Character Table for Point Group D35d

Character table for the D_35d point group

D35d E 2 C35 2 C35^2 2 C35^3 2 C35^4 2 C7 2 C35^6 2 C5 2 C35^8 2 C35^9 2 C7^2 2 C35^11 2 C35^12 2 C35^13 2 C5^2 2 C7^3 2 C35^16 2 C35^17 35 C2' i 2 S70 2 S70^3 2 S14 2 S10 2 S70^9 2 S70^11 2 S70^13 2 S14^3 2 S70^17 2 S70^19 2 S10^3 2 S70^23 2 S14^5 2 S70^27 2 S70^29 2 S70^31 2 S70^33 35 sd <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> A1g 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 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 A2g 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 -1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 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 ... ..... ....... ......... ........... ............. E1g 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 0.00000 2.00000 -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 0.00000 TT. ... ..TT. ....... ......TT. ........... ..........TT. E2g 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 0.00000 2.00000 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 0.00000 ... ... TT... ....... ....TT... ........... ........TT... E3g 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 0.00000 2.00000 -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 0.00000 ... ... ..... ....... ..TT..... ........... ......TT..... E4g 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 0.00000 2.00000 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 0.00000 ... ... ..... ....... TT....... ........... ....TT....... E5g 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 0.00000 2.00000 -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 0.00000 ... ... ..... ....... ......... ........... ..TT......... E6g 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 0.00000 2.00000 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 0.00000 ... ... ..... ....... ......... ........... TT........... E7g 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 0.00000 2.00000 -1.61803 0.61803 2.00000 0.61803 -1.61803 -1.61803 0.61803 2.00000 0.61803 -1.61803 -1.61803 0.61803 2.00000 0.61803 -1.61803 -1.61803 0.61803 0.00000 ... ... ..... ....... ......... ........... ............. E8g 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 0.00000 2.00000 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 0.00000 ... ... ..... ....... ......... ........... ............. E9g 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 0.00000 2.00000 -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 0.00000 ... ... ..... ....... ......... ........... ............. E10g 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 0.00000 2.00000 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 0.00000 ... ... ..... ....... ......... ........... ............. E11g 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 0.00000 2.00000 -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 0.00000 ... ... ..... ....... ......... ........... ............. E12g 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 0.00000 2.00000 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 0.00000 ... ... ..... ....... ......... ........... ............. E13g 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 0.00000 2.00000 -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 0.00000 ... ... ..... ....... ......... ........... ............. E14g 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 0.00000 2.00000 0.61803 -1.61803 2.00000 -1.61803 0.61803 0.61803 -1.61803 2.00000 -1.61803 0.61803 0.61803 -1.61803 2.00000 -1.61803 0.61803 0.61803 -1.61803 0.00000 ... ... ..... ....... ......... ........... ............. E15g 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 0.00000 2.00000 -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 0.00000 ... ... ..... ....... ......... ........... ............. E16g 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 0.00000 2.00000 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 0.00000 ... ... ..... ....... ......... ........... ............. E17g 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 0.00000 2.00000 -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 0.00000 ... ... ..... ....... ......... ........... ............. A1u 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 ... ... ..... ....... ......... ........... ............. A2u 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 1.00000 ... ..T ..... ......T ......... ..........T ............. E1u 2.00000 1.96786 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 0.00000 -2.00000 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 0.00000 ... TT. ..... ....TT. ......... ........TT. ............. E2u 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 0.00000 -2.00000 -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 0.00000 ... ... ..... ..TT... ......... ......TT... ............. E3u 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 0.00000 -2.00000 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 0.00000 ... ... ..... TT..... ......... ....TT..... ............. E4u 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 0.00000 -2.00000 -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 0.00000 ... ... ..... ....... ......... ..TT....... ............. E5u 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 0.00000 -2.00000 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 0.00000 ... ... ..... ....... ......... TT......... ............. E6u 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 0.00000 -2.00000 -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 0.00000 ... ... ..... ....... ......... ........... ............. E7u 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 0.00000 -2.00000 1.61803 -0.61803 -2.00000 -0.61803 1.61803 1.61803 -0.61803 -2.00000 -0.61803 1.61803 1.61803 -0.61803 -2.00000 -0.61803 1.61803 1.61803 -0.61803 0.00000 ... ... ..... ....... ......... ........... ............. E8u 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 0.00000 -2.00000 -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 0.00000 ... ... ..... ....... ......... ........... ............. E9u 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 0.00000 -2.00000 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 0.00000 ... ... ..... ....... ......... ........... ............. E10u 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 0.00000 -2.00000 -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 0.00000 ... ... ..... ....... ......... ........... ............. E11u 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 0.00000 -2.00000 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 0.00000 ... ... ..... ....... ......... ........... ............. E12u 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 0.00000 -2.00000 -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 0.00000 ... ... ..... ....... ......... ........... ............. E13u 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 0.00000 -2.00000 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 0.00000 ... ... ..... ....... ......... ........... ............. E14u 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 0.00000 -2.00000 -0.61803 1.61803 -2.00000 1.61803 -0.61803 -0.61803 1.61803 -2.00000 1.61803 -0.61803 -0.61803 1.61803 -2.00000 1.61803 -0.61803 -0.61803 1.61803 0.00000 ... ... ..... ....... ......... ........... ............. E15u 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 0.00000 -2.00000 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 0.00000 ... ... ..... ....... ......... ........... ............. E16u 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 0.00000 -2.00000 -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 0.00000 ... ... ..... ....... ......... ........... ............. E17u 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 0.00000 -2.00000 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 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 A1g d+g+i+k+m z², z⁴, z⁶ A2g R R_z 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 A2u 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 D_35d point group is 140, and the order of the principal axis (S₇₀) is 70. The group has 38 irreducible representations. β The D_35d point group is isomorphic to D_35h, C_70v and D₇₀. γ The D_35d point group is generated by two symmetry elements, S₇₀ 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). The canonical choice, however, is to use redundant generators: C₃₅, C₂^′ and i. δ The group contains one set of C₂^′ symmetry axes perpendicular to the principal (z) axis. The x axis (but not the y axis) is a member of that set. Reversely, the single set of symmetry planes denoted σ_d contains the yz plane but not the xz plane. ε The lowest nonvanishing multipole moment in D_35d 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 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.

This Character Table for the D_35d point group was created by Gernot Katzer.

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

	D_33d
	D_34d
C₃₅ C_35v C_35h D₃₅ D_35h	D_35d
	D_36d
	D_37d

Character table for the D35d point group

Character table for the D_35d point group