Character table for the D52d point group

D52d    E         2 S104    2 C52     2 S104^3  2 C26     2 S104^5  2 C52^3   2 S104^7  2 C13     2 S104^9  2 C52^5   2 S104^11 2 C26^3   2 S8      2 C52^7   2 S104^15 2 C13^2   2 S104^17 2 C52^9   2 S104^19 2 C26^5   2 S104^21 2 C52^11  2 S104^23 2 C13^3   2 S104^25 2 C4      2 S104^27 2 C26^7   2 S104^29 2 C52^15  2 S104^31 2 C13^4   2 S104^33 2 C52^17  2 S104^35 2 C26^9   2 S104^37 2 C52^19  2 S8^3    2 C13^5   2 S104^41 2 C52^21  2 S104^43 2 C26^11  2 S104^45 2 C52^23  2 S104^47 2 C13^6   2 S104^49 2 C52^25  2 S104^51 C2        52 C2'    52 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   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   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  -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  -1.00000   1.00000  -1.00000   1.00000  -1.00000   1.00000  -1.00000   1.00000     ... ..T ..... ......T ......... ..........T .............
E1       2.00000   1.99635   1.98542   1.96724   1.94188   1.90944   1.87003   1.82380   1.77091   1.71156   1.64597   1.57437   1.49702   1.41421   1.32625   1.23344   1.13613   1.03468   0.92945   0.82083   0.70921   0.59501   0.47863   0.36051   0.24107   0.12076   0.00000  -0.12076  -0.24107  -0.36051  -0.47863  -0.59501  -0.70921  -0.82083  -0.92945  -1.03468  -1.13613  -1.23344  -1.32625  -1.41421  -1.49702  -1.57437  -1.64597  -1.71156  -1.77091  -1.82380  -1.87003  -1.90944  -1.94188  -1.96724  -1.98542  -1.99635  -2.00000   0.00000   0.00000     ... TT. ..... ....TT. ......... ........TT. .............
E2       2.00000   1.98542   1.94188   1.87003   1.77091   1.64597   1.49702   1.32625   1.13613   0.92945   0.70921   0.47863   0.24107   0.00000  -0.24107  -0.47863  -0.70921  -0.92945  -1.13613  -1.32625  -1.49702  -1.64597  -1.77091  -1.87003  -1.94188  -1.98542  -2.00000  -1.98542  -1.94188  -1.87003  -1.77091  -1.64597  -1.49702  -1.32625  -1.13613  -0.92945  -0.70921  -0.47863  -0.24107   0.00000   0.24107   0.47863   0.70921   0.92945   1.13613   1.32625   1.49702   1.64597   1.77091   1.87003   1.94188   1.98542   2.00000   0.00000   0.00000     ... ... TT... ....... ....TT... ........... ........TT...
E3       2.00000   1.96724   1.87003   1.71156   1.49702   1.23344   0.92945   0.59501   0.24107  -0.12076  -0.47863  -0.82083  -1.13613  -1.41421  -1.64597  -1.82380  -1.94188  -1.99635  -1.98542  -1.90944  -1.77091  -1.57437  -1.32625  -1.03468  -0.70921  -0.36051   0.00000   0.36051   0.70921   1.03468   1.32625   1.57437   1.77091   1.90944   1.98542   1.99635   1.94188   1.82380   1.64597   1.41421   1.13613   0.82083   0.47863   0.12076  -0.24107  -0.59501  -0.92945  -1.23344  -1.49702  -1.71156  -1.87003  -1.96724  -2.00000   0.00000   0.00000     ... ... ..... TT..... ......... ....TT..... .............
E4       2.00000   1.94188   1.77091   1.49702   1.13613   0.70921   0.24107  -0.24107  -0.70921  -1.13613  -1.49702  -1.77091  -1.94188  -2.00000  -1.94188  -1.77091  -1.49702  -1.13613  -0.70921  -0.24107   0.24107   0.70921   1.13613   1.49702   1.77091   1.94188   2.00000   1.94188   1.77091   1.49702   1.13613   0.70921   0.24107  -0.24107  -0.70921  -1.13613  -1.49702  -1.77091  -1.94188  -2.00000  -1.94188  -1.77091  -1.49702  -1.13613  -0.70921  -0.24107   0.24107   0.70921   1.13613   1.49702   1.77091   1.94188   2.00000   0.00000   0.00000     ... ... ..... ....... TT....... ........... ....TT.......
E5       2.00000   1.90944   1.64597   1.23344   0.70921   0.12076  -0.47863  -1.03468  -1.49702  -1.82380  -1.98542  -1.96724  -1.77091  -1.41421  -0.92945  -0.36051   0.24107   0.82083   1.32625   1.71156   1.94188   1.99635   1.87003   1.57437   1.13613   0.59501   0.00000  -0.59501  -1.13613  -1.57437  -1.87003  -1.99635  -1.94188  -1.71156  -1.32625  -0.82083  -0.24107   0.36051   0.92945   1.41421   1.77091   1.96724   1.98542   1.82380   1.49702   1.03468   0.47863  -0.12076  -0.70921  -1.23344  -1.64597  -1.90944  -2.00000   0.00000   0.00000     ... ... ..... ....... ......... TT......... .............
E6       2.00000   1.87003   1.49702   0.92945   0.24107  -0.47863  -1.13613  -1.64597  -1.94188  -1.98542  -1.77091  -1.32625  -0.70921   0.00000   0.70921   1.32625   1.77091   1.98542   1.94188   1.64597   1.13613   0.47863  -0.24107  -0.92945  -1.49702  -1.87003  -2.00000  -1.87003  -1.49702  -0.92945  -0.24107   0.47863   1.13613   1.64597   1.94188   1.98542   1.77091   1.32625   0.70921   0.00000  -0.70921  -1.32625  -1.77091  -1.98542  -1.94188  -1.64597  -1.13613  -0.47863   0.24107   0.92945   1.49702   1.87003   2.00000   0.00000   0.00000     ... ... ..... ....... ......... ........... TT...........
E7       2.00000   1.82380   1.32625   0.59501  -0.24107  -1.03468  -1.64597  -1.96724  -1.94188  -1.57437  -0.92945  -0.12076   0.70921   1.41421   1.87003   1.99635   1.77091   1.23344   0.47863  -0.36051  -1.13613  -1.71156  -1.98542  -1.90944  -1.49702  -0.82083   0.00000   0.82083   1.49702   1.90944   1.98542   1.71156   1.13613   0.36051  -0.47863  -1.23344  -1.77091  -1.99635  -1.87003  -1.41421  -0.70921   0.12076   0.92945   1.57437   1.94188   1.96724   1.64597   1.03468   0.24107  -0.59501  -1.32625  -1.82380  -2.00000   0.00000   0.00000     ... ... ..... ....... ......... ........... .............
E8       2.00000   1.77091   1.13613   0.24107  -0.70921  -1.49702  -1.94188  -1.94188  -1.49702  -0.70921   0.24107   1.13613   1.77091   2.00000   1.77091   1.13613   0.24107  -0.70921  -1.49702  -1.94188  -1.94188  -1.49702  -0.70921   0.24107   1.13613   1.77091   2.00000   1.77091   1.13613   0.24107  -0.70921  -1.49702  -1.94188  -1.94188  -1.49702  -0.70921   0.24107   1.13613   1.77091   2.00000   1.77091   1.13613   0.24107  -0.70921  -1.49702  -1.94188  -1.94188  -1.49702  -0.70921   0.24107   1.13613   1.77091   2.00000   0.00000   0.00000     ... ... ..... ....... ......... ........... .............
E9       2.00000   1.71156   0.92945  -0.12076  -1.13613  -1.82380  -1.98542  -1.57437  -0.70921   0.36051   1.32625   1.90944   1.94188   1.41421   0.47863  -0.59501  -1.49702  -1.96724  -1.87003  -1.23344  -0.24107   0.82083   1.64597   1.99635   1.77091   1.03468   0.00000  -1.03468  -1.77091  -1.99635  -1.64597  -0.82083   0.24107   1.23344   1.87003   1.96724   1.49702   0.59501  -0.47863  -1.41421  -1.94188  -1.90944  -1.32625  -0.36051   0.70921   1.57437   1.98542   1.82380   1.13613   0.12076  -0.92945  -1.71156  -2.00000   0.00000   0.00000     ... ... ..... ....... ......... ........... .............
E10      2.00000   1.64597   0.70921  -0.47863  -1.49702  -1.98542  -1.77091  -0.92945   0.24107   1.32625   1.94188   1.87003   1.13613   0.00000  -1.13613  -1.87003  -1.94188  -1.32625  -0.24107   0.92945   1.77091   1.98542   1.49702   0.47863  -0.70921  -1.64597  -2.00000  -1.64597  -0.70921   0.47863   1.49702   1.98542   1.77091   0.92945  -0.24107  -1.32625  -1.94188  -1.87003  -1.13613   0.00000   1.13613   1.87003   1.94188   1.32625   0.24107  -0.92945  -1.77091  -1.98542  -1.49702  -0.47863   0.70921   1.64597   2.00000   0.00000   0.00000     ... ... ..... ....... ......... ........... .............
E11      2.00000   1.57437   0.47863  -0.82083  -1.77091  -1.96724  -1.32625  -0.12076   1.13613   1.90944   1.87003   1.03468  -0.24107  -1.41421  -1.98542  -1.71156  -0.70921   0.59501   1.64597   1.99635   1.49702   0.36051  -0.92945  -1.82380  -1.94188  -1.23344   0.00000   1.23344   1.94188   1.82380   0.92945  -0.36051  -1.49702  -1.99635  -1.64597  -0.59501   0.70921   1.71156   1.98542   1.41421   0.24107  -1.03468  -1.87003  -1.90944  -1.13613   0.12076   1.32625   1.96724   1.77091   0.82083  -0.47863  -1.57437  -2.00000   0.00000   0.00000     ... ... ..... ....... ......... ........... .............
E12      2.00000   1.49702   0.24107  -1.13613  -1.94188  -1.77091  -0.70921   0.70921   1.77091   1.94188   1.13613  -0.24107  -1.49702  -2.00000  -1.49702  -0.24107   1.13613   1.94188   1.77091   0.70921  -0.70921  -1.77091  -1.94188  -1.13613   0.24107   1.49702   2.00000   1.49702   0.24107  -1.13613  -1.94188  -1.77091  -0.70921   0.70921   1.77091   1.94188   1.13613  -0.24107  -1.49702  -2.00000  -1.49702  -0.24107   1.13613   1.94188   1.77091   0.70921  -0.70921  -1.77091  -1.94188  -1.13613   0.24107   1.49702   2.00000   0.00000   0.00000     ... ... ..... ....... ......... ........... .............
E13      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  -1.41421   0.00000   1.41421   2.00000   1.41421   0.00000  -1.41421  -2.00000   0.00000   0.00000     ... ... ..... ....... ......... ........... .............
E14      2.00000   1.32625  -0.24107  -1.64597  -1.94188  -0.92945   0.70921   1.87003   1.77091   0.47863  -1.13613  -1.98542  -1.49702   0.00000   1.49702   1.98542   1.13613  -0.47863  -1.77091  -1.87003  -0.70921   0.92945   1.94188   1.64597   0.24107  -1.32625  -2.00000  -1.32625   0.24107   1.64597   1.94188   0.92945  -0.70921  -1.87003  -1.77091  -0.47863   1.13613   1.98542   1.49702   0.00000  -1.49702  -1.98542  -1.13613   0.47863   1.77091   1.87003   0.70921  -0.92945  -1.94188  -1.64597  -0.24107   1.32625   2.00000   0.00000   0.00000     ... ... ..... ....... ......... ........... .............
E15      2.00000   1.23344  -0.47863  -1.82380  -1.77091  -0.36051   1.32625   1.99635   1.13613  -0.59501  -1.87003  -1.71156  -0.24107   1.41421   1.98542   1.03468  -0.70921  -1.90944  -1.64597  -0.12076   1.49702   1.96724   0.92945  -0.82083  -1.94188  -1.57437   0.00000   1.57437   1.94188   0.82083  -0.92945  -1.96724  -1.49702   0.12076   1.64597   1.90944   0.70921  -1.03468  -1.98542  -1.41421   0.24107   1.71156   1.87003   0.59501  -1.13613  -1.99635  -1.32625   0.36051   1.77091   1.82380   0.47863  -1.23344  -2.00000   0.00000   0.00000     ... ... ..... ....... ......... ........... .............
E16      2.00000   1.13613  -0.70921  -1.94188  -1.49702   0.24107   1.77091   1.77091   0.24107  -1.49702  -1.94188  -0.70921   1.13613   2.00000   1.13613  -0.70921  -1.94188  -1.49702   0.24107   1.77091   1.77091   0.24107  -1.49702  -1.94188  -0.70921   1.13613   2.00000   1.13613  -0.70921  -1.94188  -1.49702   0.24107   1.77091   1.77091   0.24107  -1.49702  -1.94188  -0.70921   1.13613   2.00000   1.13613  -0.70921  -1.94188  -1.49702   0.24107   1.77091   1.77091   0.24107  -1.49702  -1.94188  -0.70921   1.13613   2.00000   0.00000   0.00000     ... ... ..... ....... ......... ........... .............
E17      2.00000   1.03468  -0.92945  -1.99635  -1.13613   0.82083   1.98542   1.23344  -0.70921  -1.96724  -1.32625   0.59501   1.94188   1.41421  -0.47863  -1.90944  -1.49702   0.36051   1.87003   1.57437  -0.24107  -1.82380  -1.64597   0.12076   1.77091   1.71156   0.00000  -1.71156  -1.77091  -0.12076   1.64597   1.82380   0.24107  -1.57437  -1.87003  -0.36051   1.49702   1.90944   0.47863  -1.41421  -1.94188  -0.59501   1.32625   1.96724   0.70921  -1.23344  -1.98542  -0.82083   1.13613   1.99635   0.92945  -1.03468  -2.00000   0.00000   0.00000     ... ... ..... ....... ......... ........... .............
E18      2.00000   0.92945  -1.13613  -1.98542  -0.70921   1.32625   1.94188   0.47863  -1.49702  -1.87003  -0.24107   1.64597   1.77091   0.00000  -1.77091  -1.64597   0.24107   1.87003   1.49702  -0.47863  -1.94188  -1.32625   0.70921   1.98542   1.13613  -0.92945  -2.00000  -0.92945   1.13613   1.98542   0.70921  -1.32625  -1.94188  -0.47863   1.49702   1.87003   0.24107  -1.64597  -1.77091   0.00000   1.77091   1.64597  -0.24107  -1.87003  -1.49702   0.47863   1.94188   1.32625  -0.70921  -1.98542  -1.13613   0.92945   2.00000   0.00000   0.00000     ... ... ..... ....... ......... ........... .............
E19      2.00000   0.82083  -1.32625  -1.90944  -0.24107   1.71156   1.64597  -0.36051  -1.94188  -1.23344   0.92945   1.99635   0.70921  -1.41421  -1.87003  -0.12076   1.77091   1.57437  -0.47863  -1.96724  -1.13613   1.03468   1.98542   0.59501  -1.49702  -1.82380   0.00000   1.82380   1.49702  -0.59501  -1.98542  -1.03468   1.13613   1.96724   0.47863  -1.57437  -1.77091   0.12076   1.87003   1.41421  -0.70921  -1.99635  -0.92945   1.23344   1.94188   0.36051  -1.64597  -1.71156   0.24107   1.90944   1.32625  -0.82083  -2.00000   0.00000   0.00000     ... ... ..... ....... ......... ........... .............
E20      2.00000   0.70921  -1.49702  -1.77091   0.24107   1.94188   1.13613  -1.13613  -1.94188  -0.24107   1.77091   1.49702  -0.70921  -2.00000  -0.70921   1.49702   1.77091  -0.24107  -1.94188  -1.13613   1.13613   1.94188   0.24107  -1.77091  -1.49702   0.70921   2.00000   0.70921  -1.49702  -1.77091   0.24107   1.94188   1.13613  -1.13613  -1.94188  -0.24107   1.77091   1.49702  -0.70921  -2.00000  -0.70921   1.49702   1.77091  -0.24107  -1.94188  -1.13613   1.13613   1.94188   0.24107  -1.77091  -1.49702   0.70921   2.00000   0.00000   0.00000     ... ... ..... ....... ......... ........... .............
E21      2.00000   0.59501  -1.64597  -1.57437   0.70921   1.99635   0.47863  -1.71156  -1.49702   0.82083   1.98542   0.36051  -1.77091  -1.41421   0.92945   1.96724   0.24107  -1.82380  -1.32625   1.03468   1.94188   0.12076  -1.87003  -1.23344   1.13613   1.90944   0.00000  -1.90944  -1.13613   1.23344   1.87003  -0.12076  -1.94188  -1.03468   1.32625   1.82380  -0.24107  -1.96724  -0.92945   1.41421   1.77091  -0.36051  -1.98542  -0.82083   1.49702   1.71156  -0.47863  -1.99635  -0.70921   1.57437   1.64597  -0.59501  -2.00000   0.00000   0.00000     ... ... ..... ....... ......... ........... .............
E22      2.00000   0.47863  -1.77091  -1.32625   1.13613   1.87003  -0.24107  -1.98542  -0.70921   1.64597   1.49702  -0.92945  -1.94188   0.00000   1.94188   0.92945  -1.49702  -1.64597   0.70921   1.98542   0.24107  -1.87003  -1.13613   1.32625   1.77091  -0.47863  -2.00000  -0.47863   1.77091   1.32625  -1.13613  -1.87003   0.24107   1.98542   0.70921  -1.64597  -1.49702   0.92945   1.94188   0.00000  -1.94188  -0.92945   1.49702   1.64597  -0.70921  -1.98542  -0.24107   1.87003   1.13613  -1.32625  -1.77091   0.47863   2.00000   0.00000   0.00000     ... ... ..... ....... ......... ........... .............
E23      2.00000   0.36051  -1.87003  -1.03468   1.49702   1.57437  -0.92945  -1.90944   0.24107   1.99635   0.47863  -1.82380  -1.13613   1.41421   1.64597  -0.82083  -1.94188   0.12076   1.98542   0.59501  -1.77091  -1.23344   1.32625   1.71156  -0.70921  -1.96724   0.00000   1.96724   0.70921  -1.71156  -1.32625   1.23344   1.77091  -0.59501  -1.98542  -0.12076   1.94188   0.82083  -1.64597  -1.41421   1.13613   1.82380  -0.47863  -1.99635  -0.24107   1.90944   0.92945  -1.57437  -1.49702   1.03468   1.87003  -0.36051  -2.00000   0.00000   0.00000     ... ... ..... ....... ......... ........... .............
E24      2.00000   0.24107  -1.94188  -0.70921   1.77091   1.13613  -1.49702  -1.49702   1.13613   1.77091  -0.70921  -1.94188   0.24107   2.00000   0.24107  -1.94188  -0.70921   1.77091   1.13613  -1.49702  -1.49702   1.13613   1.77091  -0.70921  -1.94188   0.24107   2.00000   0.24107  -1.94188  -0.70921   1.77091   1.13613  -1.49702  -1.49702   1.13613   1.77091  -0.70921  -1.94188   0.24107   2.00000   0.24107  -1.94188  -0.70921   1.77091   1.13613  -1.49702  -1.49702   1.13613   1.77091  -0.70921  -1.94188   0.24107   2.00000   0.00000   0.00000     ... ... ..... ....... ......... ........... .............
E25      2.00000   0.12076  -1.98542  -0.36051   1.94188   0.59501  -1.87003  -0.82083   1.77091   1.03468  -1.64597  -1.23344   1.49702   1.41421  -1.32625  -1.57437   1.13613   1.71156  -0.92945  -1.82380   0.70921   1.90944  -0.47863  -1.96724   0.24107   1.99635   0.00000  -1.99635  -0.24107   1.96724   0.47863  -1.90944  -0.70921   1.82380   0.92945  -1.71156  -1.13613   1.57437   1.32625  -1.41421  -1.49702   1.23344   1.64597  -1.03468  -1.77091   0.82083   1.87003  -0.59501  -1.94188   0.36051   1.98542  -0.12076  -2.00000   0.00000   0.00000     ... ... ..... ....... ......... ........... .............
E26      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  -2.00000   0.00000   2.00000   0.00000  -2.00000   0.00000   2.00000   0.00000   0.00000     ... ... ..... ....... ......... ........... .............
E27      2.00000  -0.12076  -1.98542   0.36051   1.94188  -0.59501  -1.87003   0.82083   1.77091  -1.03468  -1.64597   1.23344   1.49702  -1.41421  -1.32625   1.57437   1.13613  -1.71156  -0.92945   1.82380   0.70921  -1.90944  -0.47863   1.96724   0.24107  -1.99635   0.00000   1.99635  -0.24107  -1.96724   0.47863   1.90944  -0.70921  -1.82380   0.92945   1.71156  -1.13613  -1.57437   1.32625   1.41421  -1.49702  -1.23344   1.64597   1.03468  -1.77091  -0.82083   1.87003   0.59501  -1.94188  -0.36051   1.98542   0.12076  -2.00000   0.00000   0.00000     ... ... ..... ....... ......... ........... .............
E28      2.00000  -0.24107  -1.94188   0.70921   1.77091  -1.13613  -1.49702   1.49702   1.13613  -1.77091  -0.70921   1.94188   0.24107  -2.00000   0.24107   1.94188  -0.70921  -1.77091   1.13613   1.49702  -1.49702  -1.13613   1.77091   0.70921  -1.94188  -0.24107   2.00000  -0.24107  -1.94188   0.70921   1.77091  -1.13613  -1.49702   1.49702   1.13613  -1.77091  -0.70921   1.94188   0.24107  -2.00000   0.24107   1.94188  -0.70921  -1.77091   1.13613   1.49702  -1.49702  -1.13613   1.77091   0.70921  -1.94188  -0.24107   2.00000   0.00000   0.00000     ... ... ..... ....... ......... ........... .............
E29      2.00000  -0.36051  -1.87003   1.03468   1.49702  -1.57437  -0.92945   1.90944   0.24107  -1.99635   0.47863   1.82380  -1.13613  -1.41421   1.64597   0.82083  -1.94188  -0.12076   1.98542  -0.59501  -1.77091   1.23344   1.32625  -1.71156  -0.70921   1.96724   0.00000  -1.96724   0.70921   1.71156  -1.32625  -1.23344   1.77091   0.59501  -1.98542   0.12076   1.94188  -0.82083  -1.64597   1.41421   1.13613  -1.82380  -0.47863   1.99635  -0.24107  -1.90944   0.92945   1.57437  -1.49702  -1.03468   1.87003   0.36051  -2.00000   0.00000   0.00000     ... ... ..... ....... ......... ........... .............
E30      2.00000  -0.47863  -1.77091   1.32625   1.13613  -1.87003  -0.24107   1.98542  -0.70921  -1.64597   1.49702   0.92945  -1.94188   0.00000   1.94188  -0.92945  -1.49702   1.64597   0.70921  -1.98542   0.24107   1.87003  -1.13613  -1.32625   1.77091   0.47863  -2.00000   0.47863   1.77091  -1.32625  -1.13613   1.87003   0.24107  -1.98542   0.70921   1.64597  -1.49702  -0.92945   1.94188   0.00000  -1.94188   0.92945   1.49702  -1.64597  -0.70921   1.98542  -0.24107  -1.87003   1.13613   1.32625  -1.77091  -0.47863   2.00000   0.00000   0.00000     ... ... ..... ....... ......... ........... .............
E31      2.00000  -0.59501  -1.64597   1.57437   0.70921  -1.99635   0.47863   1.71156  -1.49702  -0.82083   1.98542  -0.36051  -1.77091   1.41421   0.92945  -1.96724   0.24107   1.82380  -1.32625  -1.03468   1.94188  -0.12076  -1.87003   1.23344   1.13613  -1.90944   0.00000   1.90944  -1.13613  -1.23344   1.87003   0.12076  -1.94188   1.03468   1.32625  -1.82380  -0.24107   1.96724  -0.92945  -1.41421   1.77091   0.36051  -1.98542   0.82083   1.49702  -1.71156  -0.47863   1.99635  -0.70921  -1.57437   1.64597   0.59501  -2.00000   0.00000   0.00000     ... ... ..... ....... ......... ........... .............
E32      2.00000  -0.70921  -1.49702   1.77091   0.24107  -1.94188   1.13613   1.13613  -1.94188   0.24107   1.77091  -1.49702  -0.70921   2.00000  -0.70921  -1.49702   1.77091   0.24107  -1.94188   1.13613   1.13613  -1.94188   0.24107   1.77091  -1.49702  -0.70921   2.00000  -0.70921  -1.49702   1.77091   0.24107  -1.94188   1.13613   1.13613  -1.94188   0.24107   1.77091  -1.49702  -0.70921   2.00000  -0.70921  -1.49702   1.77091   0.24107  -1.94188   1.13613   1.13613  -1.94188   0.24107   1.77091  -1.49702  -0.70921   2.00000   0.00000   0.00000     ... ... ..... ....... ......... ........... .............
E33      2.00000  -0.82083  -1.32625   1.90944  -0.24107  -1.71156   1.64597   0.36051  -1.94188   1.23344   0.92945  -1.99635   0.70921   1.41421  -1.87003   0.12076   1.77091  -1.57437  -0.47863   1.96724  -1.13613  -1.03468   1.98542  -0.59501  -1.49702   1.82380   0.00000  -1.82380   1.49702   0.59501  -1.98542   1.03468   1.13613  -1.96724   0.47863   1.57437  -1.77091  -0.12076   1.87003  -1.41421  -0.70921   1.99635  -0.92945  -1.23344   1.94188  -0.36051  -1.64597   1.71156   0.24107  -1.90944   1.32625   0.82083  -2.00000   0.00000   0.00000     ... ... ..... ....... ......... ........... .............
E34      2.00000  -0.92945  -1.13613   1.98542  -0.70921  -1.32625   1.94188  -0.47863  -1.49702   1.87003  -0.24107  -1.64597   1.77091   0.00000  -1.77091   1.64597   0.24107  -1.87003   1.49702   0.47863  -1.94188   1.32625   0.70921  -1.98542   1.13613   0.92945  -2.00000   0.92945   1.13613  -1.98542   0.70921   1.32625  -1.94188   0.47863   1.49702  -1.87003   0.24107   1.64597  -1.77091   0.00000   1.77091  -1.64597  -0.24107   1.87003  -1.49702  -0.47863   1.94188  -1.32625  -0.70921   1.98542  -1.13613  -0.92945   2.00000   0.00000   0.00000     ... ... ..... ....... ......... ........... .............
E35      2.00000  -1.03468  -0.92945   1.99635  -1.13613  -0.82083   1.98542  -1.23344  -0.70921   1.96724  -1.32625  -0.59501   1.94188  -1.41421  -0.47863   1.90944  -1.49702  -0.36051   1.87003  -1.57437  -0.24107   1.82380  -1.64597  -0.12076   1.77091  -1.71156   0.00000   1.71156  -1.77091   0.12076   1.64597  -1.82380   0.24107   1.57437  -1.87003   0.36051   1.49702  -1.90944   0.47863   1.41421  -1.94188   0.59501   1.32625  -1.96724   0.70921   1.23344  -1.98542   0.82083   1.13613  -1.99635   0.92945   1.03468  -2.00000   0.00000   0.00000     ... ... ..... ....... ......... ........... .............
E36      2.00000  -1.13613  -0.70921   1.94188  -1.49702  -0.24107   1.77091  -1.77091   0.24107   1.49702  -1.94188   0.70921   1.13613  -2.00000   1.13613   0.70921  -1.94188   1.49702   0.24107  -1.77091   1.77091  -0.24107  -1.49702   1.94188  -0.70921  -1.13613   2.00000  -1.13613  -0.70921   1.94188  -1.49702  -0.24107   1.77091  -1.77091   0.24107   1.49702  -1.94188   0.70921   1.13613  -2.00000   1.13613   0.70921  -1.94188   1.49702   0.24107  -1.77091   1.77091  -0.24107  -1.49702   1.94188  -0.70921  -1.13613   2.00000   0.00000   0.00000     ... ... ..... ....... ......... ........... .............
E37      2.00000  -1.23344  -0.47863   1.82380  -1.77091   0.36051   1.32625  -1.99635   1.13613   0.59501  -1.87003   1.71156  -0.24107  -1.41421   1.98542  -1.03468  -0.70921   1.90944  -1.64597   0.12076   1.49702  -1.96724   0.92945   0.82083  -1.94188   1.57437   0.00000  -1.57437   1.94188  -0.82083  -0.92945   1.96724  -1.49702  -0.12076   1.64597  -1.90944   0.70921   1.03468  -1.98542   1.41421   0.24107  -1.71156   1.87003  -0.59501  -1.13613   1.99635  -1.32625  -0.36051   1.77091  -1.82380   0.47863   1.23344  -2.00000   0.00000   0.00000     ... ... ..... ....... ......... ........... .............
E38      2.00000  -1.32625  -0.24107   1.64597  -1.94188   0.92945   0.70921  -1.87003   1.77091  -0.47863  -1.13613   1.98542  -1.49702   0.00000   1.49702  -1.98542   1.13613   0.47863  -1.77091   1.87003  -0.70921  -0.92945   1.94188  -1.64597   0.24107   1.32625  -2.00000   1.32625   0.24107  -1.64597   1.94188  -0.92945  -0.70921   1.87003  -1.77091   0.47863   1.13613  -1.98542   1.49702   0.00000  -1.49702   1.98542  -1.13613  -0.47863   1.77091  -1.87003   0.70921   0.92945  -1.94188   1.64597  -0.24107  -1.32625   2.00000   0.00000   0.00000     ... ... ..... ....... ......... ........... .............
E39      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   1.41421   0.00000  -1.41421   2.00000  -1.41421   0.00000   1.41421  -2.00000   0.00000   0.00000     ... ... ..... ....... ......... ........... .............
E40      2.00000  -1.49702   0.24107   1.13613  -1.94188   1.77091  -0.70921  -0.70921   1.77091  -1.94188   1.13613   0.24107  -1.49702   2.00000  -1.49702   0.24107   1.13613  -1.94188   1.77091  -0.70921  -0.70921   1.77091  -1.94188   1.13613   0.24107  -1.49702   2.00000  -1.49702   0.24107   1.13613  -1.94188   1.77091  -0.70921  -0.70921   1.77091  -1.94188   1.13613   0.24107  -1.49702   2.00000  -1.49702   0.24107   1.13613  -1.94188   1.77091  -0.70921  -0.70921   1.77091  -1.94188   1.13613   0.24107  -1.49702   2.00000   0.00000   0.00000     ... ... ..... ....... ......... ........... .............
E41      2.00000  -1.57437   0.47863   0.82083  -1.77091   1.96724  -1.32625   0.12076   1.13613  -1.90944   1.87003  -1.03468  -0.24107   1.41421  -1.98542   1.71156  -0.70921  -0.59501   1.64597  -1.99635   1.49702  -0.36051  -0.92945   1.82380  -1.94188   1.23344   0.00000  -1.23344   1.94188  -1.82380   0.92945   0.36051  -1.49702   1.99635  -1.64597   0.59501   0.70921  -1.71156   1.98542  -1.41421   0.24107   1.03468  -1.87003   1.90944  -1.13613  -0.12076   1.32625  -1.96724   1.77091  -0.82083  -0.47863   1.57437  -2.00000   0.00000   0.00000     ... ... ..... ....... ......... ........... .............
E42      2.00000  -1.64597   0.70921   0.47863  -1.49702   1.98542  -1.77091   0.92945   0.24107  -1.32625   1.94188  -1.87003   1.13613   0.00000  -1.13613   1.87003  -1.94188   1.32625  -0.24107  -0.92945   1.77091  -1.98542   1.49702  -0.47863  -0.70921   1.64597  -2.00000   1.64597  -0.70921  -0.47863   1.49702  -1.98542   1.77091  -0.92945  -0.24107   1.32625  -1.94188   1.87003  -1.13613   0.00000   1.13613  -1.87003   1.94188  -1.32625   0.24107   0.92945  -1.77091   1.98542  -1.49702   0.47863   0.70921  -1.64597   2.00000   0.00000   0.00000     ... ... ..... ....... ......... ........... .............
E43      2.00000  -1.71156   0.92945   0.12076  -1.13613   1.82380  -1.98542   1.57437  -0.70921  -0.36051   1.32625  -1.90944   1.94188  -1.41421   0.47863   0.59501  -1.49702   1.96724  -1.87003   1.23344  -0.24107  -0.82083   1.64597  -1.99635   1.77091  -1.03468   0.00000   1.03468  -1.77091   1.99635  -1.64597   0.82083   0.24107  -1.23344   1.87003  -1.96724   1.49702  -0.59501  -0.47863   1.41421  -1.94188   1.90944  -1.32625   0.36051   0.70921  -1.57437   1.98542  -1.82380   1.13613  -0.12076  -0.92945   1.71156  -2.00000   0.00000   0.00000     ... ... ..... ....... ......... ........... .............
E44      2.00000  -1.77091   1.13613  -0.24107  -0.70921   1.49702  -1.94188   1.94188  -1.49702   0.70921   0.24107  -1.13613   1.77091  -2.00000   1.77091  -1.13613   0.24107   0.70921  -1.49702   1.94188  -1.94188   1.49702  -0.70921  -0.24107   1.13613  -1.77091   2.00000  -1.77091   1.13613  -0.24107  -0.70921   1.49702  -1.94188   1.94188  -1.49702   0.70921   0.24107  -1.13613   1.77091  -2.00000   1.77091  -1.13613   0.24107   0.70921  -1.49702   1.94188  -1.94188   1.49702  -0.70921  -0.24107   1.13613  -1.77091   2.00000   0.00000   0.00000     ... ... ..... ....... ......... ........... .............
E45      2.00000  -1.82380   1.32625  -0.59501  -0.24107   1.03468  -1.64597   1.96724  -1.94188   1.57437  -0.92945   0.12076   0.70921  -1.41421   1.87003  -1.99635   1.77091  -1.23344   0.47863   0.36051  -1.13613   1.71156  -1.98542   1.90944  -1.49702   0.82083   0.00000  -0.82083   1.49702  -1.90944   1.98542  -1.71156   1.13613  -0.36051  -0.47863   1.23344  -1.77091   1.99635  -1.87003   1.41421  -0.70921  -0.12076   0.92945  -1.57437   1.94188  -1.96724   1.64597  -1.03468   0.24107   0.59501  -1.32625   1.82380  -2.00000   0.00000   0.00000     ... ... ..... ....... ......... ........... .............
E46      2.00000  -1.87003   1.49702  -0.92945   0.24107   0.47863  -1.13613   1.64597  -1.94188   1.98542  -1.77091   1.32625  -0.70921   0.00000   0.70921  -1.32625   1.77091  -1.98542   1.94188  -1.64597   1.13613  -0.47863  -0.24107   0.92945  -1.49702   1.87003  -2.00000   1.87003  -1.49702   0.92945  -0.24107  -0.47863   1.13613  -1.64597   1.94188  -1.98542   1.77091  -1.32625   0.70921   0.00000  -0.70921   1.32625  -1.77091   1.98542  -1.94188   1.64597  -1.13613   0.47863   0.24107  -0.92945   1.49702  -1.87003   2.00000   0.00000   0.00000     ... ... ..... ....... ......... ........... .............
E47      2.00000  -1.90944   1.64597  -1.23344   0.70921  -0.12076  -0.47863   1.03468  -1.49702   1.82380  -1.98542   1.96724  -1.77091   1.41421  -0.92945   0.36051   0.24107  -0.82083   1.32625  -1.71156   1.94188  -1.99635   1.87003  -1.57437   1.13613  -0.59501   0.00000   0.59501  -1.13613   1.57437  -1.87003   1.99635  -1.94188   1.71156  -1.32625   0.82083  -0.24107  -0.36051   0.92945  -1.41421   1.77091  -1.96724   1.98542  -1.82380   1.49702  -1.03468   0.47863   0.12076  -0.70921   1.23344  -1.64597   1.90944  -2.00000   0.00000   0.00000     ... ... ..... ....... ......... ........... ..TT.........
E48      2.00000  -1.94188   1.77091  -1.49702   1.13613  -0.70921   0.24107   0.24107  -0.70921   1.13613  -1.49702   1.77091  -1.94188   2.00000  -1.94188   1.77091  -1.49702   1.13613  -0.70921   0.24107   0.24107  -0.70921   1.13613  -1.49702   1.77091  -1.94188   2.00000  -1.94188   1.77091  -1.49702   1.13613  -0.70921   0.24107   0.24107  -0.70921   1.13613  -1.49702   1.77091  -1.94188   2.00000  -1.94188   1.77091  -1.49702   1.13613  -0.70921   0.24107   0.24107  -0.70921   1.13613  -1.49702   1.77091  -1.94188   2.00000   0.00000   0.00000     ... ... ..... ....... ......... ..TT....... .............
E49      2.00000  -1.96724   1.87003  -1.71156   1.49702  -1.23344   0.92945  -0.59501   0.24107   0.12076  -0.47863   0.82083  -1.13613   1.41421  -1.64597   1.82380  -1.94188   1.99635  -1.98542   1.90944  -1.77091   1.57437  -1.32625   1.03468  -0.70921   0.36051   0.00000  -0.36051   0.70921  -1.03468   1.32625  -1.57437   1.77091  -1.90944   1.98542  -1.99635   1.94188  -1.82380   1.64597  -1.41421   1.13613  -0.82083   0.47863  -0.12076  -0.24107   0.59501  -0.92945   1.23344  -1.49702   1.71156  -1.87003   1.96724  -2.00000   0.00000   0.00000     ... ... ..... ....... ..TT..... ........... ......TT.....
E50      2.00000  -1.98542   1.94188  -1.87003   1.77091  -1.64597   1.49702  -1.32625   1.13613  -0.92945   0.70921  -0.47863   0.24107   0.00000  -0.24107   0.47863  -0.70921   0.92945  -1.13613   1.32625  -1.49702   1.64597  -1.77091   1.87003  -1.94188   1.98542  -2.00000   1.98542  -1.94188   1.87003  -1.77091   1.64597  -1.49702   1.32625  -1.13613   0.92945  -0.70921   0.47863  -0.24107   0.00000   0.24107  -0.47863   0.70921  -0.92945   1.13613  -1.32625   1.49702  -1.64597   1.77091  -1.87003   1.94188  -1.98542   2.00000   0.00000   0.00000     ... ... ..... ..TT... ......... ......TT... .............
E51      2.00000  -1.99635   1.98542  -1.96724   1.94188  -1.90944   1.87003  -1.82380   1.77091  -1.71156   1.64597  -1.57437   1.49702  -1.41421   1.32625  -1.23344   1.13613  -1.03468   0.92945  -0.82083   0.70921  -0.59501   0.47863  -0.36051   0.24107  -0.12076   0.00000   0.12076  -0.24107   0.36051  -0.47863   0.59501  -0.70921   0.82083  -0.92945   1.03468  -1.13613   1.23344  -1.32625   1.41421  -1.49702   1.57437  -1.64597   1.71156  -1.77091   1.82380  -1.87003   1.90944  -1.94188   1.96724  -1.98542   1.99635  -2.00000   0.00000   0.00000     TT. ... ..TT. ....... ......TT. ........... ..........TT.

 Irrational character values:  1.996351108447 = 2*cos(2*π/104) = 2*cos(π/52)
                               1.985417748196 = 2*cos(4*π/104) = 2*cos(π/26)
                               1.967239813894 = 2*cos(6*π/104) = 2*cos(3*π/52)
                               1.941883634852 = 2*cos(8*π/104) = 2*cos(π/13)
                               1.909441733017 = 2*cos(10*π/104) = 2*cos(5*π/52)
                               1.870032485371 = 2*cos(12*π/104) = 2*cos(3*π/26)
                               1.823799691984 = 2*cos(14*π/104) = 2*cos(7*π/52)
                               1.770912051306 = 2*cos(16*π/104) = 2*cos(2*π/13)
                               1.711562544603 = 2*cos(18*π/104) = 2*cos(9*π/52)
                               1.645967731787 = 2*cos(20*π/104) = 2*cos(5*π/26)
                               1.574366961218 = 2*cos(22*π/104) = 2*cos(11*π/52)
                               1.497021496342 = 2*cos(24*π/104) = 2*cos(3*π/13)
                               1.414213562373 = 2*cos(26*π/104) = 2*cos(π/4) = √2
                               1.326245316482 = 2*cos(28*π/104) = 2*cos(7*π/26)
                               1.233437745257 = 2*cos(30*π/104) = 2*cos(15*π/52)
                               1.136129493462 = 2*cos(32*π/104) = 2*cos(4*π/13)
                               1.034675628355 = 2*cos(34*π/104) = 2*cos(17*π/52)
                               0.929446344088 = 2*cos(36*π/104) = 2*cos(9*π/26)
                               0.820825610906 = 2*cos(38*π/104) = 2*cos(19*π/52)
                               0.709209774085 = 2*cos(40*π/104) = 2*cos(5*π/13)
                               0.595006107710 = 2*cos(42*π/104) = 2*cos(21*π/52)
                               0.478631328575 = 2*cos(44*π/104) = 2*cos(11*π/26)
                               0.360510075628 = 2*cos(46*π/104) = 2*cos(23*π/52)
                               0.241073360511 = 2*cos(48*π/104) = 2*cos(6*π/13)
                               0.120756994845 = 2*cos(50*π/104) = 2*cos(25*π/52)



 Symmetry of Rotations and Cartesian products

A1   d+g+i+k+m    z2, z4, z6 
A2   R            Rz 
B2   p+f+h+j+l    z, z3, z5 
E1   p+f+h+j+l    {x, y}, {xz2, yz2}, {xz4, yz4} 
E2   d+g+i+k+m    {x2y2, xy}, {z2(x2y2), xyz2}, {z4(x2y2), xyz4} 
E3   f+h+j+l      {x(x2−3y2), y(3x2y2)}, {xz2(x2−3y2), yz2(3x2y2)} 
E4   g+i+k+m      {(x2y2)2−4x2y2, xy(x2y2)}, {z2((x2y2)2−4x2y2), xyz2(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)} 
E6   i+k+m        {x2(x2−3y2)2y2(3x2y2)2, xy(x2−3y2)(3x2y2)} 
E7   j+l 
E8   k+m 
E9   l 
E10  m 
E43  m 
E44  l 
E45  k+m 
E46  j+l 
E47  i+k+m        {xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), yz((5+2√5)x2y2)((5−2√5)x2y2)} 
E48  h+j+l        {z((x2y2)2−4x2y2), xyz(x2y2)} 
E49  g+i+k+m      {xz(x2−3y2), yz(3x2y2)}, {xz3(x2−3y2), yz3(3x2y2)} 
E50  f+h+j+l      {z(x2y2), xyz}, {z3(x2y2), xyz3} 
E51  R+d+g+i+k+m  {Rx, Ry}, {xz, yz}, {xz3, yz3}, {xz5, yz5} 

 Notes:

    α  The order of the D52d point group is 208, and the order of the principal axis (S104) is 104. The group has 55 irreducible representations.

    β  The D52d point group is isomorphic to C104v and D104.

    γ  The D52d point group is generated by two symmetry elements, S104 and either a perpendicular C2 or a vertical σd.
       Also, the group may be generated from a C2 plus a σd (some pairs will yield smaller groups, though; choosing a minimum angle is safe).

    δ  The group contains one set of twofold symmetry axes (C2) perpendicular to the principal (z) axis. Both x and y axes are members of that set.

    ε  The single σd set of symmetry planes contains neither the xz nor the yz planes; but it contains the median plane (x+y)z.

    ζ  The lowest nonvanishing multipole moment in D52d is 4 (quadrupole moment).

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

    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 polygon inconstructible by the classical means of ruler and compass. Yet it becomes constructible
       if angle trisection is allowed, e.g., with neusis construction or origami. This is because the order of the principal axis is given
       by a product of any number of different Pierpont primes (...,5,7,13,17,19,37,73,97,109,163,...) times arbitrary powers of two and three.
       While some characters of this group can be expressed using real integers and square roots alone, others need complex numbers and third roots.

This Character Table for the D52d point group was created by Gernot Katzer.

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