Character table for the D85 point group

D85     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 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  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     ..T ..T ..... ......T ......... ..........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   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      {x2y2, xy}, {z(x2y2), xyz}, {z2(x2y2), xyz2}, {z3(x2y2), xyz3}, {z4(x2y2), xyz4} 
E3   f+g+h+i+j+k+l+m        {x(x2−3y2), y(3x2y2)}, {xz(x2−3y2), yz(3x2y2)}, {xz2(x2−3y2), yz2(3x2y2)}, {xz3(x2−3y2), yz3(3x2y2)} 
E4   g+h+i+j+k+l+m          {(x2y2)2−4x2y2, xy(x2y2)}, {z((x2y2)2−4x2y2), xyz(x2y2)}, {z2((x2y2)2−4x2y2), xyz2(x2y2)} 
E5   h+i+j+k+l+m            {x(x2−(5+2√5)y2)(x2−(5−2√5)y2), y((5+2√5)x2y2)((5−2√5)x2y2)}, {xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), yz((5+2√5)x2y2)((5−2√5)x2y2)} 
E6   i+j+k+l+m              {x2(x2−3y2)2y2(3x2y2)2, xy(x2−3y2)(3x2y2)} 
E7   j+k+l+m 
E8   k+l+m 
E9   l+m 
E10  m 

 Notes:

    α  The order of the D85 point group is 170, and the order of the principal axis (C85) is 85. The group has 44 irreducible representations.

    β  The D85 point group is isomorphic to C85v.

    γ  The D85 point group is generated by two symmetry elements, C85 and a perpendicular C2.
       Also, the group may be generated from two C2 axes (some pairs will yield smaller groups, though; choosing a minimum angle is safe).

    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.

    ε  The lowest nonvanishing multipole moment in D85 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.
       Some of these algebraic expressions, however, are too complicated to be included in the list above.

    κ  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.

    λ  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 D85 point group was created by Gernot Katzer.

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