Character Table for Point Group C85v

Character table for the C_85v point group

C85v E 2 C85 2 C85^2 2 C85^3 2 C85^4 2 C17 2 C85^6 2 C85^7 2 C85^8 2 C85^9 2 C17^2 2 C85^11 2 C85^12 2 C85^13 2 C85^14 2 C17^3 2 C85^16 2 C5 2 C85^18 2 C85^19 2 C17^4 2 C85^21 2 C85^22 2 C85^23 2 C85^24 2 C17^5 2 C85^26 2 C85^27 2 C85^28 2 C85^29 2 C17^6 2 C85^31 2 C85^32 2 C85^33 2 C5^2 2 C17^7 2 C85^36 2 C85^37 2 C85^38 2 C85^39 2 C17^8 2 C85^41 2 C85^42 85 sv <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 ... ..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 1.00000 1.00000 1.00000 1.00000 1.00000 -1.00000 ..T ... ..... ....... ......... ........... ............. E1 2.00000 1.99454 1.97818 1.95102 1.91321 1.86494 1.80649 1.73818 1.66037 1.57349 1.47802 1.37447 1.26342 1.14547 1.02126 0.89148 0.75682 0.61803 0.47587 0.33111 0.18454 0.03696 -0.11082 -0.25800 -0.40376 -0.54733 -0.68790 -0.82471 -0.95702 -1.08411 -1.20527 -1.31985 -1.42722 -1.52680 -1.61803 -1.70043 -1.77355 -1.83697 -1.89037 -1.93344 -1.96595 -1.98772 -1.99863 0.00000 TT. TT. ..TT. ....TT. ......TT. ........TT. ..........TT. E2 2.00000 1.97818 1.91321 1.80649 1.66037 1.47802 1.26342 1.02126 0.75682 0.47587 0.18454 -0.11082 -0.40376 -0.68790 -0.95702 -1.20527 -1.42722 -1.61803 -1.77355 -1.89037 -1.96595 -1.99863 -1.98772 -1.93344 -1.83697 -1.70043 -1.52680 -1.31985 -1.08411 -0.82471 -0.54733 -0.25800 0.03696 0.33111 0.61803 0.89148 1.14547 1.37447 1.57349 1.73818 1.86494 1.95102 1.99454 0.00000 ... ... TT... ..TT... ....TT... ......TT... ........TT... E3 2.00000 1.95102 1.80649 1.57349 1.26342 0.89148 0.47587 0.03696 -0.40376 -0.82471 -1.20527 -1.52680 -1.77355 -1.93344 -1.99863 -1.96595 -1.83697 -1.61803 -1.31985 -0.95702 -0.54733 -0.11082 0.33111 0.75682 1.14547 1.47802 1.73818 1.91321 1.99454 1.97818 1.86494 1.66037 1.37447 1.02126 0.61803 0.18454 -0.25800 -0.68790 -1.08411 -1.42722 -1.70043 -1.89037 -1.98772 0.00000 ... ... ..... TT..... ..TT..... ....TT..... ......TT..... E4 2.00000 1.91321 1.66037 1.26342 0.75682 0.18454 -0.40376 -0.95702 -1.42722 -1.77355 -1.96595 -1.98772 -1.83697 -1.52680 -1.08411 -0.54733 0.03696 0.61803 1.14547 1.57349 1.86494 1.99454 1.95102 1.73818 1.37447 0.89148 0.33111 -0.25800 -0.82471 -1.31985 -1.70043 -1.93344 -1.99863 -1.89037 -1.61803 -1.20527 -0.68790 -0.11082 0.47587 1.02126 1.47802 1.80649 1.97818 0.00000 ... ... ..... ....... TT....... ..TT....... ....TT....... E5 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 1.86494 1.47802 0.89148 0.18454 -0.54733 -1.20527 -1.70043 -1.96595 0.00000 ... ... ..... ....... ......... TT......... ..TT......... E6 2.00000 1.80649 1.26342 0.47587 -0.40376 -1.20527 -1.77355 -1.99863 -1.83697 -1.31985 -0.54733 0.33111 1.14547 1.73818 1.99454 1.86494 1.37447 0.61803 -0.25800 -1.08411 -1.70043 -1.98772 -1.89037 -1.42722 -0.68790 0.18454 1.02126 1.66037 1.97818 1.91321 1.47802 0.75682 -0.11082 -0.95702 -1.61803 -1.96595 -1.93344 -1.52680 -0.82471 0.03696 0.89148 1.57349 1.95102 0.00000 ... ... ..... ....... ......... ........... TT........... E7 2.00000 1.73818 1.02126 0.03696 -0.95702 -1.70043 -1.99863 -1.77355 -1.08411 -0.11082 0.89148 1.66037 1.99454 1.80649 1.14547 0.18454 -0.82471 -1.61803 -1.98772 -1.83697 -1.20527 -0.25800 0.75682 1.57349 1.97818 1.86494 1.26342 0.33111 -0.68790 -1.52680 -1.96595 -1.89037 -1.31985 -0.40376 0.61803 1.47802 1.95102 1.91321 1.37447 0.47587 -0.54733 -1.42722 -1.93344 0.00000 ... ... ..... ....... ......... ........... ............. E8 2.00000 1.66037 0.75682 -0.40376 -1.42722 -1.96595 -1.83697 -1.08411 0.03696 1.14547 1.86494 1.95102 1.37447 0.33111 -0.82471 -1.70043 -1.99863 -1.61803 -0.68790 0.47587 1.47802 1.97818 1.80649 1.02126 -0.11082 -1.20527 -1.89037 -1.93344 -1.31985 -0.25800 0.89148 1.73818 1.99454 1.57349 0.61803 -0.54733 -1.52680 -1.98772 -1.77355 -0.95702 0.18454 1.26342 1.91321 0.00000 ... ... ..... ....... ......... ........... ............. E9 2.00000 1.57349 0.47587 -0.82471 -1.77355 -1.96595 -1.31985 -0.11082 1.14547 1.91321 1.86494 1.02126 -0.25800 -1.42722 -1.98772 -1.70043 -0.68790 0.61803 1.66037 1.99454 1.47802 0.33111 -0.95702 -1.83697 -1.93344 -1.20527 0.03696 1.26342 1.95102 1.80649 0.89148 -0.40376 -1.52680 -1.99863 -1.61803 -0.54733 0.75682 1.73818 1.97818 1.37447 0.18454 -1.08411 -1.89037 0.00000 ... ... ..... ....... ......... ........... ............. E10 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 1.47802 0.18454 -1.20527 -1.96595 -1.70043 -0.54733 0.89148 1.86494 0.00000 ... ... ..... ....... ......... ........... ............. E11 2.00000 1.37447 -0.11082 -1.52680 -1.98772 -1.20527 0.33111 1.66037 1.95102 1.02126 -0.54733 -1.77355 -1.89037 -0.82471 0.75682 1.86494 1.80649 0.61803 -0.95702 -1.93344 -1.70043 -0.40376 1.14547 1.97818 1.57349 0.18454 -1.31985 -1.99863 -1.42722 0.03696 1.47802 1.99454 1.26342 -0.25800 -1.61803 -1.96595 -1.08411 0.47587 1.73818 1.91321 0.89148 -0.68790 -1.83697 0.00000 ... ... ..... ....... ......... ........... ............. E12 2.00000 1.26342 -0.40376 -1.77355 -1.83697 -0.54733 1.14547 1.99454 1.37447 -0.25800 -1.70043 -1.89037 -0.68790 1.02126 1.97818 1.47802 -0.11082 -1.61803 -1.93344 -0.82471 0.89148 1.95102 1.57349 0.03696 -1.52680 -1.96595 -0.95702 0.75682 1.91321 1.66037 0.18454 -1.42722 -1.98772 -1.08411 0.61803 1.86494 1.73818 0.33111 -1.31985 -1.99863 -1.20527 0.47587 1.80649 0.00000 ... ... ..... ....... ......... ........... ............. E13 2.00000 1.14547 -0.68790 -1.93344 -1.52680 0.18454 1.73818 1.80649 0.33111 -1.42722 -1.96595 -0.82471 1.02126 1.99454 1.26342 -0.54733 -1.89037 -1.61803 0.03696 1.66037 1.86494 0.47587 -1.31985 -1.98772 -0.95702 0.89148 1.97818 1.37447 -0.40376 -1.83697 -1.70043 -0.11082 1.57349 1.91321 0.61803 -1.20527 -1.99863 -1.08411 0.75682 1.95102 1.47802 -0.25800 -1.77355 0.00000 ... ... ..... ....... ......... ........... ............. E14 2.00000 1.02126 -0.95702 -1.99863 -1.08411 0.89148 1.99454 1.14547 -0.82471 -1.98772 -1.20527 0.75682 1.97818 1.26342 -0.68790 -1.96595 -1.31985 0.61803 1.95102 1.37447 -0.54733 -1.93344 -1.42722 0.47587 1.91321 1.47802 -0.40376 -1.89037 -1.52680 0.33111 1.86494 1.57349 -0.25800 -1.83697 -1.61803 0.18454 1.80649 1.66037 -0.11082 -1.77355 -1.70043 0.03696 1.73818 0.00000 ... ... ..... ....... ......... ........... ............. E15 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 0.89148 -1.20527 -1.96595 -0.54733 1.47802 1.86494 0.18454 -1.70043 0.00000 ... ... ..... ....... ......... ........... ............. E16 2.00000 0.75682 -1.42722 -1.83697 0.03696 1.86494 1.37447 -0.82471 -1.99863 -0.68790 1.47802 1.80649 -0.11082 -1.89037 -1.31985 0.89148 1.99454 0.61803 -1.52680 -1.77355 0.18454 1.91321 1.26342 -0.95702 -1.98772 -0.54733 1.57349 1.73818 -0.25800 -1.93344 -1.20527 1.02126 1.97818 0.47587 -1.61803 -1.70043 0.33111 1.95102 1.14547 -1.08411 -1.96595 -0.40376 1.66037 0.00000 ... ... ..... ....... ......... ........... ............. E17 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 0.61803 -1.61803 -1.61803 0.61803 2.00000 0.61803 -1.61803 0.00000 ... ... ..... ....... ......... ........... ............. E18 2.00000 0.47587 -1.77355 -1.31985 1.14547 1.86494 -0.25800 -1.98772 -0.68790 1.66037 1.47802 -0.95702 -1.93344 0.03696 1.95102 0.89148 -1.52680 -1.61803 0.75682 1.97818 0.18454 -1.89037 -1.08411 1.37447 1.73818 -0.54733 -1.99863 -0.40376 1.80649 1.26342 -1.20527 -1.83697 0.33111 1.99454 0.61803 -1.70043 -1.42722 1.02126 1.91321 -0.11082 -1.96595 -0.82471 1.57349 0.00000 ... ... ..... ....... ......... ........... ............. E19 2.00000 0.33111 -1.89037 -0.95702 1.57349 1.47802 -1.08411 -1.83697 0.47587 1.99454 0.18454 -1.93344 -0.82471 1.66037 1.37447 -1.20527 -1.77355 0.61803 1.97818 0.03696 -1.96595 -0.68790 1.73818 1.26342 -1.31985 -1.70043 0.75682 1.95102 -0.11082 -1.98772 -0.54733 1.80649 1.14547 -1.42722 -1.61803 0.89148 1.91321 -0.25800 -1.99863 -0.40376 1.86494 1.02126 -1.52680 0.00000 ... ... ..... ....... ......... ........... ............. E20 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 0.18454 -1.96595 -0.54733 1.86494 0.89148 -1.70043 -1.20527 1.47802 0.00000 ... ... ..... ....... ......... ........... ............. E21 2.00000 0.03696 -1.99863 -0.11082 1.99454 0.18454 -1.98772 -0.25800 1.97818 0.33111 -1.96595 -0.40376 1.95102 0.47587 -1.93344 -0.54733 1.91321 0.61803 -1.89037 -0.68790 1.86494 0.75682 -1.83697 -0.82471 1.80649 0.89148 -1.77355 -0.95702 1.73818 1.02126 -1.70043 -1.08411 1.66037 1.14547 -1.61803 -1.20527 1.57349 1.26342 -1.52680 -1.31985 1.47802 1.37447 -1.42722 0.00000 ... ... ..... ....... ......... ........... ............. E22 2.00000 -0.11082 -1.98772 0.33111 1.95102 -0.54733 -1.89037 0.75682 1.80649 -0.95702 -1.70043 1.14547 1.57349 -1.31985 -1.42722 1.47802 1.26342 -1.61803 -1.08411 1.73818 0.89148 -1.83697 -0.68790 1.91321 0.47587 -1.96595 -0.25800 1.99454 0.03696 -1.99863 0.18454 1.97818 -0.40376 -1.93344 0.61803 1.86494 -0.82471 -1.77355 1.02126 1.66037 -1.20527 -1.52680 1.37447 0.00000 ... ... ..... ....... ......... ........... ............. E23 2.00000 -0.25800 -1.93344 0.75682 1.73818 -1.20527 -1.42722 1.57349 1.02126 -1.83697 -0.54733 1.97818 0.03696 -1.98772 0.47587 1.86494 -0.95702 -1.61803 1.37447 1.26342 -1.70043 -0.82471 1.91321 0.33111 -1.99863 0.18454 1.95102 -0.68790 -1.77355 1.14547 1.47802 -1.52680 -1.08411 1.80649 0.61803 -1.96595 -0.11082 1.99454 -0.40376 -1.89037 0.89148 1.66037 -1.31985 0.00000 ... ... ..... ....... ......... ........... ............. E24 2.00000 -0.40376 -1.83697 1.14547 1.37447 -1.70043 -0.68790 1.97818 -0.11082 -1.93344 0.89148 1.57349 -1.52680 -0.95702 1.91321 0.18454 -1.98772 0.61803 1.73818 -1.31985 -1.20527 1.80649 0.47587 -1.99863 0.33111 1.86494 -1.08411 -1.42722 1.66037 0.75682 -1.96595 0.03696 1.95102 -0.82471 -1.61803 1.47802 1.02126 -1.89037 -0.25800 1.99454 -0.54733 -1.77355 1.26342 0.00000 ... ... ..... ....... ......... ........... ............. E25 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 -0.54733 -1.70043 1.47802 0.89148 -1.96595 0.18454 1.86494 -1.20527 0.00000 ... ... ..... ....... ......... ........... ............. E26 2.00000 -0.68790 -1.52680 1.73818 0.33111 -1.96595 1.02126 1.26342 -1.89037 0.03696 1.86494 -1.31985 -0.95702 1.97818 -0.40376 -1.70043 1.57349 0.61803 -1.99863 0.75682 1.47802 -1.77355 -0.25800 1.95102 -1.08411 -1.20527 1.91321 -0.11082 -1.83697 1.37447 0.89148 -1.98772 0.47587 1.66037 -1.61803 -0.54733 1.99454 -0.82471 -1.42722 1.80649 0.18454 -1.93344 1.14547 0.00000 ... ... ..... ....... ......... ........... ............. E27 2.00000 -0.82471 -1.31985 1.91321 -0.25800 -1.70043 1.66037 0.33111 -1.93344 1.26342 0.89148 -1.99863 0.75682 1.37447 -1.89037 0.18454 1.73818 -1.61803 -0.40376 1.95102 -1.20527 -0.95702 1.99454 -0.68790 -1.42722 1.86494 -0.11082 -1.77355 1.57349 0.47587 -1.96595 1.14547 1.02126 -1.98772 0.61803 1.47802 -1.83697 0.03696 1.80649 -1.52680 -0.54733 1.97818 -1.08411 0.00000 ... ... ..... ....... ......... ........... ............. E28 2.00000 -0.95702 -1.08411 1.99454 -0.82471 -1.20527 1.97818 -0.68790 -1.31985 1.95102 -0.54733 -1.42722 1.91321 -0.40376 -1.52680 1.86494 -0.25800 -1.61803 1.80649 -0.11082 -1.70043 1.73818 0.03696 -1.77355 1.66037 0.18454 -1.83697 1.57349 0.33111 -1.89037 1.47802 0.47587 -1.93344 1.37447 0.61803 -1.96595 1.26342 0.75682 -1.98772 1.14547 0.89148 -1.99863 1.02126 0.00000 ... ... ..... ....... ......... ........... ............. E29 2.00000 -1.08411 -0.82471 1.97818 -1.31985 -0.54733 1.91321 -1.52680 -0.25800 1.80649 -1.70043 0.03696 1.66037 -1.83697 0.33111 1.47802 -1.93344 0.61803 1.26342 -1.98772 0.89148 1.02126 -1.99863 1.14547 0.75682 -1.96595 1.37447 0.47587 -1.89037 1.57349 0.18454 -1.77355 1.73818 -0.11082 -1.61803 1.86494 -0.40376 -1.42722 1.95102 -0.68790 -1.20527 1.99454 -0.95702 0.00000 ... ... ..... ....... ......... ........... ............. E30 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 -1.20527 -0.54733 1.86494 -1.70043 0.18454 1.47802 -1.96595 0.89148 0.00000 ... ... ..... ....... ......... ........... ............. E31 2.00000 -1.31985 -0.25800 1.66037 -1.93344 0.89148 0.75682 -1.89037 1.73818 -0.40376 -1.20527 1.99454 -1.42722 -0.11082 1.57349 -1.96595 1.02126 0.61803 -1.83697 1.80649 -0.54733 -1.08411 1.97818 -1.52680 0.03696 1.47802 -1.98772 1.14547 0.47587 -1.77355 1.86494 -0.68790 -0.95702 1.95102 -1.61803 0.18454 1.37447 -1.99863 1.26342 0.33111 -1.70043 1.91321 -0.82471 0.00000 ... ... ..... ....... ......... ........... ............. E32 2.00000 -1.42722 0.03696 1.37447 -1.99863 1.47802 -0.11082 -1.31985 1.99454 -1.52680 0.18454 1.26342 -1.98772 1.57349 -0.25800 -1.20527 1.97818 -1.61803 0.33111 1.14547 -1.96595 1.66037 -0.40376 -1.08411 1.95102 -1.70043 0.47587 1.02126 -1.93344 1.73818 -0.54733 -0.95702 1.91321 -1.77355 0.61803 0.89148 -1.89037 1.80649 -0.68790 -0.82471 1.86494 -1.83697 0.75682 0.00000 ... ... ..... ....... ......... ........... ............. E33 2.00000 -1.52680 0.33111 1.02126 -1.89037 1.86494 -0.95702 -0.40376 1.57349 -1.99863 1.47802 -0.25800 -1.08411 1.91321 -1.83697 0.89148 0.47587 -1.61803 1.99454 -1.42722 0.18454 1.14547 -1.93344 1.80649 -0.82471 -0.54733 1.66037 -1.98772 1.37447 -0.11082 -1.20527 1.95102 -1.77355 0.75682 0.61803 -1.70043 1.97818 -1.31985 0.03696 1.26342 -1.96595 1.73818 -0.68790 0.00000 ... ... ..... ....... ......... ........... ............. E34 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 0.00000 ... ... ..... ....... ......... ........... ............. E35 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 -1.70043 0.89148 0.18454 -1.20527 1.86494 -1.96595 1.47802 -0.54733 0.00000 ... ... ..... ....... ......... ........... ............. E36 2.00000 -1.77355 1.14547 -0.25800 -0.68790 1.47802 -1.93344 1.95102 -1.52680 0.75682 0.18454 -1.08411 1.73818 -1.99863 1.80649 -1.20527 0.33111 0.61803 -1.42722 1.91321 -1.96595 1.57349 -0.82471 -0.11082 1.02126 -1.70043 1.99454 -1.83697 1.26342 -0.40376 -0.54733 1.37447 -1.89037 1.97818 -1.61803 0.89148 0.03696 -0.95702 1.66037 -1.98772 1.86494 -1.31985 0.47587 0.00000 ... ... ..... ....... ......... ........... ............. E37 2.00000 -1.83697 1.37447 -0.68790 -0.11082 0.89148 -1.52680 1.91321 -1.98772 1.73818 -1.20527 0.47587 0.33111 -1.08411 1.66037 -1.96595 1.95102 -1.61803 1.02126 -0.25800 -0.54733 1.26342 -1.77355 1.99454 -1.89037 1.47802 -0.82471 0.03696 0.75682 -1.42722 1.86494 -1.99863 1.80649 -1.31985 0.61803 0.18454 -0.95702 1.57349 -1.93344 1.97818 -1.70043 1.14547 -0.40376 0.00000 ... ... ..... ....... ......... ........... ............. E38 2.00000 -1.89037 1.57349 -1.08411 0.47587 0.18454 -0.82471 1.37447 -1.77355 1.97818 -1.96595 1.73818 -1.31985 0.75682 -0.11082 -0.54733 1.14547 -1.61803 1.91321 -1.99863 1.86494 -1.52680 1.02126 -0.40376 -0.25800 0.89148 -1.42722 1.80649 -1.98772 1.95102 -1.70043 1.26342 -0.68790 0.03696 0.61803 -1.20527 1.66037 -1.93344 1.99454 -1.83697 1.47802 -0.95702 0.33111 0.00000 ... ... ..... ....... ......... ........... ............. E39 2.00000 -1.93344 1.73818 -1.42722 1.02126 -0.54733 0.03696 0.47587 -0.95702 1.37447 -1.70043 1.91321 -1.99863 1.95102 -1.77355 1.47802 -1.08411 0.61803 -0.11082 -0.40376 0.89148 -1.31985 1.66037 -1.89037 1.99454 -1.96595 1.80649 -1.52680 1.14547 -0.68790 0.18454 0.33111 -0.82471 1.26342 -1.61803 1.86494 -1.98772 1.97818 -1.83697 1.57349 -1.20527 0.75682 -0.25800 0.00000 ... ... ..... ....... ......... ........... ............. E40 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 -1.96595 1.86494 -1.70043 1.47802 -1.20527 0.89148 -0.54733 0.18454 0.00000 ... ... ..... ....... ......... ........... ............. E41 2.00000 -1.98772 1.95102 -1.89037 1.80649 -1.70043 1.57349 -1.42722 1.26342 -1.08411 0.89148 -0.68790 0.47587 -0.25800 0.03696 0.18454 -0.40376 0.61803 -0.82471 1.02126 -1.20527 1.37447 -1.52680 1.66037 -1.77355 1.86494 -1.93344 1.97818 -1.99863 1.99454 -1.96595 1.91321 -1.83697 1.73818 -1.61803 1.47802 -1.31985 1.14547 -0.95702 0.75682 -0.54733 0.33111 -0.11082 0.00000 ... ... ..... ....... ......... ........... ............. E42 2.00000 -1.99863 1.99454 -1.98772 1.97818 -1.96595 1.95102 -1.93344 1.91321 -1.89037 1.86494 -1.83697 1.80649 -1.77355 1.73818 -1.70043 1.66037 -1.61803 1.57349 -1.52680 1.47802 -1.42722 1.37447 -1.31985 1.26342 -1.20527 1.14547 -1.08411 1.02126 -0.95702 0.89148 -0.82471 0.75682 -0.68790 0.61803 -0.54733 0.47587 -0.40376 0.33111 -0.25800 0.18454 -0.11082 0.03696 0.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.994538346772 = 2*cos(2*π/85) 1.978183216742 = 2*cos(4*π/85) 1.951023935961 = 2*cos(6*π/85) 1.913208839001 = 2*cos(8*π/85) 1.864944458809 = 2*cos(10*π/85) = 2*cos(2*π/17) = (−1+√17+√34−2*√17+2*√17+3*√17−√34−2*√17−2*√34+2*√17)/8 1.806494398692 = 2*cos(12*π/85) 1.738177892611 = 2*cos(14*π/85) 1.660368061631 = 2*cos(16*π/85) 1.573489876067 = 2*cos(18*π/85) 1.478017834441 = 2*cos(20*π/85) = 2*cos(4*π/17) = (−1+√17−√34−2*√17+2*√17+3*√17+√34−2*√17+2*√34+2*√17)/8 1.374473371939 = 2*cos(22*π/85) 1.263422012507 = 2*cos(24*π/85) 1.145470280161 = 2*cos(26*π/85) 1.021262386362 = 2*cos(28*π/85) 0.891476711553 = 2*cos(30*π/85) = 2*cos(6*π/17) = (−1−√17+√34+2*√17+2*√17−3*√17−√34+2*√17+2*√34−2*√17)/8 0.756822100085 = 2*cos(32*π/85) 0.618033988750 = 2*cos(34*π/85) = 2*cos(2*π/5) = (√5−1)/2 0.475870390085 = 2*cos(36*π/85) 0.331107752368 = 2*cos(38*π/85) 0.184536718927 = 2*cos(40*π/85) = 2*cos(8*π/17) = (−1+√17+√34−2*√17−2*√17+3*√17−√34−2*√17−2*√34+2*√17)/8 0.036957809918 = 2*cos(42*π/85) -0.110822949832 = 2*cos(44*π/85) -0.257998433060 = 2*cos(46*π/85) -0.403764818314 = 2*cos(48*π/85) -0.547325980144 = 2*cos(50*π/85) = 2*cos(10*π/17) = −(1+√17−√34+2*√17+2*√17−3*√17−√34+2*√17+2*√34−2*√17)/8 -0.687897837268 = 2*cos(52*π/85) -0.824712634948 = 2*cos(54*π/85) -0.957023138203 = 2*cos(56*π/85) -1.084106712945 = 2*cos(58*π/85) -1.205269272759 = 2*cos(60*π/85) = 2*cos(12*π/17) = −(1+√17+√34+2*√17−2*√17−3*√17+√34+2*√17−2*√34−2*√17)/8 -1.319849069757 = 2*cos(62*π/85) -1.427220308824 = 2*cos(64*π/85) -1.526796565482 = 2*cos(66*π/85) -1.618033988750 = 2*cos(68*π/85) = 2*cos(4*π/5) = −(√5+1)/2 -1.700434271459 = 2*cos(70*π/85) = 2*cos(14*π/17) = −(1+√17+√34+2*√17+2*√17−3*√17+√34+2*√17−2*√34−2*√17)/8 -1.773547371840 = 2*cos(72*π/85) -1.836973971492 = 2*cos(74*π/85) -1.890367656322 = 2*cos(76*π/85) -1.933436808538 = 2*cos(78*π/85) -1.965946199368 = 2*cos(80*π/85) = 2*cos(16*π/17) = −(1−√17+√34−2*√17+2*√17+3*√17+√34−2*√17+2*√34+2*√17)/8 -1.987718273791 = 2*cos(82*π/85) -1.998634120286 = 2*cos(84*π/85) 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 R_z E1 R+p+d+f+g+h+i+j+k+l+m {R_x, R_y}, {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 C_85v point group is 170, and the order of the principal axis (C₈₅) is 85. The group has 44 irreducible representations. β The C_85v point group is isomorphic to D₈₅. γ The C_85v point group is generated by two symmetry elements, C₈₅ and any σ_v. Also, the group may be generated from two σ_v planes (some pairs will yield smaller groups, though; choosing a minimum angle is safe). δ The group contains one set of symmetry planes σ_v intersecting in the principal (z) axis. The xz plane (but not the yz plane) is a member of that set. ε The lowest nonvanishing multipole moment in C_85v 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 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. Some of these algebraic expressions, however, are too complicated to be included in the list above. ι The fact that the regular pentagon is constructible is known since antiquity; Eukleides already discovered a construction for it. The double cosine of 2π/5 is equal to the reciprocal of the Golden Ratio of (1+√5)/2 = 1.61803. 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. The regular 85-gon can be derived from the regular 17-gon by overlaying it with a regular pentagon.

This Character Table for the C_85v point group was created by Gernot Katzer.

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

	C_83v
	C_84v
C₈₅	C_85v	C_85h D₈₅ D_85h D_85d
	C_86v
	C_87v

Character table for the C85v point group

Character table for the C_85v point group