Character table for the C82v point group

C82v    E        2 C82    2 C41    2 C82^3  2 C41^2  2 C82^5  2 C41^3  2 C82^7  2 C41^4  2 C82^9  2 C41^5  2 C82^11 2 C41^6  2 C82^13 2 C41^7  2 C82^15 2 C41^8  2 C82^17 2 C41^9  2 C82^19 2 C41^10 2 C82^21 2 C41^11 2 C82^23 2 C41^12 2 C82^25 2 C41^13 2 C82^27 2 C41^14 2 C82^29 2 C41^15 2 C82^31 2 C41^16 2 C82^33 2 C41^17 2 C82^35 2 C41^18 2 C82^37 2 C41^19 2 C82^39 2 C41^20 C2       41 sv    41 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     ... ..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 ... ..... ....... ......... ........... .............
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     ... ... ..... ....... ......... ........... .............
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     ... ... ..... ....... ......... ........... .............
E1      2.00000  1.99413  1.97656  1.94739  1.90679  1.85500  1.79233  1.71914  1.63586  1.54298  1.44104  1.33065  1.21245  1.08714  0.95544  0.81814  0.67603  0.52996  0.38078  0.22937  0.07661 -0.07661 -0.22937 -0.38078 -0.52996 -0.67603 -0.81814 -0.95544 -1.08714 -1.21245 -1.33065 -1.44104 -1.54298 -1.63586 -1.71914 -1.79233 -1.85500 -1.90679 -1.94739 -1.97656 -1.99413 -2.00000  0.00000  0.00000     TT. TT. ..TT. ....TT. ......TT. ........TT. ..........TT.
E2      2.00000  1.97656  1.90679  1.79233  1.63586  1.44104  1.21245  0.95544  0.67603  0.38078  0.07661 -0.22937 -0.52996 -0.81814 -1.08714 -1.33065 -1.54298 -1.71914 -1.85500 -1.94739 -1.99413 -1.99413 -1.94739 -1.85500 -1.71914 -1.54298 -1.33065 -1.08714 -0.81814 -0.52996 -0.22937  0.07661  0.38078  0.67603  0.95544  1.21245  1.44104  1.63586  1.79233  1.90679  1.97656  2.00000  0.00000  0.00000     ... ... TT... ..TT... ....TT... ......TT... ........TT...
E3      2.00000  1.94739  1.79233  1.54298  1.21245  0.81814  0.38078 -0.07661 -0.52996 -0.95544 -1.33065 -1.63586 -1.85500 -1.97656 -1.99413 -1.90679 -1.71914 -1.44104 -1.08714 -0.67603 -0.22937  0.22937  0.67603  1.08714  1.44104  1.71914  1.90679  1.99413  1.97656  1.85500  1.63586  1.33065  0.95544  0.52996  0.07661 -0.38078 -0.81814 -1.21245 -1.54298 -1.79233 -1.94739 -2.00000  0.00000  0.00000     ... ... ..... TT..... ..TT..... ....TT..... ......TT.....
E4      2.00000  1.90679  1.63586  1.21245  0.67603  0.07661 -0.52996 -1.08714 -1.54298 -1.85500 -1.99413 -1.94739 -1.71914 -1.33065 -0.81814 -0.22937  0.38078  0.95544  1.44104  1.79233  1.97656  1.97656  1.79233  1.44104  0.95544  0.38078 -0.22937 -0.81814 -1.33065 -1.71914 -1.94739 -1.99413 -1.85500 -1.54298 -1.08714 -0.52996  0.07661  0.67603  1.21245  1.63586  1.90679  2.00000  0.00000  0.00000     ... ... ..... ....... TT....... ..TT....... ....TT.......
E5      2.00000  1.85500  1.44104  0.81814  0.07661 -0.67603 -1.33065 -1.79233 -1.99413 -1.90679 -1.54298 -0.95544 -0.22937  0.52996  1.21245  1.71914  1.97656  1.94739  1.63586  1.08714  0.38078 -0.38078 -1.08714 -1.63586 -1.94739 -1.97656 -1.71914 -1.21245 -0.52996  0.22937  0.95544  1.54298  1.90679  1.99413  1.79233  1.33065  0.67603 -0.07661 -0.81814 -1.44104 -1.85500 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... TT......... ..TT.........
E6      2.00000  1.79233  1.21245  0.38078 -0.52996 -1.33065 -1.85500 -1.99413 -1.71914 -1.08714 -0.22937  0.67603  1.44104  1.90679  1.97656  1.63586  0.95544  0.07661 -0.81814 -1.54298 -1.94739 -1.94739 -1.54298 -0.81814  0.07661  0.95544  1.63586  1.97656  1.90679  1.44104  0.67603 -0.22937 -1.08714 -1.71914 -1.99413 -1.85500 -1.33065 -0.52996  0.38078  1.21245  1.79233  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... TT...........
E7      2.00000  1.71914  0.95544 -0.07661 -1.08714 -1.79233 -1.99413 -1.63586 -0.81814  0.22937  1.21245  1.85500  1.97656  1.54298  0.67603 -0.38078 -1.33065 -1.90679 -1.94739 -1.44104 -0.52996  0.52996  1.44104  1.94739  1.90679  1.33065  0.38078 -0.67603 -1.54298 -1.97656 -1.85500 -1.21245 -0.22937  0.81814  1.63586  1.99413  1.79233  1.08714  0.07661 -0.95544 -1.71914 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E8      2.00000  1.63586  0.67603 -0.52996 -1.54298 -1.99413 -1.71914 -0.81814  0.38078  1.44104  1.97656  1.79233  0.95544 -0.22937 -1.33065 -1.94739 -1.85500 -1.08714  0.07661  1.21245  1.90679  1.90679  1.21245  0.07661 -1.08714 -1.85500 -1.94739 -1.33065 -0.22937  0.95544  1.79233  1.97656  1.44104  0.38078 -0.81814 -1.71914 -1.99413 -1.54298 -0.52996  0.67603  1.63586  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E9      2.00000  1.54298  0.38078 -0.95544 -1.85500 -1.90679 -1.08714  0.22937  1.44104  1.99413  1.63586  0.52996 -0.81814 -1.79233 -1.94739 -1.21245  0.07661  1.33065  1.97656  1.71914  0.67603 -0.67603 -1.71914 -1.97656 -1.33065 -0.07661  1.21245  1.94739  1.79233  0.81814 -0.52996 -1.63586 -1.99413 -1.44104 -0.22937  1.08714  1.90679  1.85500  0.95544 -0.38078 -1.54298 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E10     2.00000  1.44104  0.07661 -1.33065 -1.99413 -1.54298 -0.22937  1.21245  1.97656  1.63586  0.38078 -1.08714 -1.94739 -1.71914 -0.52996  0.95544  1.90679  1.79233  0.67603 -0.81814 -1.85500 -1.85500 -0.81814  0.67603  1.79233  1.90679  0.95544 -0.52996 -1.71914 -1.94739 -1.08714  0.38078  1.63586  1.97656  1.21245 -0.22937 -1.54298 -1.99413 -1.33065  0.07661  1.44104  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E11     2.00000  1.33065 -0.22937 -1.63586 -1.94739 -0.95544  0.67603  1.85500  1.79233  0.52996 -1.08714 -1.97656 -1.54298 -0.07661  1.44104  1.99413  1.21245 -0.38078 -1.71914 -1.90679 -0.81814  0.81814  1.90679  1.71914  0.38078 -1.21245 -1.99413 -1.44104  0.07661  1.54298  1.97656  1.08714 -0.52996 -1.79233 -1.85500 -0.67603  0.95544  1.94739  1.63586  0.22937 -1.33065 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E12     2.00000  1.21245 -0.52996 -1.85500 -1.71914 -0.22937  1.44104  1.97656  0.95544 -0.81814 -1.94739 -1.54298  0.07661  1.63586  1.90679  0.67603 -1.08714 -1.99413 -1.33065  0.38078  1.79233  1.79233  0.38078 -1.33065 -1.99413 -1.08714  0.67603  1.90679  1.63586  0.07661 -1.54298 -1.94739 -0.81814  0.95544  1.97656  1.44104 -0.22937 -1.71914 -1.85500 -0.52996  1.21245  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E13     2.00000  1.08714 -0.81814 -1.97656 -1.33065  0.52996  1.90679  1.54298 -0.22937 -1.79233 -1.71914 -0.07661  1.63586  1.85500  0.38078 -1.44104 -1.94739 -0.67603  1.21245  1.99413  0.95544 -0.95544 -1.99413 -1.21245  0.67603  1.94739  1.44104 -0.38078 -1.85500 -1.63586  0.07661  1.71914  1.79233  0.22937 -1.54298 -1.90679 -0.52996  1.33065  1.97656  0.81814 -1.08714 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E14     2.00000  0.95544 -1.08714 -1.99413 -0.81814  1.21245  1.97656  0.67603 -1.33065 -1.94739 -0.52996  1.44104  1.90679  0.38078 -1.54298 -1.85500 -0.22937  1.63586  1.79233  0.07661 -1.71914 -1.71914  0.07661  1.79233  1.63586 -0.22937 -1.85500 -1.54298  0.38078  1.90679  1.44104 -0.52996 -1.94739 -1.33065  0.67603  1.97656  1.21245 -0.81814 -1.99413 -1.08714  0.95544  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E15     2.00000  0.81814 -1.33065 -1.90679 -0.22937  1.71914  1.63586 -0.38078 -1.94739 -1.21245  0.95544  1.99413  0.67603 -1.44104 -1.85500 -0.07661  1.79233  1.54298 -0.52996 -1.97656 -1.08714  1.08714  1.97656  0.52996 -1.54298 -1.79233  0.07661  1.85500  1.44104 -0.67603 -1.99413 -0.95544  1.21245  1.94739  0.38078 -1.63586 -1.71914  0.22937  1.90679  1.33065 -0.81814 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E16     2.00000  0.67603 -1.54298 -1.71914  0.38078  1.97656  0.95544 -1.33065 -1.85500  0.07661  1.90679  1.21245 -1.08714 -1.94739 -0.22937  1.79233  1.44104 -0.81814 -1.99413 -0.52996  1.63586  1.63586 -0.52996 -1.99413 -0.81814  1.44104  1.79233 -0.22937 -1.94739 -1.08714  1.21245  1.90679  0.07661 -1.85500 -1.33065  0.95544  1.97656  0.38078 -1.71914 -1.54298  0.67603  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E17     2.00000  0.52996 -1.71914 -1.44104  0.95544  1.94739  0.07661 -1.90679 -1.08714  1.33065  1.79233 -0.38078 -1.99413 -0.67603  1.63586  1.54298 -0.81814 -1.97656 -0.22937  1.85500  1.21245 -1.21245 -1.85500  0.22937  1.97656  0.81814 -1.54298 -1.63586  0.67603  1.99413  0.38078 -1.79233 -1.33065  1.08714  1.90679 -0.07661 -1.94739 -0.95544  1.44104  1.71914 -0.52996 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E18     2.00000  0.38078 -1.85500 -1.08714  1.44104  1.63586 -0.81814 -1.94739  0.07661  1.97656  0.67603 -1.71914 -1.33065  1.21245  1.79233 -0.52996 -1.99413 -0.22937  1.90679  0.95544 -1.54298 -1.54298  0.95544  1.90679 -0.22937 -1.99413 -0.52996  1.79233  1.21245 -1.33065 -1.71914  0.67603  1.97656  0.07661 -1.94739 -0.81814  1.63586  1.44104 -1.08714 -1.85500  0.38078  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E19     2.00000  0.22937 -1.94739 -0.67603  1.79233  1.08714 -1.54298 -1.44104  1.21245  1.71914 -0.81814 -1.90679  0.38078  1.99413  0.07661 -1.97656 -0.52996  1.85500  0.95544 -1.63586 -1.33065  1.33065  1.63586 -0.95544 -1.85500  0.52996  1.97656 -0.07661 -1.99413 -0.38078  1.90679  0.81814 -1.71914 -1.21245  1.44104  1.54298 -1.08714 -1.79233  0.67603  1.94739 -0.22937 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E20     2.00000  0.07661 -1.99413 -0.22937  1.97656  0.38078 -1.94739 -0.52996  1.90679  0.67603 -1.85500 -0.81814  1.79233  0.95544 -1.71914 -1.08714  1.63586  1.21245 -1.54298 -1.33065  1.44104  1.44104 -1.33065 -1.54298  1.21245  1.63586 -1.08714 -1.71914  0.95544  1.79233 -0.81814 -1.85500  0.67603  1.90679 -0.52996 -1.94739  0.38078  1.97656 -0.22937 -1.99413  0.07661  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E21     2.00000 -0.07661 -1.99413  0.22937  1.97656 -0.38078 -1.94739  0.52996  1.90679 -0.67603 -1.85500  0.81814  1.79233 -0.95544 -1.71914  1.08714  1.63586 -1.21245 -1.54298  1.33065  1.44104 -1.44104 -1.33065  1.54298  1.21245 -1.63586 -1.08714  1.71914  0.95544 -1.79233 -0.81814  1.85500  0.67603 -1.90679 -0.52996  1.94739  0.38078 -1.97656 -0.22937  1.99413  0.07661 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E22     2.00000 -0.22937 -1.94739  0.67603  1.79233 -1.08714 -1.54298  1.44104  1.21245 -1.71914 -0.81814  1.90679  0.38078 -1.99413  0.07661  1.97656 -0.52996 -1.85500  0.95544  1.63586 -1.33065 -1.33065  1.63586  0.95544 -1.85500 -0.52996  1.97656  0.07661 -1.99413  0.38078  1.90679 -0.81814 -1.71914  1.21245  1.44104 -1.54298 -1.08714  1.79233  0.67603 -1.94739 -0.22937  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E23     2.00000 -0.38078 -1.85500  1.08714  1.44104 -1.63586 -0.81814  1.94739  0.07661 -1.97656  0.67603  1.71914 -1.33065 -1.21245  1.79233  0.52996 -1.99413  0.22937  1.90679 -0.95544 -1.54298  1.54298  0.95544 -1.90679 -0.22937  1.99413 -0.52996 -1.79233  1.21245  1.33065 -1.71914 -0.67603  1.97656 -0.07661 -1.94739  0.81814  1.63586 -1.44104 -1.08714  1.85500  0.38078 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E24     2.00000 -0.52996 -1.71914  1.44104  0.95544 -1.94739  0.07661  1.90679 -1.08714 -1.33065  1.79233  0.38078 -1.99413  0.67603  1.63586 -1.54298 -0.81814  1.97656 -0.22937 -1.85500  1.21245  1.21245 -1.85500 -0.22937  1.97656 -0.81814 -1.54298  1.63586  0.67603 -1.99413  0.38078  1.79233 -1.33065 -1.08714  1.90679  0.07661 -1.94739  0.95544  1.44104 -1.71914 -0.52996  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E25     2.00000 -0.67603 -1.54298  1.71914  0.38078 -1.97656  0.95544  1.33065 -1.85500 -0.07661  1.90679 -1.21245 -1.08714  1.94739 -0.22937 -1.79233  1.44104  0.81814 -1.99413  0.52996  1.63586 -1.63586 -0.52996  1.99413 -0.81814 -1.44104  1.79233  0.22937 -1.94739  1.08714  1.21245 -1.90679  0.07661  1.85500 -1.33065 -0.95544  1.97656 -0.38078 -1.71914  1.54298  0.67603 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E26     2.00000 -0.81814 -1.33065  1.90679 -0.22937 -1.71914  1.63586  0.38078 -1.94739  1.21245  0.95544 -1.99413  0.67603  1.44104 -1.85500  0.07661  1.79233 -1.54298 -0.52996  1.97656 -1.08714 -1.08714  1.97656 -0.52996 -1.54298  1.79233  0.07661 -1.85500  1.44104  0.67603 -1.99413  0.95544  1.21245 -1.94739  0.38078  1.63586 -1.71914 -0.22937  1.90679 -1.33065 -0.81814  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E27     2.00000 -0.95544 -1.08714  1.99413 -0.81814 -1.21245  1.97656 -0.67603 -1.33065  1.94739 -0.52996 -1.44104  1.90679 -0.38078 -1.54298  1.85500 -0.22937 -1.63586  1.79233 -0.07661 -1.71914  1.71914  0.07661 -1.79233  1.63586  0.22937 -1.85500  1.54298  0.38078 -1.90679  1.44104  0.52996 -1.94739  1.33065  0.67603 -1.97656  1.21245  0.81814 -1.99413  1.08714  0.95544 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E28     2.00000 -1.08714 -0.81814  1.97656 -1.33065 -0.52996  1.90679 -1.54298 -0.22937  1.79233 -1.71914  0.07661  1.63586 -1.85500  0.38078  1.44104 -1.94739  0.67603  1.21245 -1.99413  0.95544  0.95544 -1.99413  1.21245  0.67603 -1.94739  1.44104  0.38078 -1.85500  1.63586  0.07661 -1.71914  1.79233 -0.22937 -1.54298  1.90679 -0.52996 -1.33065  1.97656 -0.81814 -1.08714  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E29     2.00000 -1.21245 -0.52996  1.85500 -1.71914  0.22937  1.44104 -1.97656  0.95544  0.81814 -1.94739  1.54298  0.07661 -1.63586  1.90679 -0.67603 -1.08714  1.99413 -1.33065 -0.38078  1.79233 -1.79233  0.38078  1.33065 -1.99413  1.08714  0.67603 -1.90679  1.63586 -0.07661 -1.54298  1.94739 -0.81814 -0.95544  1.97656 -1.44104 -0.22937  1.71914 -1.85500  0.52996  1.21245 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E30     2.00000 -1.33065 -0.22937  1.63586 -1.94739  0.95544  0.67603 -1.85500  1.79233 -0.52996 -1.08714  1.97656 -1.54298  0.07661  1.44104 -1.99413  1.21245  0.38078 -1.71914  1.90679 -0.81814 -0.81814  1.90679 -1.71914  0.38078  1.21245 -1.99413  1.44104  0.07661 -1.54298  1.97656 -1.08714 -0.52996  1.79233 -1.85500  0.67603  0.95544 -1.94739  1.63586 -0.22937 -1.33065  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E31     2.00000 -1.44104  0.07661  1.33065 -1.99413  1.54298 -0.22937 -1.21245  1.97656 -1.63586  0.38078  1.08714 -1.94739  1.71914 -0.52996 -0.95544  1.90679 -1.79233  0.67603  0.81814 -1.85500  1.85500 -0.81814 -0.67603  1.79233 -1.90679  0.95544  0.52996 -1.71914  1.94739 -1.08714 -0.38078  1.63586 -1.97656  1.21245  0.22937 -1.54298  1.99413 -1.33065 -0.07661  1.44104 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E32     2.00000 -1.54298  0.38078  0.95544 -1.85500  1.90679 -1.08714 -0.22937  1.44104 -1.99413  1.63586 -0.52996 -0.81814  1.79233 -1.94739  1.21245  0.07661 -1.33065  1.97656 -1.71914  0.67603  0.67603 -1.71914  1.97656 -1.33065  0.07661  1.21245 -1.94739  1.79233 -0.81814 -0.52996  1.63586 -1.99413  1.44104 -0.22937 -1.08714  1.90679 -1.85500  0.95544  0.38078 -1.54298  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E33     2.00000 -1.63586  0.67603  0.52996 -1.54298  1.99413 -1.71914  0.81814  0.38078 -1.44104  1.97656 -1.79233  0.95544  0.22937 -1.33065  1.94739 -1.85500  1.08714  0.07661 -1.21245  1.90679 -1.90679  1.21245 -0.07661 -1.08714  1.85500 -1.94739  1.33065 -0.22937 -0.95544  1.79233 -1.97656  1.44104 -0.38078 -0.81814  1.71914 -1.99413  1.54298 -0.52996 -0.67603  1.63586 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E34     2.00000 -1.71914  0.95544  0.07661 -1.08714  1.79233 -1.99413  1.63586 -0.81814 -0.22937  1.21245 -1.85500  1.97656 -1.54298  0.67603  0.38078 -1.33065  1.90679 -1.94739  1.44104 -0.52996 -0.52996  1.44104 -1.94739  1.90679 -1.33065  0.38078  0.67603 -1.54298  1.97656 -1.85500  1.21245 -0.22937 -0.81814  1.63586 -1.99413  1.79233 -1.08714  0.07661  0.95544 -1.71914  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E35     2.00000 -1.79233  1.21245 -0.38078 -0.52996  1.33065 -1.85500  1.99413 -1.71914  1.08714 -0.22937 -0.67603  1.44104 -1.90679  1.97656 -1.63586  0.95544 -0.07661 -0.81814  1.54298 -1.94739  1.94739 -1.54298  0.81814  0.07661 -0.95544  1.63586 -1.97656  1.90679 -1.44104  0.67603  0.22937 -1.08714  1.71914 -1.99413  1.85500 -1.33065  0.52996  0.38078 -1.21245  1.79233 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E36     2.00000 -1.85500  1.44104 -0.81814  0.07661  0.67603 -1.33065  1.79233 -1.99413  1.90679 -1.54298  0.95544 -0.22937 -0.52996  1.21245 -1.71914  1.97656 -1.94739  1.63586 -1.08714  0.38078  0.38078 -1.08714  1.63586 -1.94739  1.97656 -1.71914  1.21245 -0.52996 -0.22937  0.95544 -1.54298  1.90679 -1.99413  1.79233 -1.33065  0.67603  0.07661 -0.81814  1.44104 -1.85500  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E37     2.00000 -1.90679  1.63586 -1.21245  0.67603 -0.07661 -0.52996  1.08714 -1.54298  1.85500 -1.99413  1.94739 -1.71914  1.33065 -0.81814  0.22937  0.38078 -0.95544  1.44104 -1.79233  1.97656 -1.97656  1.79233 -1.44104  0.95544 -0.38078 -0.22937  0.81814 -1.33065  1.71914 -1.94739  1.99413 -1.85500  1.54298 -1.08714  0.52996  0.07661 -0.67603  1.21245 -1.63586  1.90679 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E38     2.00000 -1.94739  1.79233 -1.54298  1.21245 -0.81814  0.38078  0.07661 -0.52996  0.95544 -1.33065  1.63586 -1.85500  1.97656 -1.99413  1.90679 -1.71914  1.44104 -1.08714  0.67603 -0.22937 -0.22937  0.67603 -1.08714  1.44104 -1.71914  1.90679 -1.99413  1.97656 -1.85500  1.63586 -1.33065  0.95544 -0.52996  0.07661  0.38078 -0.81814  1.21245 -1.54298  1.79233 -1.94739  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E39     2.00000 -1.97656  1.90679 -1.79233  1.63586 -1.44104  1.21245 -0.95544  0.67603 -0.38078  0.07661  0.22937 -0.52996  0.81814 -1.08714  1.33065 -1.54298  1.71914 -1.85500  1.94739 -1.99413  1.99413 -1.94739  1.85500 -1.71914  1.54298 -1.33065  1.08714 -0.81814  0.52996 -0.22937 -0.07661  0.38078 -0.67603  0.95544 -1.21245  1.44104 -1.63586  1.79233 -1.90679  1.97656 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E40     2.00000 -1.99413  1.97656 -1.94739  1.90679 -1.85500  1.79233 -1.71914  1.63586 -1.54298  1.44104 -1.33065  1.21245 -1.08714  0.95544 -0.81814  0.67603 -0.52996  0.38078 -0.22937  0.07661  0.07661 -0.22937  0.38078 -0.52996  0.67603 -0.81814  0.95544 -1.08714  1.21245 -1.33065  1.44104 -1.54298  1.63586 -1.71914  1.79233 -1.85500  1.90679 -1.94739  1.97656 -1.99413  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............

 Irrational character values:  1.994131602367 = 2*cos(2*π/82) = 2*cos(π/41)
                               1.976560847561 = 2*cos(4*π/82) = 2*cos(2*π/41)
                               1.947390847756 = 2*cos(6*π/82) = 2*cos(3*π/41)
                               1.906792784110 = 2*cos(8*π/82) = 2*cos(4*π/41)
                               1.855004902204 = 2*cos(10*π/82) = 2*cos(5*π/41)
                               1.792331113922 = 2*cos(12*π/82) = 2*cos(6*π/41)
                               1.719139213974 = 2*cos(14*π/82) = 2*cos(7*π/41)
                               1.635858721533 = 2*cos(16*π/82) = 2*cos(8*π/41)
                               1.542978359644 = 2*cos(18*π/82) = 2*cos(9*π/41)
                               1.441043187202 = 2*cos(20*π/82) = 2*cos(10*π/41)
                               1.330651400331 = 2*cos(22*π/82) = 2*cos(11*π/41)
                               1.212450821933 = 2*cos(24*π/82) = 2*cos(12*π/41)
                               1.087135100002 = 2*cos(26*π/82) = 2*cos(13*π/41)
                               0.955439637025 = 2*cos(28*π/82) = 2*cos(14*π/41)
                               0.818137274343 = 2*cos(30*π/82) = 2*cos(15*π/41)
                               0.676033756817 = 2*cos(32*π/82) = 2*cos(16*π/41)
                               0.529963004393 = 2*cos(34*π/82) = 2*cos(17*π/41)
                               0.380782218329 = 2*cos(36*π/82) = 2*cos(18*π/41)
                               0.229366850797 = 2*cos(38*π/82) = 2*cos(19*π/41)
                               0.076605467380 = 2*cos(40*π/82) = 2*cos(20*π/41)



 Symmetry of Rotations and Cartesian products

A1   p+d+f+g+h+i+j+k+l+m    z, z2, z3, z4, z5, z6 
A2   R                      Rz 
E1   R+p+d+f+g+h+i+j+k+l+m  {Rx, Ry}, {x, y}, {xz, yz}, {xz2, yz2}, {xz3, yz3}, {xz4, yz4}, {xz5, yz5} 
E2   d+f+g+h+i+j+k+l+m      {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 C82v point group is 164, and the order of the principal axis (C82) is 82. The group has 44 irreducible representations.

    β  The C82v point group is isomorphic to D41d, D41h and D82.

    γ  The C82v point group is generated by two symmetry elements, C82 and any σv (or, non-canonically, any σd).
       Also, the group may be generated from a σv plus a σd (some pairs will yield smaller groups, though; choosing a minimum angle is safe).

    δ  There are two different sets of symmetry planes containing the principal axis (z axis in standard orientation).
       By convention, the set denoted as σv has the xz plane as a member, while the yz plane is a member of the σd set.

    ε  The lowest nonvanishing multipole moment in C82v 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.
       For this group, however, none of the irrational characters can be expressed by a closed algebraic form using real numbers only.

This Character Table for the C82v point group was created by Gernot Katzer.

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