Character table for the D33h point group

D33h     E        2 C33    2 C33^2  2 C11    2 C33^4  2 C33^5  2 C11^2  2 C33^7  2 C33^8  2 C11^3  2 C33^10 2 C3     2 C11^4  2 C33^13 2 C33^14 2 C11^5  2 C33^16 33 C2'   sh       2 S33    2 S11    2 S33^5  2 S33^7  2 S11^3  2 S3     2 S33^13 2 S11^5  2 S33^17 2 S33^19 2 S11^7  2 S33^23 2 S33^25 2 S11^9  2 S33^29 2 S33^31 33 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     ... ... ....T ....... ........T ........... ............T
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     ... ... ..... ....... ......... ........... .............
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     ..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     ... ..T ..... ......T ......... ..........T .............
E1'     2.00000  1.96386  1.85674  1.68251  1.44747  1.16011  0.83083  0.47152  0.09516 -0.28463 -0.65414 -1.00000 -1.30972 -1.57211 -1.77767 -1.91899 -1.99094  0.00000  2.00000  1.96386  1.68251  1.16011  0.47152 -0.28463 -1.00000 -1.57211 -1.91899 -1.99094 -1.77767 -1.30972 -0.65414  0.09516  0.83083  1.44747  1.85674  0.00000     ... TT. ..... ....TT. ......... ........TT. .............
E1"     2.00000  1.96386  1.85674  1.68251  1.44747  1.16011  0.83083  0.47152  0.09516 -0.28463 -0.65414 -1.00000 -1.30972 -1.57211 -1.77767 -1.91899 -1.99094  0.00000 -2.00000 -1.96386 -1.68251 -1.16011 -0.47152  0.28463  1.00000  1.57211  1.91899  1.99094  1.77767  1.30972  0.65414 -0.09516 -0.83083 -1.44747 -1.85674  0.00000     TT. ... ..TT. ....... ......TT. ........... ..........TT.
E2'     2.00000  1.85674  1.44747  0.83083  0.09516 -0.65414 -1.30972 -1.77767 -1.99094 -1.91899 -1.57211 -1.00000 -0.28463  0.47152  1.16011  1.68251  1.96386  0.00000  2.00000  1.85674  0.83083 -0.65414 -1.77767 -1.91899 -1.00000  0.47152  1.68251  1.96386  1.16011 -0.28463 -1.57211 -1.99094 -1.30972  0.09516  1.44747  0.00000     ... ... TT... ....... ....TT... ........... ........TT...
E2"     2.00000  1.85674  1.44747  0.83083  0.09516 -0.65414 -1.30972 -1.77767 -1.99094 -1.91899 -1.57211 -1.00000 -0.28463  0.47152  1.16011  1.68251  1.96386  0.00000 -2.00000 -1.85674 -0.83083  0.65414  1.77767  1.91899  1.00000 -0.47152 -1.68251 -1.96386 -1.16011  0.28463  1.57211  1.99094  1.30972 -0.09516 -1.44747  0.00000     ... ... ..... ..TT... ......... ......TT... .............
E3'     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  0.00000  2.00000  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  0.00000     ... ... ..... TT..... ......... ....TT..... .............
E3"     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  0.00000 -2.00000 -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  0.00000     ... ... ..... ....... ..TT..... ........... ......TT.....
E4'     2.00000  1.44747  0.09516 -1.30972 -1.99094 -1.57211 -0.28463  1.16011  1.96386  1.68251  0.47152 -1.00000 -1.91899 -1.77767 -0.65414  0.83083  1.85674  0.00000  2.00000  1.44747 -1.30972 -1.57211  1.16011  1.68251 -1.00000 -1.77767  0.83083  1.85674 -0.65414 -1.91899  0.47152  1.96386 -0.28463 -1.99094  0.09516  0.00000     ... ... ..... ....... TT....... ........... ....TT.......
E4"     2.00000  1.44747  0.09516 -1.30972 -1.99094 -1.57211 -0.28463  1.16011  1.96386  1.68251  0.47152 -1.00000 -1.91899 -1.77767 -0.65414  0.83083  1.85674  0.00000 -2.00000 -1.44747  1.30972  1.57211 -1.16011 -1.68251  1.00000  1.77767 -0.83083 -1.85674  0.65414  1.91899 -0.47152 -1.96386  0.28463  1.99094 -0.09516  0.00000     ... ... ..... ....... ......... ..TT....... .............
E5'     2.00000  1.16011 -0.65414 -1.91899 -1.57211  0.09516  1.68251  1.85674  0.47152 -1.30972 -1.99094 -1.00000  0.83083  1.96386  1.44747 -0.28463 -1.77767  0.00000  2.00000  1.16011 -1.91899  0.09516  1.85674 -1.30972 -1.00000  1.96386 -0.28463 -1.77767  1.44747  0.83083 -1.99094  0.47152  1.68251 -1.57211 -0.65414  0.00000     ... ... ..... ....... ......... TT......... .............
E5"     2.00000  1.16011 -0.65414 -1.91899 -1.57211  0.09516  1.68251  1.85674  0.47152 -1.30972 -1.99094 -1.00000  0.83083  1.96386  1.44747 -0.28463 -1.77767  0.00000 -2.00000 -1.16011  1.91899 -0.09516 -1.85674  1.30972  1.00000 -1.96386  0.28463  1.77767 -1.44747 -0.83083  1.99094 -0.47152 -1.68251  1.57211  0.65414  0.00000     ... ... ..... ....... ......... ........... ..TT.........
E6'     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  0.00000  2.00000  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  0.00000     ... ... ..... ....... ......... ........... TT...........
E6"     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  0.00000 -2.00000 -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  0.00000     ... ... ..... ....... ......... ........... .............
E7'     2.00000  0.47152 -1.77767 -1.30972  1.16011  1.85674 -0.28463 -1.99094 -0.65414  1.68251  1.44747 -1.00000 -1.91899  0.09516  1.96386  0.83083 -1.57211  0.00000  2.00000  0.47152 -1.30972  1.85674 -1.99094  1.68251 -1.00000  0.09516  0.83083 -1.57211  1.96386 -1.91899  1.44747 -0.65414 -0.28463  1.16011 -1.77767  0.00000     ... ... ..... ....... ......... ........... .............
E7"     2.00000  0.47152 -1.77767 -1.30972  1.16011  1.85674 -0.28463 -1.99094 -0.65414  1.68251  1.44747 -1.00000 -1.91899  0.09516  1.96386  0.83083 -1.57211  0.00000 -2.00000 -0.47152  1.30972 -1.85674  1.99094 -1.68251  1.00000 -0.09516 -0.83083  1.57211 -1.96386  1.91899 -1.44747  0.65414  0.28463 -1.16011  1.77767  0.00000     ... ... ..... ....... ......... ........... .............
E8'     2.00000  0.09516 -1.99094 -0.28463  1.96386  0.47152 -1.91899 -0.65414  1.85674  0.83083 -1.77767 -1.00000  1.68251  1.16011 -1.57211 -1.30972  1.44747  0.00000  2.00000  0.09516 -0.28463  0.47152 -0.65414  0.83083 -1.00000  1.16011 -1.30972  1.44747 -1.57211  1.68251 -1.77767  1.85674 -1.91899  1.96386 -1.99094  0.00000     ... ... ..... ....... ......... ........... .............
E8"     2.00000  0.09516 -1.99094 -0.28463  1.96386  0.47152 -1.91899 -0.65414  1.85674  0.83083 -1.77767 -1.00000  1.68251  1.16011 -1.57211 -1.30972  1.44747  0.00000 -2.00000 -0.09516  0.28463 -0.47152  0.65414 -0.83083  1.00000 -1.16011  1.30972 -1.44747  1.57211 -1.68251  1.77767 -1.85674  1.91899 -1.96386  1.99094  0.00000     ... ... ..... ....... ......... ........... .............
E9'     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  0.00000  2.00000 -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  0.00000     ... ... ..... ....... ......... ........... .............
E9"     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  0.00000 -2.00000  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  0.00000     ... ... ..... ....... ......... ........... .............
E10'    2.00000 -0.65414 -1.57211  1.68251  0.47152 -1.99094  0.83083  1.44747 -1.77767 -0.28463  1.96386 -1.00000 -1.30972  1.85674  0.09516 -1.91899  1.16011  0.00000  2.00000 -0.65414  1.68251 -1.99094  1.44747 -0.28463 -1.00000  1.85674 -1.91899  1.16011  0.09516 -1.30972  1.96386 -1.77767  0.83083  0.47152 -1.57211  0.00000     ... ... ..... ....... ......... ........... .............
E10"    2.00000 -0.65414 -1.57211  1.68251  0.47152 -1.99094  0.83083  1.44747 -1.77767 -0.28463  1.96386 -1.00000 -1.30972  1.85674  0.09516 -1.91899  1.16011  0.00000 -2.00000  0.65414 -1.68251  1.99094 -1.44747  0.28463  1.00000 -1.85674  1.91899 -1.16011 -0.09516  1.30972 -1.96386  1.77767 -0.83083 -0.47152  1.57211  0.00000     ... ... ..... ....... ......... ........... .............
E11'    2.00000 -1.00000 -1.00000  2.00000 -1.00000 -1.00000  2.00000 -1.00000 -1.00000  2.00000 -1.00000 -1.00000  2.00000 -1.00000 -1.00000  2.00000 -1.00000  0.00000  2.00000 -1.00000  2.00000 -1.00000 -1.00000  2.00000 -1.00000 -1.00000  2.00000 -1.00000 -1.00000  2.00000 -1.00000 -1.00000  2.00000 -1.00000 -1.00000  0.00000     ... ... ..... ....... ......... ........... .............
E11"    2.00000 -1.00000 -1.00000  2.00000 -1.00000 -1.00000  2.00000 -1.00000 -1.00000  2.00000 -1.00000 -1.00000  2.00000 -1.00000 -1.00000  2.00000 -1.00000  0.00000 -2.00000  1.00000 -2.00000  1.00000  1.00000 -2.00000  1.00000  1.00000 -2.00000  1.00000  1.00000 -2.00000  1.00000  1.00000 -2.00000  1.00000  1.00000  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  0.00000  2.00000 -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  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  0.00000 -2.00000  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  0.00000     ... ... ..... ....... ......... ........... .............
E13'    2.00000 -1.57211  0.47152  0.83083 -1.77767  1.96386 -1.30972  0.09516  1.16011 -1.91899  1.85674 -1.00000 -0.28463  1.44747 -1.99094  1.68251 -0.65414  0.00000  2.00000 -1.57211  0.83083  1.96386  0.09516 -1.91899 -1.00000  1.44747  1.68251 -0.65414 -1.99094 -0.28463  1.85674  1.16011 -1.30972 -1.77767  0.47152  0.00000     ... ... ..... ....... ......... ........... .............
E13"    2.00000 -1.57211  0.47152  0.83083 -1.77767  1.96386 -1.30972  0.09516  1.16011 -1.91899  1.85674 -1.00000 -0.28463  1.44747 -1.99094  1.68251 -0.65414  0.00000 -2.00000  1.57211 -0.83083 -1.96386 -0.09516  1.91899  1.00000 -1.44747 -1.68251  0.65414  1.99094  0.28463 -1.85674 -1.16011  1.30972  1.77767 -0.47152  0.00000     ... ... ..... ....... ......... ........... .............
E14'    2.00000 -1.77767  1.16011 -0.28463 -0.65414  1.44747 -1.91899  1.96386 -1.57211  0.83083  0.09516 -1.00000  1.68251 -1.99094  1.85674 -1.30972  0.47152  0.00000  2.00000 -1.77767 -0.28463  1.44747  1.96386  0.83083 -1.00000 -1.99094 -1.30972  0.47152  1.85674  1.68251  0.09516 -1.57211 -1.91899 -0.65414  1.16011  0.00000     ... ... ..... ....... ......... ........... .............
E14"    2.00000 -1.77767  1.16011 -0.28463 -0.65414  1.44747 -1.91899  1.96386 -1.57211  0.83083  0.09516 -1.00000  1.68251 -1.99094  1.85674 -1.30972  0.47152  0.00000 -2.00000  1.77767  0.28463 -1.44747 -1.96386 -0.83083  1.00000  1.99094  1.30972 -0.47152 -1.85674 -1.68251 -0.09516  1.57211  1.91899  0.65414 -1.16011  0.00000     ... ... ..... ....... ......... ........... .............
E15'    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  0.00000  2.00000 -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  0.00000     ... ... ..... ....... ......... ........... .............
E15"    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  0.00000 -2.00000  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  0.00000     ... ... ..... ....... ......... ........... .............
E16'    2.00000 -1.99094  1.96386 -1.91899  1.85674 -1.77767  1.68251 -1.57211  1.44747 -1.30972  1.16011 -1.00000  0.83083 -0.65414  0.47152 -0.28463  0.09516  0.00000  2.00000 -1.99094 -1.91899 -1.77767 -1.57211 -1.30972 -1.00000 -0.65414 -0.28463  0.09516  0.47152  0.83083  1.16011  1.44747  1.68251  1.85674  1.96386  0.00000     ... ... ..... ....... ......... ........... .............
E16"    2.00000 -1.99094  1.96386 -1.91899  1.85674 -1.77767  1.68251 -1.57211  1.44747 -1.30972  1.16011 -1.00000  0.83083 -0.65414  0.47152 -0.28463  0.09516  0.00000 -2.00000  1.99094  1.91899  1.77767  1.57211  1.30972  1.00000  0.65414  0.28463 -0.09516 -0.47152 -0.83083 -1.16011 -1.44747 -1.68251 -1.85674 -1.96386  0.00000     ... ... ..... ....... ......... ........... .............

 Irrational character values:  1.990943845146 = 2*cos(2*π/66) = 2*cos(π/33)
                               1.963857394525 = 2*cos(4*π/66) = 2*cos(2*π/33)
                               1.918985947229 = 2*cos(6*π/66) = 2*cos(π/11)
                               1.856735866032 = 2*cos(8*π/66) = 2*cos(4*π/33)
                               1.777670897310 = 2*cos(10*π/66) = 2*cos(5*π/33)
                               1.682507065662 = 2*cos(12*π/66) = 2*cos(2*π/11)
                               1.572106189486 = 2*cos(14*π/66) = 2*cos(7*π/33)
                               1.447468076210 = 2*cos(16*π/66) = 2*cos(8*π/33)
                               1.309721467891 = 2*cos(18*π/66) = 2*cos(3*π/11)
                               1.160113819142 = 2*cos(20*π/66) = 2*cos(10*π/33)
                               0.830830026004 = 2*cos(24*π/66) = 2*cos(4*π/11)
                               0.654135926635 = 2*cos(26*π/66) = 2*cos(13*π/33)
                               0.471517871019 = 2*cos(28*π/66) = 2*cos(14*π/33)
                               0.284629676547 = 2*cos(30*π/66) = 2*cos(5*π/11)
                               0.095163831647 = 2*cos(32*π/66) = 2*cos(16*π/33)



 Symmetry of Rotations and Cartesian products

A1'  d+g+i+k+m    z2, z4, z6 
A2'  R            Rz 
A2"  p+f+h+j+l    z, z3, z5 
E1'  p+f+h+j+l    {x, y}, {xz2, yz2}, {xz4, yz4} 
E1"  R+d+g+i+k+m  {Rx, Ry}, {xz, yz}, {xz3, yz3}, {xz5, yz5} 
E2'  d+g+i+k+m    {x2y2, xy}, {z2(x2y2), xyz2}, {z4(x2y2), xyz4} 
E2"  f+h+j+l      {z(x2y2), xyz}, {z3(x2y2), xyz3} 
E3'  f+h+j+l      {x(x2−3y2), y(3x2y2)}, {xz2(x2−3y2), yz2(3x2y2)} 
E3"  g+i+k+m      {xz(x2−3y2), yz(3x2y2)}, {xz3(x2−3y2), yz3(3x2y2)} 
E4'  g+i+k+m      {(x2y2)2−4x2y2, xy(x2y2)}, {z2((x2y2)2−4x2y2), xyz2(x2y2)} 
E4"  h+j+l        {z((x2y2)2−4x2y2), xyz(x2y2)} 
E5'  h+j+l        {x(x2−(5+2√5)y2)(x2−(5−2√5)y2), y((5+2√5)x2y2)((5−2√5)x2y2)} 
E5"  i+k+m        {xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), yz((5+2√5)x2y2)((5−2√5)x2y2)} 
E6'  i+k+m        {x2(x2−3y2)2y2(3x2y2)2, xy(x2−3y2)(3x2y2)} 
E6"  j+l 
E7'  j+l 
E7"  k+m 
E8'  k+m 
E8"  l 
E9'  l 
E9"  m 
E10' m 

 Notes:

    α  The order of the D33h point group is 132, and the order of the principal axis (S33) is 66. The group has 36 irreducible representations.

    β  The D33h point group is isomorphic to D33d, C66v and D66.

    γ  The D33h point group is generated by two symmetry elements, S33 and either a perpendicular C2 or a vertical σv.
       Also, the group may be generated from two σv planes, or a σv and a C2 (some pairs will yield smaller groups, though; choosing a minimum angle is safe).
       The canonical choice, however, is to use redundant generators: C33, C2 and σh.

    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.
       Similarly, the single set of symmetry planes denoted σd contains the xz plane but not the yz plane.

    ε  The lowest nonvanishing multipole moment in D33h is 4 (quadrupole moment).

    ζ  This point group is non-Abelian (some symmetry operations are not commutative).
       Therefore, the character table contains multi-membered classes and degenerate irreducible representations.

    η  Some of the characters in the table are irrational because the order of the principal axis is neither 1,2,3,4 nor 6.
       These irrational values can be expressed as cosine values, or as solutions of algebraic equations with a leading coefficient of 1.
       All characters are algebraic integers of a degree much less than half the order of the principal axis.
       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 D33h point group was created by Gernot Katzer.

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