C70v E 2 C70 2 C35 2 C70^3 2 C35^2 2 C14 2 C35^3 2 C10 2 C35^4 2 C70^9 2 C7 2 C70^11 2 C35^6 2 C70^13 2 C5 2 C14^3 2 C35^8 2 C70^17 2 C35^9 2 C70^19 2 C7^2 2 C10^3 2 C35^11 2 C70^23 2 C35^12 2 C14^5 2 C35^13 2 C70^27 2 C5^2 2 C70^29 2 C7^3 2 C70^31 2 C35^16 2 C70^33 2 C35^17 C2 35 sv 35 sd <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> A1 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 ... ..T ....T ......T ........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 -1.00000 ..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 -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 1.00000 ... ... ..... ....... ......... ........... ............. E1 2.00000 1.99195 1.96786 1.92793 1.87247 1.80194 1.71690 1.61803 1.50614 1.38213 1.24698 1.10179 0.94774 0.78605 0.61803 0.44504 0.26847 0.08973 -0.08973 -0.26847 -0.44504 -0.61803 -0.78605 -0.94774 -1.10179 -1.24698 -1.38213 -1.50614 -1.61803 -1.71690 -1.80194 -1.87247 -1.92793 -1.96786 -1.99195 -2.00000 0.00000 0.00000 TT. TT. ..TT. ....TT. ......TT. ........TT. ..........TT. E2 2.00000 1.96786 1.87247 1.71690 1.50614 1.24698 0.94774 0.61803 0.26847 -0.08973 -0.44504 -0.78605 -1.10179 -1.38213 -1.61803 -1.80194 -1.92793 -1.99195 -1.99195 -1.92793 -1.80194 -1.61803 -1.38213 -1.10179 -0.78605 -0.44504 -0.08973 0.26847 0.61803 0.94774 1.24698 1.50614 1.71690 1.87247 1.96786 2.00000 0.00000 0.00000 ... ... TT... ..TT... ....TT... ......TT... ........TT... E3 2.00000 1.92793 1.71690 1.38213 0.94774 0.44504 -0.08973 -0.61803 -1.10179 -1.50614 -1.80194 -1.96786 -1.99195 -1.87247 -1.61803 -1.24698 -0.78605 -0.26847 0.26847 0.78605 1.24698 1.61803 1.87247 1.99195 1.96786 1.80194 1.50614 1.10179 0.61803 0.08973 -0.44504 -0.94774 -1.38213 -1.71690 -1.92793 -2.00000 0.00000 0.00000 ... ... ..... TT..... ..TT..... ....TT..... ......TT..... E4 2.00000 1.87247 1.50614 0.94774 0.26847 -0.44504 -1.10179 -1.61803 -1.92793 -1.99195 -1.80194 -1.38213 -0.78605 -0.08973 0.61803 1.24698 1.71690 1.96786 1.96786 1.71690 1.24698 0.61803 -0.08973 -0.78605 -1.38213 -1.80194 -1.99195 -1.92793 -1.61803 -1.10179 -0.44504 0.26847 0.94774 1.50614 1.87247 2.00000 0.00000 0.00000 ... ... ..... ....... TT....... ..TT....... ....TT....... E5 2.00000 1.80194 1.24698 0.44504 -0.44504 -1.24698 -1.80194 -2.00000 -1.80194 -1.24698 -0.44504 0.44504 1.24698 1.80194 2.00000 1.80194 1.24698 0.44504 -0.44504 -1.24698 -1.80194 -2.00000 -1.80194 -1.24698 -0.44504 0.44504 1.24698 1.80194 2.00000 1.80194 1.24698 0.44504 -0.44504 -1.24698 -1.80194 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... TT......... ..TT......... E6 2.00000 1.71690 0.94774 -0.08973 -1.10179 -1.80194 -1.99195 -1.61803 -0.78605 0.26847 1.24698 1.87247 1.96786 1.50614 0.61803 -0.44504 -1.38213 -1.92793 -1.92793 -1.38213 -0.44504 0.61803 1.50614 1.96786 1.87247 1.24698 0.26847 -0.78605 -1.61803 -1.99195 -1.80194 -1.10179 -0.08973 0.94774 1.71690 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... TT........... E7 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 0.61803 1.61803 2.00000 1.61803 0.61803 -0.61803 -1.61803 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E8 2.00000 1.50614 0.26847 -1.10179 -1.92793 -1.80194 -0.78605 0.61803 1.71690 1.96786 1.24698 -0.08973 -1.38213 -1.99195 -1.61803 -0.44504 0.94774 1.87247 1.87247 0.94774 -0.44504 -1.61803 -1.99195 -1.38213 -0.08973 1.24698 1.96786 1.71690 0.61803 -0.78605 -1.80194 -1.92793 -1.10179 0.26847 1.50614 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E9 2.00000 1.38213 -0.08973 -1.50614 -1.99195 -1.24698 0.26847 1.61803 1.96786 1.10179 -0.44504 -1.71690 -1.92793 -0.94774 0.61803 1.80194 1.87247 0.78605 -0.78605 -1.87247 -1.80194 -0.61803 0.94774 1.92793 1.71690 0.44504 -1.10179 -1.96786 -1.61803 -0.26847 1.24698 1.99195 1.50614 0.08973 -1.38213 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E10 2.00000 1.24698 -0.44504 -1.80194 -1.80194 -0.44504 1.24698 2.00000 1.24698 -0.44504 -1.80194 -1.80194 -0.44504 1.24698 2.00000 1.24698 -0.44504 -1.80194 -1.80194 -0.44504 1.24698 2.00000 1.24698 -0.44504 -1.80194 -1.80194 -0.44504 1.24698 2.00000 1.24698 -0.44504 -1.80194 -1.80194 -0.44504 1.24698 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E11 2.00000 1.10179 -0.78605 -1.96786 -1.38213 0.44504 1.87247 1.61803 -0.08973 -1.71690 -1.80194 -0.26847 1.50614 1.92793 0.61803 -1.24698 -1.99195 -0.94774 0.94774 1.99195 1.24698 -0.61803 -1.92793 -1.50614 0.26847 1.80194 1.71690 0.08973 -1.61803 -1.87247 -0.44504 1.38213 1.96786 0.78605 -1.10179 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E12 2.00000 0.94774 -1.10179 -1.99195 -0.78605 1.24698 1.96786 0.61803 -1.38213 -1.92793 -0.44504 1.50614 1.87247 0.26847 -1.61803 -1.80194 -0.08973 1.71690 1.71690 -0.08973 -1.80194 -1.61803 0.26847 1.87247 1.50614 -0.44504 -1.92793 -1.38213 0.61803 1.96786 1.24698 -0.78605 -1.99195 -1.10179 0.94774 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E13 2.00000 0.78605 -1.38213 -1.87247 -0.08973 1.80194 1.50614 -0.61803 -1.99195 -0.94774 1.24698 1.92793 0.26847 -1.71690 -1.61803 0.44504 1.96786 1.10179 -1.10179 -1.96786 -0.44504 1.61803 1.71690 -0.26847 -1.92793 -1.24698 0.94774 1.99195 0.61803 -1.50614 -1.80194 0.08973 1.87247 1.38213 -0.78605 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E14 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 -1.61803 0.61803 2.00000 0.61803 -1.61803 -1.61803 0.61803 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E15 2.00000 0.44504 -1.80194 -1.24698 1.24698 1.80194 -0.44504 -2.00000 -0.44504 1.80194 1.24698 -1.24698 -1.80194 0.44504 2.00000 0.44504 -1.80194 -1.24698 1.24698 1.80194 -0.44504 -2.00000 -0.44504 1.80194 1.24698 -1.24698 -1.80194 0.44504 2.00000 0.44504 -1.80194 -1.24698 1.24698 1.80194 -0.44504 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E16 2.00000 0.26847 -1.92793 -0.78605 1.71690 1.24698 -1.38213 -1.61803 0.94774 1.87247 -0.44504 -1.99195 -0.08973 1.96786 0.61803 -1.80194 -1.10179 1.50614 1.50614 -1.10179 -1.80194 0.61803 1.96786 -0.08973 -1.99195 -0.44504 1.87247 0.94774 -1.61803 -1.38213 1.24698 1.71690 -0.78605 -1.92793 0.26847 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E17 2.00000 0.08973 -1.99195 -0.26847 1.96786 0.44504 -1.92793 -0.61803 1.87247 0.78605 -1.80194 -0.94774 1.71690 1.10179 -1.61803 -1.24698 1.50614 1.38213 -1.38213 -1.50614 1.24698 1.61803 -1.10179 -1.71690 0.94774 1.80194 -0.78605 -1.87247 0.61803 1.92793 -0.44504 -1.96786 0.26847 1.99195 -0.08973 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E18 2.00000 -0.08973 -1.99195 0.26847 1.96786 -0.44504 -1.92793 0.61803 1.87247 -0.78605 -1.80194 0.94774 1.71690 -1.10179 -1.61803 1.24698 1.50614 -1.38213 -1.38213 1.50614 1.24698 -1.61803 -1.10179 1.71690 0.94774 -1.80194 -0.78605 1.87247 0.61803 -1.92793 -0.44504 1.96786 0.26847 -1.99195 -0.08973 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E19 2.00000 -0.26847 -1.92793 0.78605 1.71690 -1.24698 -1.38213 1.61803 0.94774 -1.87247 -0.44504 1.99195 -0.08973 -1.96786 0.61803 1.80194 -1.10179 -1.50614 1.50614 1.10179 -1.80194 -0.61803 1.96786 0.08973 -1.99195 0.44504 1.87247 -0.94774 -1.61803 1.38213 1.24698 -1.71690 -0.78605 1.92793 0.26847 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E20 2.00000 -0.44504 -1.80194 1.24698 1.24698 -1.80194 -0.44504 2.00000 -0.44504 -1.80194 1.24698 1.24698 -1.80194 -0.44504 2.00000 -0.44504 -1.80194 1.24698 1.24698 -1.80194 -0.44504 2.00000 -0.44504 -1.80194 1.24698 1.24698 -1.80194 -0.44504 2.00000 -0.44504 -1.80194 1.24698 1.24698 -1.80194 -0.44504 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E21 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 -1.61803 -0.61803 2.00000 -0.61803 -1.61803 1.61803 0.61803 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E22 2.00000 -0.78605 -1.38213 1.87247 -0.08973 -1.80194 1.50614 0.61803 -1.99195 0.94774 1.24698 -1.92793 0.26847 1.71690 -1.61803 -0.44504 1.96786 -1.10179 -1.10179 1.96786 -0.44504 -1.61803 1.71690 0.26847 -1.92793 1.24698 0.94774 -1.99195 0.61803 1.50614 -1.80194 -0.08973 1.87247 -1.38213 -0.78605 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E23 2.00000 -0.94774 -1.10179 1.99195 -0.78605 -1.24698 1.96786 -0.61803 -1.38213 1.92793 -0.44504 -1.50614 1.87247 -0.26847 -1.61803 1.80194 -0.08973 -1.71690 1.71690 0.08973 -1.80194 1.61803 0.26847 -1.87247 1.50614 0.44504 -1.92793 1.38213 0.61803 -1.96786 1.24698 0.78605 -1.99195 1.10179 0.94774 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E24 2.00000 -1.10179 -0.78605 1.96786 -1.38213 -0.44504 1.87247 -1.61803 -0.08973 1.71690 -1.80194 0.26847 1.50614 -1.92793 0.61803 1.24698 -1.99195 0.94774 0.94774 -1.99195 1.24698 0.61803 -1.92793 1.50614 0.26847 -1.80194 1.71690 -0.08973 -1.61803 1.87247 -0.44504 -1.38213 1.96786 -0.78605 -1.10179 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E25 2.00000 -1.24698 -0.44504 1.80194 -1.80194 0.44504 1.24698 -2.00000 1.24698 0.44504 -1.80194 1.80194 -0.44504 -1.24698 2.00000 -1.24698 -0.44504 1.80194 -1.80194 0.44504 1.24698 -2.00000 1.24698 0.44504 -1.80194 1.80194 -0.44504 -1.24698 2.00000 -1.24698 -0.44504 1.80194 -1.80194 0.44504 1.24698 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E26 2.00000 -1.38213 -0.08973 1.50614 -1.99195 1.24698 0.26847 -1.61803 1.96786 -1.10179 -0.44504 1.71690 -1.92793 0.94774 0.61803 -1.80194 1.87247 -0.78605 -0.78605 1.87247 -1.80194 0.61803 0.94774 -1.92793 1.71690 -0.44504 -1.10179 1.96786 -1.61803 0.26847 1.24698 -1.99195 1.50614 -0.08973 -1.38213 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E27 2.00000 -1.50614 0.26847 1.10179 -1.92793 1.80194 -0.78605 -0.61803 1.71690 -1.96786 1.24698 0.08973 -1.38213 1.99195 -1.61803 0.44504 0.94774 -1.87247 1.87247 -0.94774 -0.44504 1.61803 -1.99195 1.38213 -0.08973 -1.24698 1.96786 -1.71690 0.61803 0.78605 -1.80194 1.92793 -1.10179 -0.26847 1.50614 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E28 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 0.61803 -1.61803 2.00000 -1.61803 0.61803 0.61803 -1.61803 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E29 2.00000 -1.71690 0.94774 0.08973 -1.10179 1.80194 -1.99195 1.61803 -0.78605 -0.26847 1.24698 -1.87247 1.96786 -1.50614 0.61803 0.44504 -1.38213 1.92793 -1.92793 1.38213 -0.44504 -0.61803 1.50614 -1.96786 1.87247 -1.24698 0.26847 0.78605 -1.61803 1.99195 -1.80194 1.10179 -0.08973 -0.94774 1.71690 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E30 2.00000 -1.80194 1.24698 -0.44504 -0.44504 1.24698 -1.80194 2.00000 -1.80194 1.24698 -0.44504 -0.44504 1.24698 -1.80194 2.00000 -1.80194 1.24698 -0.44504 -0.44504 1.24698 -1.80194 2.00000 -1.80194 1.24698 -0.44504 -0.44504 1.24698 -1.80194 2.00000 -1.80194 1.24698 -0.44504 -0.44504 1.24698 -1.80194 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E31 2.00000 -1.87247 1.50614 -0.94774 0.26847 0.44504 -1.10179 1.61803 -1.92793 1.99195 -1.80194 1.38213 -0.78605 0.08973 0.61803 -1.24698 1.71690 -1.96786 1.96786 -1.71690 1.24698 -0.61803 -0.08973 0.78605 -1.38213 1.80194 -1.99195 1.92793 -1.61803 1.10179 -0.44504 -0.26847 0.94774 -1.50614 1.87247 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E32 2.00000 -1.92793 1.71690 -1.38213 0.94774 -0.44504 -0.08973 0.61803 -1.10179 1.50614 -1.80194 1.96786 -1.99195 1.87247 -1.61803 1.24698 -0.78605 0.26847 0.26847 -0.78605 1.24698 -1.61803 1.87247 -1.99195 1.96786 -1.80194 1.50614 -1.10179 0.61803 -0.08973 -0.44504 0.94774 -1.38213 1.71690 -1.92793 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E33 2.00000 -1.96786 1.87247 -1.71690 1.50614 -1.24698 0.94774 -0.61803 0.26847 0.08973 -0.44504 0.78605 -1.10179 1.38213 -1.61803 1.80194 -1.92793 1.99195 -1.99195 1.92793 -1.80194 1.61803 -1.38213 1.10179 -0.78605 0.44504 -0.08973 -0.26847 0.61803 -0.94774 1.24698 -1.50614 1.71690 -1.87247 1.96786 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E34 2.00000 -1.99195 1.96786 -1.92793 1.87247 -1.80194 1.71690 -1.61803 1.50614 -1.38213 1.24698 -1.10179 0.94774 -0.78605 0.61803 -0.44504 0.26847 -0.08973 -0.08973 0.26847 -0.44504 0.61803 -0.78605 0.94774 -1.10179 1.24698 -1.38213 1.50614 -1.61803 1.71690 -1.80194 1.87247 -1.92793 1.96786 -1.99195 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.991948587990 = 2*cos(2*π/70) = 2*cos(π/35) 1.967859177197 = 2*cos(4*π/70) = 2*cos(2*π/35) 1.927925721392 = 2*cos(6*π/70) = 2*cos(3*π/35) 1.872469741279 = 2*cos(8*π/70) = 2*cos(4*π/35) 1.801937735805 = 2*cos(10*π/70) = 2*cos(π/7) 1.716897587204 = 2*cos(12*π/70) = 2*cos(6*π/35) 1.618033988750 = 2*cos(14*π/70) = 2*cos(π/5) = (√5+1)/2 1.506142932007 = 2*cos(16*π/70) = 2*cos(8*π/35) 1.382125297974 = 2*cos(18*π/70) = 2*cos(9*π/35) 1.246979603717 = 2*cos(20*π/70) = 2*cos(2*π/7) 1.101793962904 = 2*cos(22*π/70) = 2*cos(11*π/35) 0.947737324946 = 2*cos(24*π/70) = 2*cos(12*π/35) 0.786050063308 = 2*cos(26*π/70) = 2*cos(13*π/35) 0.618033988750 = 2*cos(28*π/70) = 2*cos(2*π/5) = (√5−1)/2 0.445041867913 = 2*cos(30*π/70) = 2*cos(3*π/7) 0.268466531635 = 2*cos(32*π/70) = 2*cos(16*π/35) 0.089729660701 = 2*cos(34*π/70) = 2*cos(17*π/35) Symmetry of Rotations and Cartesian products A1 p+d+f+g+h+i+j+k+l+m z, z2, z3, z4, z5, z6 A2 R Rz 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 C70v point group is 140, and the order of the principal axis (C70) is 70. The group has 38 irreducible representations. β The C70v point group is isomorphic to D35d, D35h and D70. γ The C70v point group is generated by two symmetry elements, C70 and any σv (or, non-canonically, any σd). Also, the group may be generated from a σv plus a σd (some pairs will yield smaller groups, though; choosing a minimum angle is safe). δ There are two different sets of symmetry planes containing the principal axis (z axis in standard orientation). By convention, the set denoted as σv has the xz plane as a member, while the yz plane is a member of the σd set. ε The lowest nonvanishing multipole moment in C70v is 2 (dipole moment). ζ This point group is non-Abelian (some symmetry operations are not commutative). Therefore, the character table contains multi-membered classes and degenerate irreducible representations. η Some of the characters in the table are irrational because the order of the principal axis is neither 1,2,3,4 nor 6. These irrational values can be expressed as cosine values, or as solutions of algebraic equations with a leading coefficient of 1. All characters are algebraic integers of a degree much less than half the order of the principal axis. θ The point group corresponds to a polygon inconstructible by the classical means of ruler and compass. Yet it becomes constructible if angle trisection is allowed, e.g., with neusis construction or origami. This is because the order of the principal axis is given by a product of any number of different Pierpont primes (...,5,7,13,17,19,37,73,97,109,163,...) times arbitrary powers of two and three. While some characters of this group can be expressed using real integers and square roots alone, others need complex numbers and third roots.
C68v | ||
C69v | ||
C70 | C70v | C70h D70 D70h D70d S70 |
C71v | ||
C72v |
This Character Table for the C70v point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.