*π/34) = √2*√17−√17−√34−2*√17−2*√17+3*√17−√34−2*√17−2*√34+2*√17/4 0.184536718927 = 2*cos(32*π/68) = 2*cos(8*π/17) = (−1+√17+√34−2*√17−2*√17+3*√17−√34−2*√17−2*√34+2*√17)/8 Symmetry of Rotations and Cartesian products Ag R+d+g+i+k+m R_z, z², z⁴, z⁶ E1g R+d+g+i+k+m {R_x, R_y}, {xz, yz}, {xz³, yz³}, {xz⁵, yz⁵} E2g d+g+i+k+m {x²−y², xy}, {z²(x²−y²), xyz²}, {z⁴(x²−y²), xyz⁴} E3g g+i+k+m {xz(x²−3y²), yz(3x²−y²)}, {xz³(x²−3y²), yz³(3x²−y²)} E4g g+i+k+m {(x²−y²)²−4x²y², xy(x²−y²)}, {z²((x²−y²)²−4x²y²), xyz²(x²−y²)} E5g 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)} E6g i+k+m {x²(x²−3y²)²−y²(3x²−y²)², xy(x²−3y²)(3x²−y²)} E7g k+m E8g k+m E9g m E10g m Au p+f+h+j+l z, z³, z⁵ E1u p+f+h+j+l {x, y}, {xz², yz²}, {xz⁴, yz⁴} E2u f+h+j+l {z(x²−y²), xyz}, {z³(x²−y²), xyz³} E3u f+h+j+l {x(x²−3y²), y(3x²−y²)}, {xz²(x²−3y²), yz²(3x²−y²)} E4u h+j+l {z((x²−y²)²−4x²y²), xyz(x²−y²)} E5u 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)} E6u j+l E7u j+l E8u l E9u l Notes: α The order of the C_68h point group is 136, and the order of the principal axis (C₆₈) is 68. The group has 70 irreducible representations. β The C_68h point group is generated by two symmetry elements, which are canonically chosen as C₆₈ and i. Other possible choices are C₆₈ and σ_h, or less commonly S₆₈ with either i or σ_h. γ The lowest nonvanishing multipole moment in C_68h is 4 (quadrupole moment). δ This is an Abelian point group (the commutative law holds between all symmetry operations). The C_68h group is Abelian because all its symmetry operations are coaxial. This is a sufficient condition. In Abelian groups, all symmetry operations form a class of their own, and all irreducible representations are one-dimensional. ε Because the group is Abelian and the maximum order of rotation is >2, some irreducible representations have complex characters. These 132 cases have been combined into 66 two-dimensional representations that are no longer irreducible but have real-valued characters. Accordingly, 66 pairs of left and right rotations have been combined into one two-membered pseudo-class each. ζ The 66 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. 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 constructible polygon, as the order of the principal axis is a product of any number of different Fermat primes (3,5,17,257,65537) times an arbitrary power of two. Therefore, all characters have an algebraic degree which is a power of two and can be expressed as radicals involving only square roots and integer numbers. ι That a regular 17-gon can be constructed with compass and ruler was unknown to mathematicians until Gauss proved it in 1796. The first actual construction was performed thirty years later by Johannes Erchinger in 1825. Regular polygons of order 34,68,136 etc are easily derived from the regular 17-gon by successive halving of angles.

This Character Table for the C_68h point group was created by Gernot Katzer.

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

	C_66h
	C_67h
C₆₈ C_68v	C_68h	D₆₈ D_68h D_68d S₆₈
	C_69h
	C_70h

Character table for the C68h point group

Character table for the C_68h point group