Character table for the C88v point group

C88v    E        2 C88    2 C44    2 C88^3  2 C22    2 C88^5  2 C44^3  2 C88^7  2 C11    2 C88^9  2 C44^5  2 C8     2 C22^3  2 C88^13 2 C44^7  2 C88^15 2 C11^2  2 C88^17 2 C44^9  2 C88^19 2 C22^5  2 C88^21 2 C4     2 C88^23 2 C11^3  2 C88^25 2 C44^13 2 C88^27 2 C22^7  2 C88^29 2 C44^15 2 C88^31 2 C11^4  2 C8^3   2 C44^17 2 C88^35 2 C22^9  2 C88^37 2 C44^19 2 C88^39 2 C11^5  2 C88^41 2 C44^21 2 C88^43 C2       44 sv    44 sd       <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> 
A1      1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000     ... ..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  1.00000 -1.00000 -1.00000     ..T ... ..... ....... ......... ........... .............
B1      1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000  1.00000 -1.00000     ... ... ..... ....... ......... ........... .............
B2      1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000     ... ... ..... ....... ......... ........... .............
E1      2.00000  1.99490  1.97964  1.95429  1.91899  1.87390  1.81926  1.75536  1.68251  1.60108  1.51150  1.41421  1.30972  1.19856  1.08128  0.95850  0.83083  0.69893  0.56347  0.42513  0.28463  0.14268  0.00000 -0.14268 -0.28463 -0.42513 -0.56347 -0.69893 -0.83083 -0.95850 -1.08128 -1.19856 -1.30972 -1.41421 -1.51150 -1.60108 -1.68251 -1.75536 -1.81926 -1.87390 -1.91899 -1.95429 -1.97964 -1.99490 -2.00000  0.00000  0.00000     TT. TT. ..TT. ....TT. ......TT. ........TT. ..........TT.
E2      2.00000  1.97964  1.91899  1.81926  1.68251  1.51150  1.30972  1.08128  0.83083  0.56347  0.28463  0.00000 -0.28463 -0.56347 -0.83083 -1.08128 -1.30972 -1.51150 -1.68251 -1.81926 -1.91899 -1.97964 -2.00000 -1.97964 -1.91899 -1.81926 -1.68251 -1.51150 -1.30972 -1.08128 -0.83083 -0.56347 -0.28463  0.00000  0.28463  0.56347  0.83083  1.08128  1.30972  1.51150  1.68251  1.81926  1.91899  1.97964  2.00000  0.00000  0.00000     ... ... TT... ..TT... ....TT... ......TT... ........TT...
E3      2.00000  1.95429  1.81926  1.60108  1.30972  0.95850  0.56347  0.14268 -0.28463 -0.69893 -1.08128 -1.41421 -1.68251 -1.87390 -1.97964 -1.99490 -1.91899 -1.75536 -1.51150 -1.19856 -0.83083 -0.42513  0.00000  0.42513  0.83083  1.19856  1.51150  1.75536  1.91899  1.99490  1.97964  1.87390  1.68251  1.41421  1.08128  0.69893  0.28463 -0.14268 -0.56347 -0.95850 -1.30972 -1.60108 -1.81926 -1.95429 -2.00000  0.00000  0.00000     ... ... ..... TT..... ..TT..... ....TT..... ......TT.....
E4      2.00000  1.91899  1.68251  1.30972  0.83083  0.28463 -0.28463 -0.83083 -1.30972 -1.68251 -1.91899 -2.00000 -1.91899 -1.68251 -1.30972 -0.83083 -0.28463  0.28463  0.83083  1.30972  1.68251  1.91899  2.00000  1.91899  1.68251  1.30972  0.83083  0.28463 -0.28463 -0.83083 -1.30972 -1.68251 -1.91899 -2.00000 -1.91899 -1.68251 -1.30972 -0.83083 -0.28463  0.28463  0.83083  1.30972  1.68251  1.91899  2.00000  0.00000  0.00000     ... ... ..... ....... TT....... ..TT....... ....TT.......
E5      2.00000  1.87390  1.51150  0.95850  0.28463 -0.42513 -1.08128 -1.60108 -1.91899 -1.99490 -1.81926 -1.41421 -0.83083 -0.14268  0.56347  1.19856  1.68251  1.95429  1.97964  1.75536  1.30972  0.69893  0.00000 -0.69893 -1.30972 -1.75536 -1.97964 -1.95429 -1.68251 -1.19856 -0.56347  0.14268  0.83083  1.41421  1.81926  1.99490  1.91899  1.60108  1.08128  0.42513 -0.28463 -0.95850 -1.51150 -1.87390 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... TT......... ..TT.........
E6      2.00000  1.81926  1.30972  0.56347 -0.28463 -1.08128 -1.68251 -1.97964 -1.91899 -1.51150 -0.83083  0.00000  0.83083  1.51150  1.91899  1.97964  1.68251  1.08128  0.28463 -0.56347 -1.30972 -1.81926 -2.00000 -1.81926 -1.30972 -0.56347  0.28463  1.08128  1.68251  1.97964  1.91899  1.51150  0.83083  0.00000 -0.83083 -1.51150 -1.91899 -1.97964 -1.68251 -1.08128 -0.28463  0.56347  1.30972  1.81926  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... TT...........
E7      2.00000  1.75536  1.08128  0.14268 -0.83083 -1.60108 -1.97964 -1.87390 -1.30972 -0.42513  0.56347  1.41421  1.91899  1.95429  1.51150  0.69893 -0.28463 -1.19856 -1.81926 -1.99490 -1.68251 -0.95850  0.00000  0.95850  1.68251  1.99490  1.81926  1.19856  0.28463 -0.69893 -1.51150 -1.95429 -1.91899 -1.41421 -0.56347  0.42513  1.30972  1.87390  1.97964  1.60108  0.83083 -0.14268 -1.08128 -1.75536 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E8      2.00000  1.68251  0.83083 -0.28463 -1.30972 -1.91899 -1.91899 -1.30972 -0.28463  0.83083  1.68251  2.00000  1.68251  0.83083 -0.28463 -1.30972 -1.91899 -1.91899 -1.30972 -0.28463  0.83083  1.68251  2.00000  1.68251  0.83083 -0.28463 -1.30972 -1.91899 -1.91899 -1.30972 -0.28463  0.83083  1.68251  2.00000  1.68251  0.83083 -0.28463 -1.30972 -1.91899 -1.91899 -1.30972 -0.28463  0.83083  1.68251  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E9      2.00000  1.60108  0.56347 -0.69893 -1.68251 -1.99490 -1.51150 -0.42513  0.83083  1.75536  1.97964  1.41421  0.28463 -0.95850 -1.81926 -1.95429 -1.30972 -0.14268  1.08128  1.87390  1.91899  1.19856  0.00000 -1.19856 -1.91899 -1.87390 -1.08128  0.14268  1.30972  1.95429  1.81926  0.95850 -0.28463 -1.41421 -1.97964 -1.75536 -0.83083  0.42513  1.51150  1.99490  1.68251  0.69893 -0.56347 -1.60108 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E10     2.00000  1.51150  0.28463 -1.08128 -1.91899 -1.81926 -0.83083  0.56347  1.68251  1.97964  1.30972  0.00000 -1.30972 -1.97964 -1.68251 -0.56347  0.83083  1.81926  1.91899  1.08128 -0.28463 -1.51150 -2.00000 -1.51150 -0.28463  1.08128  1.91899  1.81926  0.83083 -0.56347 -1.68251 -1.97964 -1.30972  0.00000  1.30972  1.97964  1.68251  0.56347 -0.83083 -1.81926 -1.91899 -1.08128  0.28463  1.51150  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E11     2.00000  1.41421  0.00000 -1.41421 -2.00000 -1.41421  0.00000  1.41421  2.00000  1.41421  0.00000 -1.41421 -2.00000 -1.41421  0.00000  1.41421  2.00000  1.41421  0.00000 -1.41421 -2.00000 -1.41421  0.00000  1.41421  2.00000  1.41421  0.00000 -1.41421 -2.00000 -1.41421  0.00000  1.41421  2.00000  1.41421  0.00000 -1.41421 -2.00000 -1.41421  0.00000  1.41421  2.00000  1.41421  0.00000 -1.41421 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E12     2.00000  1.30972 -0.28463 -1.68251 -1.91899 -0.83083  0.83083  1.91899  1.68251  0.28463 -1.30972 -2.00000 -1.30972  0.28463  1.68251  1.91899  0.83083 -0.83083 -1.91899 -1.68251 -0.28463  1.30972  2.00000  1.30972 -0.28463 -1.68251 -1.91899 -0.83083  0.83083  1.91899  1.68251  0.28463 -1.30972 -2.00000 -1.30972  0.28463  1.68251  1.91899  0.83083 -0.83083 -1.91899 -1.68251 -0.28463  1.30972  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E13     2.00000  1.19856 -0.56347 -1.87390 -1.68251 -0.14268  1.51150  1.95429  0.83083 -0.95850 -1.97964 -1.41421  0.28463  1.75536  1.81926  0.42513 -1.30972 -1.99490 -1.08128  0.69893  1.91899  1.60108  0.00000 -1.60108 -1.91899 -0.69893  1.08128  1.99490  1.30972 -0.42513 -1.81926 -1.75536 -0.28463  1.41421  1.97964  0.95850 -0.83083 -1.95429 -1.51150  0.14268  1.68251  1.87390  0.56347 -1.19856 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E14     2.00000  1.08128 -0.83083 -1.97964 -1.30972  0.56347  1.91899  1.51150 -0.28463 -1.81926 -1.68251  0.00000  1.68251  1.81926  0.28463 -1.51150 -1.91899 -0.56347  1.30972  1.97964  0.83083 -1.08128 -2.00000 -1.08128  0.83083  1.97964  1.30972 -0.56347 -1.91899 -1.51150  0.28463  1.81926  1.68251  0.00000 -1.68251 -1.81926 -0.28463  1.51150  1.91899  0.56347 -1.30972 -1.97964 -0.83083  1.08128  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E15     2.00000  0.95850 -1.08128 -1.99490 -0.83083  1.19856  1.97964  0.69893 -1.30972 -1.95429 -0.56347  1.41421  1.91899  0.42513 -1.51150 -1.87390 -0.28463  1.60108  1.81926  0.14268 -1.68251 -1.75536  0.00000  1.75536  1.68251 -0.14268 -1.81926 -1.60108  0.28463  1.87390  1.51150 -0.42513 -1.91899 -1.41421  0.56347  1.95429  1.30972 -0.69893 -1.97964 -1.19856  0.83083  1.99490  1.08128 -0.95850 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E16     2.00000  0.83083 -1.30972 -1.91899 -0.28463  1.68251  1.68251 -0.28463 -1.91899 -1.30972  0.83083  2.00000  0.83083 -1.30972 -1.91899 -0.28463  1.68251  1.68251 -0.28463 -1.91899 -1.30972  0.83083  2.00000  0.83083 -1.30972 -1.91899 -0.28463  1.68251  1.68251 -0.28463 -1.91899 -1.30972  0.83083  2.00000  0.83083 -1.30972 -1.91899 -0.28463  1.68251  1.68251 -0.28463 -1.91899 -1.30972  0.83083  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E17     2.00000  0.69893 -1.51150 -1.75536  0.28463  1.95429  1.08128 -1.19856 -1.91899 -0.14268  1.81926  1.41421 -0.83083 -1.99490 -0.56347  1.60108  1.68251 -0.42513 -1.97964 -0.95850  1.30972  1.87390  0.00000 -1.87390 -1.30972  0.95850  1.97964  0.42513 -1.68251 -1.60108  0.56347  1.99490  0.83083 -1.41421 -1.81926  0.14268  1.91899  1.19856 -1.08128 -1.95429 -0.28463  1.75536  1.51150 -0.69893 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E18     2.00000  0.56347 -1.68251 -1.51150  0.83083  1.97964  0.28463 -1.81926 -1.30972  1.08128  1.91899  0.00000 -1.91899 -1.08128  1.30972  1.81926 -0.28463 -1.97964 -0.83083  1.51150  1.68251 -0.56347 -2.00000 -0.56347  1.68251  1.51150 -0.83083 -1.97964 -0.28463  1.81926  1.30972 -1.08128 -1.91899  0.00000  1.91899  1.08128 -1.30972 -1.81926  0.28463  1.97964  0.83083 -1.51150 -1.68251  0.56347  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E19     2.00000  0.42513 -1.81926 -1.19856  1.30972  1.75536 -0.56347 -1.99490 -0.28463  1.87390  1.08128 -1.41421 -1.68251  0.69893  1.97964  0.14268 -1.91899 -0.95850  1.51150  1.60108 -0.83083 -1.95429  0.00000  1.95429  0.83083 -1.60108 -1.51150  0.95850  1.91899 -0.14268 -1.97964 -0.69893  1.68251  1.41421 -1.08128 -1.87390  0.28463  1.99490  0.56347 -1.75536 -1.30972  1.19856  1.81926 -0.42513 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E20     2.00000  0.28463 -1.91899 -0.83083  1.68251  1.30972 -1.30972 -1.68251  0.83083  1.91899 -0.28463 -2.00000 -0.28463  1.91899  0.83083 -1.68251 -1.30972  1.30972  1.68251 -0.83083 -1.91899  0.28463  2.00000  0.28463 -1.91899 -0.83083  1.68251  1.30972 -1.30972 -1.68251  0.83083  1.91899 -0.28463 -2.00000 -0.28463  1.91899  0.83083 -1.68251 -1.30972  1.30972  1.68251 -0.83083 -1.91899  0.28463  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E21     2.00000  0.14268 -1.97964 -0.42513  1.91899  0.69893 -1.81926 -0.95850  1.68251  1.19856 -1.51150 -1.41421  1.30972  1.60108 -1.08128 -1.75536  0.83083  1.87390 -0.56347 -1.95429  0.28463  1.99490  0.00000 -1.99490 -0.28463  1.95429  0.56347 -1.87390 -0.83083  1.75536  1.08128 -1.60108 -1.30972  1.41421  1.51150 -1.19856 -1.68251  0.95850  1.81926 -0.69893 -1.91899  0.42513  1.97964 -0.14268 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E22     2.00000  0.00000 -2.00000  0.00000  2.00000  0.00000 -2.00000  0.00000  2.00000  0.00000 -2.00000  0.00000  2.00000  0.00000 -2.00000  0.00000  2.00000  0.00000 -2.00000  0.00000  2.00000  0.00000 -2.00000  0.00000  2.00000  0.00000 -2.00000  0.00000  2.00000  0.00000 -2.00000  0.00000  2.00000  0.00000 -2.00000  0.00000  2.00000  0.00000 -2.00000  0.00000  2.00000  0.00000 -2.00000  0.00000  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E23     2.00000 -0.14268 -1.97964  0.42513  1.91899 -0.69893 -1.81926  0.95850  1.68251 -1.19856 -1.51150  1.41421  1.30972 -1.60108 -1.08128  1.75536  0.83083 -1.87390 -0.56347  1.95429  0.28463 -1.99490  0.00000  1.99490 -0.28463 -1.95429  0.56347  1.87390 -0.83083 -1.75536  1.08128  1.60108 -1.30972 -1.41421  1.51150  1.19856 -1.68251 -0.95850  1.81926  0.69893 -1.91899 -0.42513  1.97964  0.14268 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E24     2.00000 -0.28463 -1.91899  0.83083  1.68251 -1.30972 -1.30972  1.68251  0.83083 -1.91899 -0.28463  2.00000 -0.28463 -1.91899  0.83083  1.68251 -1.30972 -1.30972  1.68251  0.83083 -1.91899 -0.28463  2.00000 -0.28463 -1.91899  0.83083  1.68251 -1.30972 -1.30972  1.68251  0.83083 -1.91899 -0.28463  2.00000 -0.28463 -1.91899  0.83083  1.68251 -1.30972 -1.30972  1.68251  0.83083 -1.91899 -0.28463  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E25     2.00000 -0.42513 -1.81926  1.19856  1.30972 -1.75536 -0.56347  1.99490 -0.28463 -1.87390  1.08128  1.41421 -1.68251 -0.69893  1.97964 -0.14268 -1.91899  0.95850  1.51150 -1.60108 -0.83083  1.95429  0.00000 -1.95429  0.83083  1.60108 -1.51150 -0.95850  1.91899  0.14268 -1.97964  0.69893  1.68251 -1.41421 -1.08128  1.87390  0.28463 -1.99490  0.56347  1.75536 -1.30972 -1.19856  1.81926  0.42513 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E26     2.00000 -0.56347 -1.68251  1.51150  0.83083 -1.97964  0.28463  1.81926 -1.30972 -1.08128  1.91899  0.00000 -1.91899  1.08128  1.30972 -1.81926 -0.28463  1.97964 -0.83083 -1.51150  1.68251  0.56347 -2.00000  0.56347  1.68251 -1.51150 -0.83083  1.97964 -0.28463 -1.81926  1.30972  1.08128 -1.91899  0.00000  1.91899 -1.08128 -1.30972  1.81926  0.28463 -1.97964  0.83083  1.51150 -1.68251 -0.56347  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E27     2.00000 -0.69893 -1.51150  1.75536  0.28463 -1.95429  1.08128  1.19856 -1.91899  0.14268  1.81926 -1.41421 -0.83083  1.99490 -0.56347 -1.60108  1.68251  0.42513 -1.97964  0.95850  1.30972 -1.87390  0.00000  1.87390 -1.30972 -0.95850  1.97964 -0.42513 -1.68251  1.60108  0.56347 -1.99490  0.83083  1.41421 -1.81926 -0.14268  1.91899 -1.19856 -1.08128  1.95429 -0.28463 -1.75536  1.51150  0.69893 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E28     2.00000 -0.83083 -1.30972  1.91899 -0.28463 -1.68251  1.68251  0.28463 -1.91899  1.30972  0.83083 -2.00000  0.83083  1.30972 -1.91899  0.28463  1.68251 -1.68251 -0.28463  1.91899 -1.30972 -0.83083  2.00000 -0.83083 -1.30972  1.91899 -0.28463 -1.68251  1.68251  0.28463 -1.91899  1.30972  0.83083 -2.00000  0.83083  1.30972 -1.91899  0.28463  1.68251 -1.68251 -0.28463  1.91899 -1.30972 -0.83083  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E29     2.00000 -0.95850 -1.08128  1.99490 -0.83083 -1.19856  1.97964 -0.69893 -1.30972  1.95429 -0.56347 -1.41421  1.91899 -0.42513 -1.51150  1.87390 -0.28463 -1.60108  1.81926 -0.14268 -1.68251  1.75536  0.00000 -1.75536  1.68251  0.14268 -1.81926  1.60108  0.28463 -1.87390  1.51150  0.42513 -1.91899  1.41421  0.56347 -1.95429  1.30972  0.69893 -1.97964  1.19856  0.83083 -1.99490  1.08128  0.95850 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E30     2.00000 -1.08128 -0.83083  1.97964 -1.30972 -0.56347  1.91899 -1.51150 -0.28463  1.81926 -1.68251  0.00000  1.68251 -1.81926  0.28463  1.51150 -1.91899  0.56347  1.30972 -1.97964  0.83083  1.08128 -2.00000  1.08128  0.83083 -1.97964  1.30972  0.56347 -1.91899  1.51150  0.28463 -1.81926  1.68251  0.00000 -1.68251  1.81926 -0.28463 -1.51150  1.91899 -0.56347 -1.30972  1.97964 -0.83083 -1.08128  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E31     2.00000 -1.19856 -0.56347  1.87390 -1.68251  0.14268  1.51150 -1.95429  0.83083  0.95850 -1.97964  1.41421  0.28463 -1.75536  1.81926 -0.42513 -1.30972  1.99490 -1.08128 -0.69893  1.91899 -1.60108  0.00000  1.60108 -1.91899  0.69893  1.08128 -1.99490  1.30972  0.42513 -1.81926  1.75536 -0.28463 -1.41421  1.97964 -0.95850 -0.83083  1.95429 -1.51150 -0.14268  1.68251 -1.87390  0.56347  1.19856 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E32     2.00000 -1.30972 -0.28463  1.68251 -1.91899  0.83083  0.83083 -1.91899  1.68251 -0.28463 -1.30972  2.00000 -1.30972 -0.28463  1.68251 -1.91899  0.83083  0.83083 -1.91899  1.68251 -0.28463 -1.30972  2.00000 -1.30972 -0.28463  1.68251 -1.91899  0.83083  0.83083 -1.91899  1.68251 -0.28463 -1.30972  2.00000 -1.30972 -0.28463  1.68251 -1.91899  0.83083  0.83083 -1.91899  1.68251 -0.28463 -1.30972  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E33     2.00000 -1.41421  0.00000  1.41421 -2.00000  1.41421  0.00000 -1.41421  2.00000 -1.41421  0.00000  1.41421 -2.00000  1.41421  0.00000 -1.41421  2.00000 -1.41421  0.00000  1.41421 -2.00000  1.41421  0.00000 -1.41421  2.00000 -1.41421  0.00000  1.41421 -2.00000  1.41421  0.00000 -1.41421  2.00000 -1.41421  0.00000  1.41421 -2.00000  1.41421  0.00000 -1.41421  2.00000 -1.41421  0.00000  1.41421 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E34     2.00000 -1.51150  0.28463  1.08128 -1.91899  1.81926 -0.83083 -0.56347  1.68251 -1.97964  1.30972  0.00000 -1.30972  1.97964 -1.68251  0.56347  0.83083 -1.81926  1.91899 -1.08128 -0.28463  1.51150 -2.00000  1.51150 -0.28463 -1.08128  1.91899 -1.81926  0.83083  0.56347 -1.68251  1.97964 -1.30972  0.00000  1.30972 -1.97964  1.68251 -0.56347 -0.83083  1.81926 -1.91899  1.08128  0.28463 -1.51150  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E35     2.00000 -1.60108  0.56347  0.69893 -1.68251  1.99490 -1.51150  0.42513  0.83083 -1.75536  1.97964 -1.41421  0.28463  0.95850 -1.81926  1.95429 -1.30972  0.14268  1.08128 -1.87390  1.91899 -1.19856  0.00000  1.19856 -1.91899  1.87390 -1.08128 -0.14268  1.30972 -1.95429  1.81926 -0.95850 -0.28463  1.41421 -1.97964  1.75536 -0.83083 -0.42513  1.51150 -1.99490  1.68251 -0.69893 -0.56347  1.60108 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E36     2.00000 -1.68251  0.83083  0.28463 -1.30972  1.91899 -1.91899  1.30972 -0.28463 -0.83083  1.68251 -2.00000  1.68251 -0.83083 -0.28463  1.30972 -1.91899  1.91899 -1.30972  0.28463  0.83083 -1.68251  2.00000 -1.68251  0.83083  0.28463 -1.30972  1.91899 -1.91899  1.30972 -0.28463 -0.83083  1.68251 -2.00000  1.68251 -0.83083 -0.28463  1.30972 -1.91899  1.91899 -1.30972  0.28463  0.83083 -1.68251  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E37     2.00000 -1.75536  1.08128 -0.14268 -0.83083  1.60108 -1.97964  1.87390 -1.30972  0.42513  0.56347 -1.41421  1.91899 -1.95429  1.51150 -0.69893 -0.28463  1.19856 -1.81926  1.99490 -1.68251  0.95850  0.00000 -0.95850  1.68251 -1.99490  1.81926 -1.19856  0.28463  0.69893 -1.51150  1.95429 -1.91899  1.41421 -0.56347 -0.42513  1.30972 -1.87390  1.97964 -1.60108  0.83083  0.14268 -1.08128  1.75536 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E38     2.00000 -1.81926  1.30972 -0.56347 -0.28463  1.08128 -1.68251  1.97964 -1.91899  1.51150 -0.83083  0.00000  0.83083 -1.51150  1.91899 -1.97964  1.68251 -1.08128  0.28463  0.56347 -1.30972  1.81926 -2.00000  1.81926 -1.30972  0.56347  0.28463 -1.08128  1.68251 -1.97964  1.91899 -1.51150  0.83083  0.00000 -0.83083  1.51150 -1.91899  1.97964 -1.68251  1.08128 -0.28463 -0.56347  1.30972 -1.81926  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E39     2.00000 -1.87390  1.51150 -0.95850  0.28463  0.42513 -1.08128  1.60108 -1.91899  1.99490 -1.81926  1.41421 -0.83083  0.14268  0.56347 -1.19856  1.68251 -1.95429  1.97964 -1.75536  1.30972 -0.69893  0.00000  0.69893 -1.30972  1.75536 -1.97964  1.95429 -1.68251  1.19856 -0.56347 -0.14268  0.83083 -1.41421  1.81926 -1.99490  1.91899 -1.60108  1.08128 -0.42513 -0.28463  0.95850 -1.51150  1.87390 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E40     2.00000 -1.91899  1.68251 -1.30972  0.83083 -0.28463 -0.28463  0.83083 -1.30972  1.68251 -1.91899  2.00000 -1.91899  1.68251 -1.30972  0.83083 -0.28463 -0.28463  0.83083 -1.30972  1.68251 -1.91899  2.00000 -1.91899  1.68251 -1.30972  0.83083 -0.28463 -0.28463  0.83083 -1.30972  1.68251 -1.91899  2.00000 -1.91899  1.68251 -1.30972  0.83083 -0.28463 -0.28463  0.83083 -1.30972  1.68251 -1.91899  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E41     2.00000 -1.95429  1.81926 -1.60108  1.30972 -0.95850  0.56347 -0.14268 -0.28463  0.69893 -1.08128  1.41421 -1.68251  1.87390 -1.97964  1.99490 -1.91899  1.75536 -1.51150  1.19856 -0.83083  0.42513  0.00000 -0.42513  0.83083 -1.19856  1.51150 -1.75536  1.91899 -1.99490  1.97964 -1.87390  1.68251 -1.41421  1.08128 -0.69893  0.28463  0.14268 -0.56347  0.95850 -1.30972  1.60108 -1.81926  1.95429 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E42     2.00000 -1.97964  1.91899 -1.81926  1.68251 -1.51150  1.30972 -1.08128  0.83083 -0.56347  0.28463  0.00000 -0.28463  0.56347 -0.83083  1.08128 -1.30972  1.51150 -1.68251  1.81926 -1.91899  1.97964 -2.00000  1.97964 -1.91899  1.81926 -1.68251  1.51150 -1.30972  1.08128 -0.83083  0.56347 -0.28463  0.00000  0.28463 -0.56347  0.83083 -1.08128  1.30972 -1.51150  1.68251 -1.81926  1.91899 -1.97964  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E43     2.00000 -1.99490  1.97964 -1.95429  1.91899 -1.87390  1.81926 -1.75536  1.68251 -1.60108  1.51150 -1.41421  1.30972 -1.19856  1.08128 -0.95850  0.83083 -0.69893  0.56347 -0.42513  0.28463 -0.14268  0.00000  0.14268 -0.28463  0.42513 -0.56347  0.69893 -0.83083  0.95850 -1.08128  1.19856 -1.30972  1.41421 -1.51150  1.60108 -1.68251  1.75536 -1.81926  1.87390 -1.91899  1.95429 -1.97964  1.99490 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............

 Irrational character values:  1.994904229221 = 2*cos(2*π/88) = 2*cos(π/44)
                               1.979642883762 = 2*cos(4*π/88) = 2*cos(π/22)
                               1.954293731942 = 2*cos(6*π/88) = 2*cos(3*π/44)
                               1.918985947229 = 2*cos(8*π/88) = 2*cos(π/11)
                               1.873899450000 = 2*cos(10*π/88) = 2*cos(5*π/44)
                               1.819263990709 = 2*cos(12*π/88) = 2*cos(3*π/22)
                               1.755357979135 = 2*cos(14*π/88) = 2*cos(7*π/44)
                               1.682507065662 = 2*cos(16*π/88) = 2*cos(2*π/11)
                               1.601082481849 = 2*cos(18*π/88) = 2*cos(9*π/44)
                               1.511499148709 = 2*cos(20*π/88) = 2*cos(5*π/22)
                               1.414213562373 = 2*cos(22*π/88) = 2*cos(π/4) = √2
                               1.309721467891 = 2*cos(24*π/88) = 2*cos(3*π/11)
                               1.198555333023 = 2*cos(26*π/88) = 2*cos(13*π/44)
                               1.081281634911 = 2*cos(28*π/88) = 2*cos(7*π/22)
                               0.958497973440 = 2*cos(30*π/88) = 2*cos(15*π/44)
                               0.830830026004 = 2*cos(32*π/88) = 2*cos(4*π/11)
                               0.698928359198 = 2*cos(34*π/88) = 2*cos(17*π/44)
                               0.563465113683 = 2*cos(36*π/88) = 2*cos(9*π/22)
                               0.425130579106 = 2*cos(38*π/88) = 2*cos(19*π/44)
                               0.284629676547 = 2*cos(40*π/88) = 2*cos(5*π/11)
                               0.142678366398 = 2*cos(42*π/88) = 2*cos(21*π/44)



 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 C88v point group is 176, and the order of the principal axis (C88) is 88. The group has 47 irreducible representations.

    β  The C88v point group is isomorphic to D44d and D88.

    γ  The C88v point group is generated by two symmetry elements, C88 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 contains both the xz and the yz planes.

    ε  The lowest nonvanishing multipole moment in C88v 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, some of the irrational characters cannot be expressed by a closed algebraic form using real numbers only.

This Character Table for the C88v point group was created by Gernot Katzer.

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