Character table for the D32h point group

D32h     E        2 C32    2 C16    2 C32^3  2 C8     2 C32^5  2 C16^3  2 C32^7  2 C4     2 C32^9  2 C16^5  2 C32^11 2 C8^3   2 C32^13 2 C16^7  2 C32^15 C2       16 C2'   16 C2"   i        2 S32    2 S16    2 S32^3  2 S8     2 S32^5  2 S16^3  2 S32^7  2 S4     2 S32^9  2 S16^5  2 S32^11 2 S8^3   2 S32^13 2 S16^7  2 S32^15 sh       16 sv    16 sd      <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> 
A1g     1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  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
A2g     1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000 -1.00000 -1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  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 ... ..... ....... ......... ........... .............
B1g     1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000  1.00000 -1.00000     ... ... ..... ....... ......... ........... .............
B2g     1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000     ... ... ..... ....... ......... ........... .............
E1g     2.00000  1.96157  1.84776  1.66294  1.41421  1.11114  0.76537  0.39018  0.00000 -0.39018 -0.76537 -1.11114 -1.41421 -1.66294 -1.84776 -1.96157 -2.00000  0.00000  0.00000  2.00000 -1.96157 -1.84776 -1.66294 -1.41421 -1.11114 -0.76537 -0.39018  0.00000  0.39018  0.76537  1.11114  1.41421  1.66294  1.84776  1.96157 -2.00000  0.00000  0.00000     TT. ... ..TT. ....... ......TT. ........... ..........TT.
E2g     2.00000  1.84776  1.41421  0.76537  0.00000 -0.76537 -1.41421 -1.84776 -2.00000 -1.84776 -1.41421 -0.76537  0.00000  0.76537  1.41421  1.84776  2.00000  0.00000  0.00000  2.00000  1.84776  1.41421  0.76537  0.00000 -0.76537 -1.41421 -1.84776 -2.00000 -1.84776 -1.41421 -0.76537  0.00000  0.76537  1.41421  1.84776  2.00000  0.00000  0.00000     ... ... TT... ....... ....TT... ........... ........TT...
E3g     2.00000  1.66294  0.76537 -0.39018 -1.41421 -1.96157 -1.84776 -1.11114  0.00000  1.11114  1.84776  1.96157  1.41421  0.39018 -0.76537 -1.66294 -2.00000  0.00000  0.00000  2.00000 -1.66294 -0.76537  0.39018  1.41421  1.96157  1.84776  1.11114  0.00000 -1.11114 -1.84776 -1.96157 -1.41421 -0.39018  0.76537  1.66294 -2.00000  0.00000  0.00000     ... ... ..... ....... ..TT..... ........... ......TT.....
E4g     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  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     ... ... ..... ....... TT....... ........... ....TT.......
E5g     2.00000  1.11114 -0.76537 -1.96157 -1.41421  0.39018  1.84776  1.66294  0.00000 -1.66294 -1.84776 -0.39018  1.41421  1.96157  0.76537 -1.11114 -2.00000  0.00000  0.00000  2.00000 -1.11114  0.76537  1.96157  1.41421 -0.39018 -1.84776 -1.66294  0.00000  1.66294  1.84776  0.39018 -1.41421 -1.96157 -0.76537  1.11114 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... ..TT.........
E6g     2.00000  0.76537 -1.41421 -1.84776  0.00000  1.84776  1.41421 -0.76537 -2.00000 -0.76537  1.41421  1.84776  0.00000 -1.84776 -1.41421  0.76537  2.00000  0.00000  0.00000  2.00000  0.76537 -1.41421 -1.84776  0.00000  1.84776  1.41421 -0.76537 -2.00000 -0.76537  1.41421  1.84776  0.00000 -1.84776 -1.41421  0.76537  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... TT...........
E7g     2.00000  0.39018 -1.84776 -1.11114  1.41421  1.66294 -0.76537 -1.96157  0.00000  1.96157  0.76537 -1.66294 -1.41421  1.11114  1.84776 -0.39018 -2.00000  0.00000  0.00000  2.00000 -0.39018  1.84776  1.11114 -1.41421 -1.66294  0.76537  1.96157  0.00000 -1.96157 -0.76537  1.66294  1.41421 -1.11114 -1.84776  0.39018 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E8g     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  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     ... ... ..... ....... ......... ........... .............
E9g     2.00000 -0.39018 -1.84776  1.11114  1.41421 -1.66294 -0.76537  1.96157  0.00000 -1.96157  0.76537  1.66294 -1.41421 -1.11114  1.84776  0.39018 -2.00000  0.00000  0.00000  2.00000  0.39018  1.84776 -1.11114 -1.41421  1.66294  0.76537 -1.96157  0.00000  1.96157 -0.76537 -1.66294  1.41421  1.11114 -1.84776 -0.39018 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E10g    2.00000 -0.76537 -1.41421  1.84776  0.00000 -1.84776  1.41421  0.76537 -2.00000  0.76537  1.41421 -1.84776  0.00000  1.84776 -1.41421 -0.76537  2.00000  0.00000  0.00000  2.00000 -0.76537 -1.41421  1.84776  0.00000 -1.84776  1.41421  0.76537 -2.00000  0.76537  1.41421 -1.84776  0.00000  1.84776 -1.41421 -0.76537  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E11g    2.00000 -1.11114 -0.76537  1.96157 -1.41421 -0.39018  1.84776 -1.66294  0.00000  1.66294 -1.84776  0.39018  1.41421 -1.96157  0.76537  1.11114 -2.00000  0.00000  0.00000  2.00000  1.11114  0.76537 -1.96157  1.41421  0.39018 -1.84776  1.66294  0.00000 -1.66294  1.84776 -0.39018 -1.41421  1.96157 -0.76537 -1.11114 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E12g    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  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     ... ... ..... ....... ......... ........... .............
E13g    2.00000 -1.66294  0.76537  0.39018 -1.41421  1.96157 -1.84776  1.11114  0.00000 -1.11114  1.84776 -1.96157  1.41421 -0.39018 -0.76537  1.66294 -2.00000  0.00000  0.00000  2.00000  1.66294 -0.76537 -0.39018  1.41421 -1.96157  1.84776 -1.11114  0.00000  1.11114 -1.84776  1.96157 -1.41421  0.39018  0.76537 -1.66294 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E14g    2.00000 -1.84776  1.41421 -0.76537  0.00000  0.76537 -1.41421  1.84776 -2.00000  1.84776 -1.41421  0.76537  0.00000 -0.76537  1.41421 -1.84776  2.00000  0.00000  0.00000  2.00000 -1.84776  1.41421 -0.76537  0.00000  0.76537 -1.41421  1.84776 -2.00000  1.84776 -1.41421  0.76537  0.00000 -0.76537  1.41421 -1.84776  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E15g    2.00000 -1.96157  1.84776 -1.66294  1.41421 -1.11114  0.76537 -0.39018  0.00000  0.39018 -0.76537  1.11114 -1.41421  1.66294 -1.84776  1.96157 -2.00000  0.00000  0.00000  2.00000  1.96157 -1.84776  1.66294 -1.41421  1.11114 -0.76537  0.39018  0.00000 -0.39018  0.76537 -1.11114  1.41421 -1.66294  1.84776 -1.96157 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
A1u     1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000     ... ... ..... ....... ......... ........... .............
A2u     1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -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 .............
B1u     1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000  1.00000 -1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000 -1.00000  1.00000     ... ... ..... ....... ......... ........... .............
B2u     1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000     ... ... ..... ....... ......... ........... .............
E1u     2.00000  1.96157  1.84776  1.66294  1.41421  1.11114  0.76537  0.39018  0.00000 -0.39018 -0.76537 -1.11114 -1.41421 -1.66294 -1.84776 -1.96157 -2.00000  0.00000  0.00000 -2.00000  1.96157  1.84776  1.66294  1.41421  1.11114  0.76537  0.39018  0.00000 -0.39018 -0.76537 -1.11114 -1.41421 -1.66294 -1.84776 -1.96157  2.00000  0.00000  0.00000     ... TT. ..... ....TT. ......... ........TT. .............
E2u     2.00000  1.84776  1.41421  0.76537  0.00000 -0.76537 -1.41421 -1.84776 -2.00000 -1.84776 -1.41421 -0.76537  0.00000  0.76537  1.41421  1.84776  2.00000  0.00000  0.00000 -2.00000 -1.84776 -1.41421 -0.76537  0.00000  0.76537  1.41421  1.84776  2.00000  1.84776  1.41421  0.76537  0.00000 -0.76537 -1.41421 -1.84776 -2.00000  0.00000  0.00000     ... ... ..... ..TT... ......... ......TT... .............
E3u     2.00000  1.66294  0.76537 -0.39018 -1.41421 -1.96157 -1.84776 -1.11114  0.00000  1.11114  1.84776  1.96157  1.41421  0.39018 -0.76537 -1.66294 -2.00000  0.00000  0.00000 -2.00000  1.66294  0.76537 -0.39018 -1.41421 -1.96157 -1.84776 -1.11114  0.00000  1.11114  1.84776  1.96157  1.41421  0.39018 -0.76537 -1.66294  2.00000  0.00000  0.00000     ... ... ..... TT..... ......... ....TT..... .............
E4u     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 -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     ... ... ..... ....... ......... ..TT....... .............
E5u     2.00000  1.11114 -0.76537 -1.96157 -1.41421  0.39018  1.84776  1.66294  0.00000 -1.66294 -1.84776 -0.39018  1.41421  1.96157  0.76537 -1.11114 -2.00000  0.00000  0.00000 -2.00000  1.11114 -0.76537 -1.96157 -1.41421  0.39018  1.84776  1.66294  0.00000 -1.66294 -1.84776 -0.39018  1.41421  1.96157  0.76537 -1.11114  2.00000  0.00000  0.00000     ... ... ..... ....... ......... TT......... .............
E6u     2.00000  0.76537 -1.41421 -1.84776  0.00000  1.84776  1.41421 -0.76537 -2.00000 -0.76537  1.41421  1.84776  0.00000 -1.84776 -1.41421  0.76537  2.00000  0.00000  0.00000 -2.00000 -0.76537  1.41421  1.84776  0.00000 -1.84776 -1.41421  0.76537  2.00000  0.76537 -1.41421 -1.84776  0.00000  1.84776  1.41421 -0.76537 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E7u     2.00000  0.39018 -1.84776 -1.11114  1.41421  1.66294 -0.76537 -1.96157  0.00000  1.96157  0.76537 -1.66294 -1.41421  1.11114  1.84776 -0.39018 -2.00000  0.00000  0.00000 -2.00000  0.39018 -1.84776 -1.11114  1.41421  1.66294 -0.76537 -1.96157  0.00000  1.96157  0.76537 -1.66294 -1.41421  1.11114  1.84776 -0.39018  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E8u     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 -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     ... ... ..... ....... ......... ........... .............
E9u     2.00000 -0.39018 -1.84776  1.11114  1.41421 -1.66294 -0.76537  1.96157  0.00000 -1.96157  0.76537  1.66294 -1.41421 -1.11114  1.84776  0.39018 -2.00000  0.00000  0.00000 -2.00000 -0.39018 -1.84776  1.11114  1.41421 -1.66294 -0.76537  1.96157  0.00000 -1.96157  0.76537  1.66294 -1.41421 -1.11114  1.84776  0.39018  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E10u    2.00000 -0.76537 -1.41421  1.84776  0.00000 -1.84776  1.41421  0.76537 -2.00000  0.76537  1.41421 -1.84776  0.00000  1.84776 -1.41421 -0.76537  2.00000  0.00000  0.00000 -2.00000  0.76537  1.41421 -1.84776  0.00000  1.84776 -1.41421 -0.76537  2.00000 -0.76537 -1.41421  1.84776  0.00000 -1.84776  1.41421  0.76537 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E11u    2.00000 -1.11114 -0.76537  1.96157 -1.41421 -0.39018  1.84776 -1.66294  0.00000  1.66294 -1.84776  0.39018  1.41421 -1.96157  0.76537  1.11114 -2.00000  0.00000  0.00000 -2.00000 -1.11114 -0.76537  1.96157 -1.41421 -0.39018  1.84776 -1.66294  0.00000  1.66294 -1.84776  0.39018  1.41421 -1.96157  0.76537  1.11114  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E12u    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 -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     ... ... ..... ....... ......... ........... .............
E13u    2.00000 -1.66294  0.76537  0.39018 -1.41421  1.96157 -1.84776  1.11114  0.00000 -1.11114  1.84776 -1.96157  1.41421 -0.39018 -0.76537  1.66294 -2.00000  0.00000  0.00000 -2.00000 -1.66294  0.76537  0.39018 -1.41421  1.96157 -1.84776  1.11114  0.00000 -1.11114  1.84776 -1.96157  1.41421 -0.39018 -0.76537  1.66294  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E14u    2.00000 -1.84776  1.41421 -0.76537  0.00000  0.76537 -1.41421  1.84776 -2.00000  1.84776 -1.41421  0.76537  0.00000 -0.76537  1.41421 -1.84776  2.00000  0.00000  0.00000 -2.00000  1.84776 -1.41421  0.76537  0.00000 -0.76537  1.41421 -1.84776  2.00000 -1.84776  1.41421 -0.76537  0.00000  0.76537 -1.41421  1.84776 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E15u    2.00000 -1.96157  1.84776 -1.66294  1.41421 -1.11114  0.76537 -0.39018  0.00000  0.39018 -0.76537  1.11114 -1.41421  1.66294 -1.84776  1.96157 -2.00000  0.00000  0.00000 -2.00000 -1.96157  1.84776 -1.66294  1.41421 -1.11114  0.76537 -0.39018  0.00000  0.39018 -0.76537  1.11114 -1.41421  1.66294 -1.84776  1.96157  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............

 Irrational character values:  1.961570560806 = 2*cos(2*π/32) = 2*cos(π/16) = √2+√2+√2
                               1.847759065023 = 2*cos(4*π/32) = 2*cos(π/8) = √2+√2
                               1.662939224605 = 2*cos(6*π/32) = 2*cos(3*π/16) = √2+√2−√2
                               1.414213562373 = 2*cos(8*π/32) = 2*cos(π/4) = √2
                               1.111140466039 = 2*cos(10*π/32) = 2*cos(5*π/16) = √2−√2−√2
                               0.765366864730 = 2*cos(12*π/32) = 2*cos(3*π/8) = √2−√2
                               0.390180644032 = 2*cos(14*π/32) = 2*cos(7*π/16) = √2−√2+√2



 Symmetry of Rotations and Cartesian products

A1g  d+g+i+k+m    z2, z4, z6 
A2g  R            Rz 
E1g  R+d+g+i+k+m  {Rx, Ry}, {xz, yz}, {xz3, yz3}, {xz5, yz5} 
E2g  d+g+i+k+m    {x2y2, xy}, {z2(x2y2), xyz2}, {z4(x2y2), xyz4} 
E3g  g+i+k+m      {xz(x2−3y2), yz(3x2y2)}, {xz3(x2−3y2), yz3(3x2y2)} 
E4g  g+i+k+m      {(x2y2)2−4x2y2, xy(x2y2)}, {z2((x2y2)2−4x2y2), xyz2(x2y2)} 
E5g  i+k+m        {xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), yz((5+2√5)x2y2)((5−2√5)x2y2)} 
E6g  i+k+m        {x2(x2−3y2)2y2(3x2y2)2, xy(x2−3y2)(3x2y2)} 
E7g  k+m 
E8g  k+m 
E9g  m 
E10g m 
A2u  p+f+h+j+l    z, z3, z5 
E1u  p+f+h+j+l    {x, y}, {xz2, yz2}, {xz4, yz4} 
E2u  f+h+j+l      {z(x2y2), xyz}, {z3(x2y2), xyz3} 
E3u  f+h+j+l      {x(x2−3y2), y(3x2y2)}, {xz2(x2−3y2), yz2(3x2y2)} 
E4u  h+j+l        {z((x2y2)2−4x2y2), xyz(x2y2)} 
E5u  h+j+l        {x(x2−(5+2√5)y2)(x2−(5−2√5)y2), y((5+2√5)x2y2)((5−2√5)x2y2)} 
E6u  j+l 
E7u  j+l 
E8u  l 
E9u  l 

 Notes:

    α  The order of the D32h point group is 128, and the order of the principal axis (C32) is 32. The group has 38 irreducible representations.

    β  The D32h point group is generated by three symmetry elements that are canonically chosen C32, C2 and i.
       Other choices include σh instead of i, or any of C2, σv or σd instead of C2. Also, some ternary combinations of C2, C2, σv and σd act as generators.
       Lastly, the S32 can be chosen, together with i or σh and any one of C2, C2, σv or σd.

    γ  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 contains both the x and y axes.

    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 D32h 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.

    θ  The point group corresponds to a constructible polygon, as the order of the principal axis is a product of any number
       of different Fermat primes (3,5,17,257,65537) times an arbitrary power of two. Therefore, all characters have an
       algebraic degree which is a power of two and can be expressed as radicals involving only square roots and integer numbers.

This Character Table for the D32h point group was created by Gernot Katzer.

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