D68 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 34 C2' 34 C2" <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> A1 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 ... ... ....T ....... ........T ........... ............T A2 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 -1.00000 -1.00000 ..T ..T ..... ......T ......... ..........T ............. B1 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 1.00000 -1.00000 ... ... ..... ....... ......... ........... ............. B2 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 ... ... ..... ....... ......... ........... ............. E1 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 0.00000 0.00000 TT. TT. ..TT. ....TT. ......TT. ........TT. ..........TT. E2 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 0.00000 0.00000 ... ... TT... ..TT... ....TT... ......TT... ........TT... E3 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 0.00000 0.00000 ... ... ..... TT..... ..TT..... ....TT..... ......TT..... E4 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 0.00000 0.00000 ... ... ..... ....... TT....... ..TT....... ....TT....... E5 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 0.00000 0.00000 ... ... ..... ....... ......... TT......... ..TT......... E6 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 0.00000 0.00000 ... ... ..... ....... ......... ........... TT........... E7 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E8 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E9 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E10 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E11 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E12 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E13 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E14 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E15 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E16 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E17 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E18 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E19 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E20 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E21 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E22 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E23 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E24 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E25 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E26 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E27 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E28 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E29 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E30 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E31 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E32 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E33 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. 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 A1 d+g+i+k+m z2, z4, z6 A2 R+p+f+h+j+l Rz, z, z3, z5 E1 R+p+d+f+g+h+i+j+k+l+m {Rx, Ry}, {x, y}, {xz, yz}, {xz2, yz2}, {xz3, yz3}, {xz4, yz4}, {xz5, yz5} E2 d+f+g+h+i+j+k+l+m {x2−y2, xy}, {z(x2−y2), xyz}, {z2(x2−y2), xyz2}, {z3(x2−y2), xyz3}, {z4(x2−y2), xyz4} E3 f+g+h+i+j+k+l+m {x(x2−3y2), y(3x2−y2)}, {xz(x2−3y2), yz(3x2−y2)}, {xz2(x2−3y2), yz2(3x2−y2)}, {xz3(x2−3y2), yz3(3x2−y2)} E4 g+h+i+j+k+l+m {(x2−y2)2−4x2y2, xy(x2−y2)}, {z((x2−y2)2−4x2y2), xyz(x2−y2)}, {z2((x2−y2)2−4x2y2), xyz2(x2−y2)} E5 h+i+j+k+l+m {x(x2−(5+2√5)y2)(x2−(5−2√5)y2), y((5+2√5)x2−y2)((5−2√5)x2−y2)}, {xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), yz((5+2√5)x2−y2)((5−2√5)x2−y2)} E6 i+j+k+l+m {x2(x2−3y2)2−y2(3x2−y2)2, xy(x2−3y2)(3x2−y2)} E7 j+k+l+m E8 k+l+m E9 l+m E10 m Notes: α The order of the D68 point group is 136, and the order of the principal axis (C68) is 68. The group has 37 irreducible representations. β The D68 point group is isomorphic to D34d and C68v. γ The D68 point group is generated by two symmetry elements, C68 and a perpendicular C2′ (or, non-canonically, C2″). Also, the group may be generated from a C2′ plus a C2″ (some pairs will yield smaller groups, though; choosing a minimum angle is safe). δ There are two different sets of twofold symmetry axes perpendicular to the principal axis (z axis in standard orientation). By convention, the set denoted as C2′ contains both the x and y axes. ε The lowest nonvanishing multipole moment in D68 is 4 (quadrupole moment). ζ This point group is non-Abelian (some symmetry operations are not commutative). Therefore, the character table contains multi-membered classes and degenerate irreducible representations. η The point group is chiral, as it does not contain any mirroring operation. θ 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. ι 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. κ 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.
| D66 | ||
| D67 | ||
| C68 C68v C68h | D68 | D68h D68d S68 |
| D69 | ||
| D70 |
This Character Table for the D68 point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.