Character table for the D66 point group

D66     E        2 C66    2 C33    2 C22    2 C33^2  2 C66^5  2 C11    2 C66^7  2 C33^4  2 C22^3  2 C33^5  2 C6     2 C11^2  2 C66^13 2 C33^7  2 C22^5  2 C33^8  2 C66^17 2 C11^3  2 C66^19 2 C33^10 2 C22^7  2 C3     2 C66^23 2 C11^4  2 C66^25 2 C33^13 2 C22^9  2 C33^14 2 C66^29 2 C11^5  2 C66^31 2 C33^16 C2       33 C2'   33 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     ... ... ....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     ..T ..T ..... ......T ......... ..........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     ... ... ..... ....... ......... ........... .............
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     ... ... ..... ....... ......... ........... .............
E1      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 -0.09516 -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 -2.00000  0.00000  0.00000     TT. TT. ..TT. ....TT. ......TT. ........TT. ..........TT.
E2      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 -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  2.00000  0.00000  0.00000     ... ... TT... ..TT... ....TT... ......TT... ........TT...
E3      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..... ......TT.....
E4      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  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  2.00000  0.00000  0.00000     ... ... ..... ....... TT....... ..TT....... ....TT.......
E5      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 -0.47152 -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 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... TT......... ..TT.........
E6      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     ... ... ..... ....... ......... ........... TT...........
E7      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  0.65414  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 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E8      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  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  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E9      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     ... ... ..... ....... ......... ........... .............
E10     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 -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  2.00000  0.00000  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  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 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E12     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     ... ... ..... ....... ......... ........... .............
E13     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 -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 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E14     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 -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  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E15     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     ... ... ..... ....... ......... ........... .............
E16     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  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  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E17     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 -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 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E18     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     ... ... ..... ....... ......... ........... .............
E19     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  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 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E20     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  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  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E21     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     ... ... ..... ....... ......... ........... .............
E22     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 -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  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E23     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  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 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E24     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     ... ... ..... ....... ......... ........... .............
E25     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 -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 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E26     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 -0.65414  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  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E27     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     ... ... ..... ....... ......... ........... .............
E28     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  0.47152 -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  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E29     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 -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 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E30     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     ... ... ..... ....... ......... ........... .............
E31     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  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 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E32     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  0.09516 -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  2.00000  0.00000  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+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 D66 point group is 132, and the order of the principal axis (C66) is 66. The group has 36 irreducible representations.

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

    γ  The D66 point group is generated by two symmetry elements, C66 and a perpendicular C2 (or, non-canonically, C2).
       Also, the group may be generated from a C2 plus a C2 (some pairs will yield smaller groups, though; choosing a minimum angle is safe).

    δ  There are two different sets of twofold symmetry axes perpendicular to the principal axis (z axis in standard orientation).
       By convention, the set denoted as C2 has the x axis as a member, while the y axis is a member of the C2 set.

    ε  The lowest nonvanishing multipole moment in D66 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.
       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 D66 point group was created by Gernot Katzer.

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