Character Table for Point Group D70d

Character table for the D_70d point group

D70d E 2 S140 2 C70 2 S140^3 2 C35 2 S28 2 C70^3 2 S20 2 C35^2 2 S140^9 2 C14 2 S140^11 2 C35^3 2 S140^13 2 C10 2 S28^3 2 C35^4 2 S140^17 2 C70^9 2 S140^19 2 C7 2 S20^3 2 C70^11 2 S140^23 2 C35^6 2 S28^5 2 C70^13 2 S140^27 2 C5 2 S140^29 2 C14^3 2 S140^31 2 C35^8 2 S140^33 2 C70^17 2 S4 2 C35^9 2 S140^37 2 C70^19 2 S140^39 2 C7^2 2 S140^41 2 C10^3 2 S140^43 2 C35^11 2 S28^9 2 C70^23 2 S140^47 2 C35^12 2 S20^7 2 C14^5 2 S140^51 2 C35^13 2 S140^53 2 C70^27 2 S28^11 2 C5^2 2 S140^57 2 C70^29 2 S140^59 2 C7^3 2 S140^61 2 C70^31 2 S20^9 2 C35^16 2 S28^13 2 C70^33 2 S140^67 2 C35^17 2 S140^69 C2 70 C2' 70 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 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 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 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 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 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -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 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 ... ..T ..... ......T ......... ..........T ............. E1 2.00000 1.99799 1.99195 1.98190 1.96786 1.94986 1.92793 1.90211 1.87247 1.83906 1.80194 1.76119 1.71690 1.66915 1.61803 1.56366 1.50614 1.44559 1.38213 1.31588 1.24698 1.17557 1.10179 1.02580 0.94774 0.86777 0.78605 0.70275 0.61803 0.53207 0.44504 0.35711 0.26847 0.17928 0.08973 0.00000 -0.08973 -0.17928 -0.26847 -0.35711 -0.44504 -0.53207 -0.61803 -0.70275 -0.78605 -0.86777 -0.94774 -1.02580 -1.10179 -1.17557 -1.24698 -1.31588 -1.38213 -1.44559 -1.50614 -1.56366 -1.61803 -1.66915 -1.71690 -1.76119 -1.80194 -1.83906 -1.87247 -1.90211 -1.92793 -1.94986 -1.96786 -1.98190 -1.99195 -1.99799 -2.00000 0.00000 0.00000 ... TT. ..... ....TT. ......... ........TT. ............. E2 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 -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... E3 2.00000 1.98190 1.92793 1.83906 1.71690 1.56366 1.38213 1.17557 0.94774 0.70275 0.44504 0.17928 -0.08973 -0.35711 -0.61803 -0.86777 -1.10179 -1.31588 -1.50614 -1.66915 -1.80194 -1.90211 -1.96786 -1.99799 -1.99195 -1.94986 -1.87247 -1.76119 -1.61803 -1.44559 -1.24698 -1.02580 -0.78605 -0.53207 -0.26847 0.00000 0.26847 0.53207 0.78605 1.02580 1.24698 1.44559 1.61803 1.76119 1.87247 1.94986 1.99195 1.99799 1.96786 1.90211 1.80194 1.66915 1.50614 1.31588 1.10179 0.86777 0.61803 0.35711 0.08973 -0.17928 -0.44504 -0.70275 -0.94774 -1.17557 -1.38213 -1.56366 -1.71690 -1.83906 -1.92793 -1.98190 -2.00000 0.00000 0.00000 ... ... ..... TT..... ......... ....TT..... ............. E4 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 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....... E5 2.00000 1.94986 1.80194 1.56366 1.24698 0.86777 0.44504 0.00000 -0.44504 -0.86777 -1.24698 -1.56366 -1.80194 -1.94986 -2.00000 -1.94986 -1.80194 -1.56366 -1.24698 -0.86777 -0.44504 0.00000 0.44504 0.86777 1.24698 1.56366 1.80194 1.94986 2.00000 1.94986 1.80194 1.56366 1.24698 0.86777 0.44504 0.00000 -0.44504 -0.86777 -1.24698 -1.56366 -1.80194 -1.94986 -2.00000 -1.94986 -1.80194 -1.56366 -1.24698 -0.86777 -0.44504 0.00000 0.44504 0.86777 1.24698 1.56366 1.80194 1.94986 2.00000 1.94986 1.80194 1.56366 1.24698 0.86777 0.44504 0.00000 -0.44504 -0.86777 -1.24698 -1.56366 -1.80194 -1.94986 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... TT......... ............. E6 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 -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........... E7 2.00000 1.90211 1.61803 1.17557 0.61803 0.00000 -0.61803 -1.17557 -1.61803 -1.90211 -2.00000 -1.90211 -1.61803 -1.17557 -0.61803 0.00000 0.61803 1.17557 1.61803 1.90211 2.00000 1.90211 1.61803 1.17557 0.61803 0.00000 -0.61803 -1.17557 -1.61803 -1.90211 -2.00000 -1.90211 -1.61803 -1.17557 -0.61803 0.00000 0.61803 1.17557 1.61803 1.90211 2.00000 1.90211 1.61803 1.17557 0.61803 0.00000 -0.61803 -1.17557 -1.61803 -1.90211 -2.00000 -1.90211 -1.61803 -1.17557 -0.61803 0.00000 0.61803 1.17557 1.61803 1.90211 2.00000 1.90211 1.61803 1.17557 0.61803 0.00000 -0.61803 -1.17557 -1.61803 -1.90211 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E8 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 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 ... ... ..... ....... ......... ........... ............. E9 2.00000 1.83906 1.38213 0.70275 -0.08973 -0.86777 -1.50614 -1.90211 -1.99195 -1.76119 -1.24698 -0.53207 0.26847 1.02580 1.61803 1.94986 1.96786 1.66915 1.10179 0.35711 -0.44504 -1.17557 -1.71690 -1.98190 -1.92793 -1.56366 -0.94774 -0.17928 0.61803 1.31588 1.80194 1.99799 1.87247 1.44559 0.78605 0.00000 -0.78605 -1.44559 -1.87247 -1.99799 -1.80194 -1.31588 -0.61803 0.17928 0.94774 1.56366 1.92793 1.98190 1.71690 1.17557 0.44504 -0.35711 -1.10179 -1.66915 -1.96786 -1.94986 -1.61803 -1.02580 -0.26847 0.53207 1.24698 1.76119 1.99195 1.90211 1.50614 0.86777 0.08973 -0.70275 -1.38213 -1.83906 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E10 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 -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 ... ... ..... ....... ......... ........... ............. E11 2.00000 1.76119 1.10179 0.17928 -0.78605 -1.56366 -1.96786 -1.90211 -1.38213 -0.53207 0.44504 1.31588 1.87247 1.98190 1.61803 0.86777 -0.08973 -1.02580 -1.71690 -1.99799 -1.80194 -1.17557 -0.26847 0.70275 1.50614 1.94986 1.92793 1.44559 0.61803 -0.35711 -1.24698 -1.83906 -1.99195 -1.66915 -0.94774 0.00000 0.94774 1.66915 1.99195 1.83906 1.24698 0.35711 -0.61803 -1.44559 -1.92793 -1.94986 -1.50614 -0.70275 0.26847 1.17557 1.80194 1.99799 1.71690 1.02580 0.08973 -0.86777 -1.61803 -1.98190 -1.87247 -1.31588 -0.44504 0.53207 1.38213 1.90211 1.96786 1.56366 0.78605 -0.17928 -1.10179 -1.76119 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E12 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 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 ... ... ..... ....... ......... ........... ............. E13 2.00000 1.66915 0.78605 -0.35711 -1.38213 -1.94986 -1.87247 -1.17557 -0.08973 1.02580 1.80194 1.98190 1.50614 0.53207 -0.61803 -1.56366 -1.99195 -1.76119 -0.94774 0.17928 1.24698 1.90211 1.92793 1.31588 0.26847 -0.86777 -1.71690 -1.99799 -1.61803 -0.70275 0.44504 1.44559 1.96786 1.83906 1.10179 0.00000 -1.10179 -1.83906 -1.96786 -1.44559 -0.44504 0.70275 1.61803 1.99799 1.71690 0.86777 -0.26847 -1.31588 -1.92793 -1.90211 -1.24698 -0.17928 0.94774 1.76119 1.99195 1.56366 0.61803 -0.53207 -1.50614 -1.98190 -1.80194 -1.02580 0.08973 1.17557 1.87247 1.94986 1.38213 0.35711 -0.78605 -1.66915 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E14 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 -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 ... ... ..... ....... ......... ........... ............. E15 2.00000 1.56366 0.44504 -0.86777 -1.80194 -1.94986 -1.24698 0.00000 1.24698 1.94986 1.80194 0.86777 -0.44504 -1.56366 -2.00000 -1.56366 -0.44504 0.86777 1.80194 1.94986 1.24698 0.00000 -1.24698 -1.94986 -1.80194 -0.86777 0.44504 1.56366 2.00000 1.56366 0.44504 -0.86777 -1.80194 -1.94986 -1.24698 0.00000 1.24698 1.94986 1.80194 0.86777 -0.44504 -1.56366 -2.00000 -1.56366 -0.44504 0.86777 1.80194 1.94986 1.24698 0.00000 -1.24698 -1.94986 -1.80194 -0.86777 0.44504 1.56366 2.00000 1.56366 0.44504 -0.86777 -1.80194 -1.94986 -1.24698 0.00000 1.24698 1.94986 1.80194 0.86777 -0.44504 -1.56366 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E16 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 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 ... ... ..... ....... ......... ........... ............. E17 2.00000 1.44559 0.08973 -1.31588 -1.99195 -1.56366 -0.26847 1.17557 1.96786 1.66915 0.44504 -1.02580 -1.92793 -1.76119 -0.61803 0.86777 1.87247 1.83906 0.78605 -0.70275 -1.80194 -1.90211 -0.94774 0.53207 1.71690 1.94986 1.10179 -0.35711 -1.61803 -1.98190 -1.24698 0.17928 1.50614 1.99799 1.38213 0.00000 -1.38213 -1.99799 -1.50614 -0.17928 1.24698 1.98190 1.61803 0.35711 -1.10179 -1.94986 -1.71690 -0.53207 0.94774 1.90211 1.80194 0.70275 -0.78605 -1.83906 -1.87247 -0.86777 0.61803 1.76119 1.92793 1.02580 -0.44504 -1.66915 -1.96786 -1.17557 0.26847 1.56366 1.99195 1.31588 -0.08973 -1.44559 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. 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E63 2.00000 -1.90211 1.61803 -1.17557 0.61803 0.00000 -0.61803 1.17557 -1.61803 1.90211 -2.00000 1.90211 -1.61803 1.17557 -0.61803 0.00000 0.61803 -1.17557 1.61803 -1.90211 2.00000 -1.90211 1.61803 -1.17557 0.61803 0.00000 -0.61803 1.17557 -1.61803 1.90211 -2.00000 1.90211 -1.61803 1.17557 -0.61803 0.00000 0.61803 -1.17557 1.61803 -1.90211 2.00000 -1.90211 1.61803 -1.17557 0.61803 0.00000 -0.61803 1.17557 -1.61803 1.90211 -2.00000 1.90211 -1.61803 1.17557 -0.61803 0.00000 0.61803 -1.17557 1.61803 -1.90211 2.00000 -1.90211 1.61803 -1.17557 0.61803 0.00000 -0.61803 1.17557 -1.61803 1.90211 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E64 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 -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 ... ... ..... ....... ......... ........... ............. E65 2.00000 -1.94986 1.80194 -1.56366 1.24698 -0.86777 0.44504 0.00000 -0.44504 0.86777 -1.24698 1.56366 -1.80194 1.94986 -2.00000 1.94986 -1.80194 1.56366 -1.24698 0.86777 -0.44504 0.00000 0.44504 -0.86777 1.24698 -1.56366 1.80194 -1.94986 2.00000 -1.94986 1.80194 -1.56366 1.24698 -0.86777 0.44504 0.00000 -0.44504 0.86777 -1.24698 1.56366 -1.80194 1.94986 -2.00000 1.94986 -1.80194 1.56366 -1.24698 0.86777 -0.44504 0.00000 0.44504 -0.86777 1.24698 -1.56366 1.80194 -1.94986 2.00000 -1.94986 1.80194 -1.56366 1.24698 -0.86777 0.44504 0.00000 -0.44504 0.86777 -1.24698 1.56366 -1.80194 1.94986 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ..TT......... E66 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 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....... ............. E67 2.00000 -1.98190 1.92793 -1.83906 1.71690 -1.56366 1.38213 -1.17557 0.94774 -0.70275 0.44504 -0.17928 -0.08973 0.35711 -0.61803 0.86777 -1.10179 1.31588 -1.50614 1.66915 -1.80194 1.90211 -1.96786 1.99799 -1.99195 1.94986 -1.87247 1.76119 -1.61803 1.44559 -1.24698 1.02580 -0.78605 0.53207 -0.26847 0.00000 0.26847 -0.53207 0.78605 -1.02580 1.24698 -1.44559 1.61803 -1.76119 1.87247 -1.94986 1.99195 -1.99799 1.96786 -1.90211 1.80194 -1.66915 1.50614 -1.31588 1.10179 -0.86777 0.61803 -0.35711 0.08973 0.17928 -0.44504 0.70275 -0.94774 1.17557 -1.38213 1.56366 -1.71690 1.83906 -1.92793 1.98190 -2.00000 0.00000 0.00000 ... ... ..... ....... ..TT..... ........... ......TT..... E68 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 -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... ............. E69 2.00000 -1.99799 1.99195 -1.98190 1.96786 -1.94986 1.92793 -1.90211 1.87247 -1.83906 1.80194 -1.76119 1.71690 -1.66915 1.61803 -1.56366 1.50614 -1.44559 1.38213 -1.31588 1.24698 -1.17557 1.10179 -1.02580 0.94774 -0.86777 0.78605 -0.70275 0.61803 -0.53207 0.44504 -0.35711 0.26847 -0.17928 0.08973 0.00000 -0.08973 0.17928 -0.26847 0.35711 -0.44504 0.53207 -0.61803 0.70275 -0.78605 0.86777 -0.94774 1.02580 -1.10179 1.17557 -1.24698 1.31588 -1.38213 1.44559 -1.50614 1.56366 -1.61803 1.66915 -1.71690 1.76119 -1.80194 1.83906 -1.87247 1.90211 -1.92793 1.94986 -1.96786 1.98190 -1.99195 1.99799 -2.00000 0.00000 0.00000 TT. ... ..TT. ....... ......TT. ........... ..........TT. Irrational character values: 1.997986133083 = 2*cos(2*π/140) = 2*cos(π/70) 1.991948587990 = 2*cos(4*π/140) = 2*cos(π/35) 1.981899523536 = 2*cos(6*π/140) = 2*cos(3*π/70) 1.967859177197 = 2*cos(8*π/140) = 2*cos(2*π/35) 1.949855824364 = 2*cos(10*π/140) = 2*cos(π/14) 1.927925721392 = 2*cos(12*π/140) = 2*cos(3*π/35) 1.902113032590 = 2*cos(14*π/140) = 2*cos(π/10) = (√10+2*√5)/2 1.872469741279 = 2*cos(16*π/140) = 2*cos(4*π/35) 1.839055545103 = 2*cos(18*π/140) = 2*cos(9*π/70) 1.801937735805 = 2*cos(20*π/140) = 2*cos(π/7) 1.761191063713 = 2*cos(22*π/140) = 2*cos(11*π/70) 1.716897587204 = 2*cos(24*π/140) = 2*cos(6*π/35) 1.669146507443 = 2*cos(26*π/140) = 2*cos(13*π/70) 1.618033988750 = 2*cos(28*π/140) = 2*cos(π/5) = (√5+1)/2 1.563662964936 = 2*cos(30*π/140) = 2*cos(3*π/14) 1.506142932007 = 2*cos(32*π/140) = 2*cos(8*π/35) 1.445589727655 = 2*cos(34*π/140) = 2*cos(17*π/70) 1.382125297974 = 2*cos(36*π/140) = 2*cos(9*π/35) 1.315877451879 = 2*cos(38*π/140) = 2*cos(19*π/70) 1.246979603717 = 2*cos(40*π/140) = 2*cos(2*π/7) 1.175570504585 = 2*cos(42*π/140) = 2*cos(3*π/10) = (√10−2*√5)/2 1.101793962904 = 2*cos(44*π/140) = 2*cos(11*π/35) 1.025798554812 = 2*cos(46*π/140) = 2*cos(23*π/70) 0.947737324946 = 2*cos(48*π/140) = 2*cos(12*π/35) 0.867767478235 = 2*cos(50*π/140) = 2*cos(5*π/14) 0.786050063308 = 2*cos(52*π/140) = 2*cos(13*π/35) 0.702749648163 = 2*cos(54*π/140) = 2*cos(27*π/70) 0.618033988750 = 2*cos(56*π/140) = 2*cos(2*π/5) = (√5−1)/2 0.532073691133 = 2*cos(58*π/140) = 2*cos(29*π/70) 0.445041867913 = 2*cos(60*π/140) = 2*cos(3*π/7) 0.357113789597 = 2*cos(62*π/140) = 2*cos(31*π/70) 0.268466531635 = 2*cos(64*π/140) = 2*cos(16*π/35) 0.179278617807 = 2*cos(66*π/140) = 2*cos(33*π/70) 0.089729660701 = 2*cos(68*π/140) = 2*cos(17*π/35) Symmetry of Rotations and Cartesian products A1 d+g+i+k+m z², z⁴, z⁶ A2 R R_z B2 p+f+h+j+l z, z³, z⁵ E1 p+f+h+j+l {x, y}, {xz², yz²}, {xz⁴, yz⁴} E2 d+g+i+k+m {x²−y², xy}, {z²(x²−y²), xyz²}, {z⁴(x²−y²), xyz⁴} E3 f+h+j+l {x(x²−3y²), y(3x²−y²)}, {xz²(x²−3y²), yz²(3x²−y²)} E4 g+i+k+m {(x²−y²)²−4x²y², xy(x²−y²)}, {z²((x²−y²)²−4x²y²), xyz²(x²−y²)} E5 h+j+l {x(x²−(5+2√5)y²)(x²−(5−2√5)y²), y((5+2√5)x²−y²)((5−2√5)x²−y²)} E6 i+k+m {x²(x²−3y²)²−y²(3x²−y²)², xy(x²−3y²)(3x²−y²)} E7 j+l E8 k+m E9 l E10 m E61 m E62 l E63 k+m E64 j+l E65 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)} E66 h+j+l {z((x²−y²)²−4x²y²), xyz(x²−y²)} E67 g+i+k+m {xz(x²−3y²), yz(3x²−y²)}, {xz³(x²−3y²), yz³(3x²−y²)} E68 f+h+j+l {z(x²−y²), xyz}, {z³(x²−y²), xyz³} E69 R+d+g+i+k+m {R_x, R_y}, {xz, yz}, {xz³, yz³}, {xz⁵, yz⁵} Notes: α The order of the D_70d point group is 280, and the order of the principal axis (S₁₄₀) is 140. The group has 73 irreducible representations. β The D_70d point group is isomorphic to C_140v and D₁₄₀. γ The D_70d point group is generated by two symmetry elements, S₁₄₀ and either a perpendicular C₂^′ or a vertical σ_d. Also, the group may be generated from a C₂^′ plus a σ_d (some pairs will yield smaller groups, though; choosing a minimum angle is safe). δ The group contains one set of twofold symmetry axes (C₂^′) perpendicular to the principal (z) axis. Both x and y axes are members of that set. ε The single σ_d set of symmetry planes contains neither the xz nor the yz planes; but it contains the median plane (x+y)z. ζ The lowest nonvanishing multipole moment in D_70d is 4 (quadrupole moment). η This point group is non-Abelian (some symmetry operations are not commutative). Therefore, the character table contains multi-membered classes and degenerate irreducible representations. 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 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.

This Character Table for the D_70d point group was created by Gernot Katzer.

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

	D_68d
	D_69d
C₇₀ C_70v C_70h D₇₀ D_70h	D_70d	S₇₀
	D_71d
	D_72d

Character table for the D70d point group

Character table for the D_70d point group