Character table for the D42h point group

D42h     E        2 C42    2 C21    2 C14    2 C21^2  2 C42^5  2 C7     2 C6     2 C21^4  2 C14^3  2 C21^5  2 C42^11 2 C7^2   2 C42^13 2 C3     2 C14^5  2 C21^8  2 C42^17 2 C7^3   2 C42^19 2 C21^10 C2       21 C2'   21 C2"   i        2 S42    2 S21    2 S14    2 S21^2  2 S42^5  2 S7     2 S6     2 S21^4  2 S14^3  2 S21^5  2 S42^11 2 S7^2   2 S42^13 2 S3     2 S14^5  2 S21^8  2 S42^17 2 S7^3   2 S42^19 2 S21^10 sh       21 sv    21 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  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  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 -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 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000 -1.00000  1.00000 -1.00000     ... ... ..... ....... ......... ........... .............
E1g     2.00000  1.97766  1.91115  1.80194  1.65248  1.46610  1.24698  1.00000  0.73068  0.44504  0.14946 -0.14946 -0.44504 -0.73068 -1.00000 -1.24698 -1.46610 -1.65248 -1.80194 -1.91115 -1.97766 -2.00000  0.00000  0.00000  2.00000 -1.97766 -1.91115 -1.80194 -1.65248 -1.46610 -1.24698 -1.00000 -0.73068 -0.44504 -0.14946  0.14946  0.44504  0.73068  1.00000  1.24698  1.46610  1.65248  1.80194  1.91115  1.97766 -2.00000  0.00000  0.00000     TT. ... ..TT. ....... ......TT. ........... ..........TT.
E2g     2.00000  1.91115  1.65248  1.24698  0.73068  0.14946 -0.44504 -1.00000 -1.46610 -1.80194 -1.97766 -1.97766 -1.80194 -1.46610 -1.00000 -0.44504  0.14946  0.73068  1.24698  1.65248  1.91115  2.00000  0.00000  0.00000  2.00000  1.91115  1.65248  1.24698  0.73068  0.14946 -0.44504 -1.00000 -1.46610 -1.80194 -1.97766 -1.97766 -1.80194 -1.46610 -1.00000 -0.44504  0.14946  0.73068  1.24698  1.65248  1.91115  2.00000  0.00000  0.00000     ... ... TT... ....... ....TT... ........... ........TT...
E3g     2.00000  1.80194  1.24698  0.44504 -0.44504 -1.24698 -1.80194 -2.00000 -1.80194 -1.24698 -0.44504  0.44504  1.24698  1.80194  2.00000  1.80194  1.24698  0.44504 -0.44504 -1.24698 -1.80194 -2.00000  0.00000  0.00000  2.00000 -1.80194 -1.24698 -0.44504  0.44504  1.24698  1.80194  2.00000  1.80194  1.24698  0.44504 -0.44504 -1.24698 -1.80194 -2.00000 -1.80194 -1.24698 -0.44504  0.44504  1.24698  1.80194 -2.00000  0.00000  0.00000     ... ... ..... ....... ..TT..... ........... ......TT.....
E4g     2.00000  1.65248  0.73068 -0.44504 -1.46610 -1.97766 -1.80194 -1.00000  0.14946  1.24698  1.91115  1.91115  1.24698  0.14946 -1.00000 -1.80194 -1.97766 -1.46610 -0.44504  0.73068  1.65248  2.00000  0.00000  0.00000  2.00000  1.65248  0.73068 -0.44504 -1.46610 -1.97766 -1.80194 -1.00000  0.14946  1.24698  1.91115  1.91115  1.24698  0.14946 -1.00000 -1.80194 -1.97766 -1.46610 -0.44504  0.73068  1.65248  2.00000  0.00000  0.00000     ... ... ..... ....... TT....... ........... ....TT.......
E5g     2.00000  1.46610  0.14946 -1.24698 -1.97766 -1.65248 -0.44504  1.00000  1.91115  1.80194  0.73068 -0.73068 -1.80194 -1.91115 -1.00000  0.44504  1.65248  1.97766  1.24698 -0.14946 -1.46610 -2.00000  0.00000  0.00000  2.00000 -1.46610 -0.14946  1.24698  1.97766  1.65248  0.44504 -1.00000 -1.91115 -1.80194 -0.73068  0.73068  1.80194  1.91115  1.00000 -0.44504 -1.65248 -1.97766 -1.24698  0.14946  1.46610 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... ..TT.........
E6g     2.00000  1.24698 -0.44504 -1.80194 -1.80194 -0.44504  1.24698  2.00000  1.24698 -0.44504 -1.80194 -1.80194 -0.44504  1.24698  2.00000  1.24698 -0.44504 -1.80194 -1.80194 -0.44504  1.24698  2.00000  0.00000  0.00000  2.00000  1.24698 -0.44504 -1.80194 -1.80194 -0.44504  1.24698  2.00000  1.24698 -0.44504 -1.80194 -1.80194 -0.44504  1.24698  2.00000  1.24698 -0.44504 -1.80194 -1.80194 -0.44504  1.24698  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... TT...........
E7g     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  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     ... ... ..... ....... ......... ........... .............
E8g     2.00000  0.73068 -1.46610 -1.80194  0.14946  1.91115  1.24698 -1.00000 -1.97766 -0.44504  1.65248  1.65248 -0.44504 -1.97766 -1.00000  1.24698  1.91115  0.14946 -1.80194 -1.46610  0.73068  2.00000  0.00000  0.00000  2.00000  0.73068 -1.46610 -1.80194  0.14946  1.91115  1.24698 -1.00000 -1.97766 -0.44504  1.65248  1.65248 -0.44504 -1.97766 -1.00000  1.24698  1.91115  0.14946 -1.80194 -1.46610  0.73068  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E9g     2.00000  0.44504 -1.80194 -1.24698  1.24698  1.80194 -0.44504 -2.00000 -0.44504  1.80194  1.24698 -1.24698 -1.80194  0.44504  2.00000  0.44504 -1.80194 -1.24698  1.24698  1.80194 -0.44504 -2.00000  0.00000  0.00000  2.00000 -0.44504  1.80194  1.24698 -1.24698 -1.80194  0.44504  2.00000  0.44504 -1.80194 -1.24698  1.24698  1.80194 -0.44504 -2.00000 -0.44504  1.80194  1.24698 -1.24698 -1.80194  0.44504 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E10g    2.00000  0.14946 -1.97766 -0.44504  1.91115  0.73068 -1.80194 -1.00000  1.65248  1.24698 -1.46610 -1.46610  1.24698  1.65248 -1.00000 -1.80194  0.73068  1.91115 -0.44504 -1.97766  0.14946  2.00000  0.00000  0.00000  2.00000  0.14946 -1.97766 -0.44504  1.91115  0.73068 -1.80194 -1.00000  1.65248  1.24698 -1.46610 -1.46610  1.24698  1.65248 -1.00000 -1.80194  0.73068  1.91115 -0.44504 -1.97766  0.14946  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E11g    2.00000 -0.14946 -1.97766  0.44504  1.91115 -0.73068 -1.80194  1.00000  1.65248 -1.24698 -1.46610  1.46610  1.24698 -1.65248 -1.00000  1.80194  0.73068 -1.91115 -0.44504  1.97766  0.14946 -2.00000  0.00000  0.00000  2.00000  0.14946  1.97766 -0.44504 -1.91115  0.73068  1.80194 -1.00000 -1.65248  1.24698  1.46610 -1.46610 -1.24698  1.65248  1.00000 -1.80194 -0.73068  1.91115  0.44504 -1.97766 -0.14946 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E12g    2.00000 -0.44504 -1.80194  1.24698  1.24698 -1.80194 -0.44504  2.00000 -0.44504 -1.80194  1.24698  1.24698 -1.80194 -0.44504  2.00000 -0.44504 -1.80194  1.24698  1.24698 -1.80194 -0.44504  2.00000  0.00000  0.00000  2.00000 -0.44504 -1.80194  1.24698  1.24698 -1.80194 -0.44504  2.00000 -0.44504 -1.80194  1.24698  1.24698 -1.80194 -0.44504  2.00000 -0.44504 -1.80194  1.24698  1.24698 -1.80194 -0.44504  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E13g    2.00000 -0.73068 -1.46610  1.80194  0.14946 -1.91115  1.24698  1.00000 -1.97766  0.44504  1.65248 -1.65248 -0.44504  1.97766 -1.00000 -1.24698  1.91115 -0.14946 -1.80194  1.46610  0.73068 -2.00000  0.00000  0.00000  2.00000  0.73068  1.46610 -1.80194 -0.14946  1.91115 -1.24698 -1.00000  1.97766 -0.44504 -1.65248  1.65248  0.44504 -1.97766  1.00000  1.24698 -1.91115  0.14946  1.80194 -1.46610 -0.73068 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E14g    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  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     ... ... ..... ....... ......... ........... .............
E15g    2.00000 -1.24698 -0.44504  1.80194 -1.80194  0.44504  1.24698 -2.00000  1.24698  0.44504 -1.80194  1.80194 -0.44504 -1.24698  2.00000 -1.24698 -0.44504  1.80194 -1.80194  0.44504  1.24698 -2.00000  0.00000  0.00000  2.00000  1.24698  0.44504 -1.80194  1.80194 -0.44504 -1.24698  2.00000 -1.24698 -0.44504  1.80194 -1.80194  0.44504  1.24698 -2.00000  1.24698  0.44504 -1.80194  1.80194 -0.44504 -1.24698 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E16g    2.00000 -1.46610  0.14946  1.24698 -1.97766  1.65248 -0.44504 -1.00000  1.91115 -1.80194  0.73068  0.73068 -1.80194  1.91115 -1.00000 -0.44504  1.65248 -1.97766  1.24698  0.14946 -1.46610  2.00000  0.00000  0.00000  2.00000 -1.46610  0.14946  1.24698 -1.97766  1.65248 -0.44504 -1.00000  1.91115 -1.80194  0.73068  0.73068 -1.80194  1.91115 -1.00000 -0.44504  1.65248 -1.97766  1.24698  0.14946 -1.46610  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E17g    2.00000 -1.65248  0.73068  0.44504 -1.46610  1.97766 -1.80194  1.00000  0.14946 -1.24698  1.91115 -1.91115  1.24698 -0.14946 -1.00000  1.80194 -1.97766  1.46610 -0.44504 -0.73068  1.65248 -2.00000  0.00000  0.00000  2.00000  1.65248 -0.73068 -0.44504  1.46610 -1.97766  1.80194 -1.00000 -0.14946  1.24698 -1.91115  1.91115 -1.24698  0.14946  1.00000 -1.80194  1.97766 -1.46610  0.44504  0.73068 -1.65248 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E18g    2.00000 -1.80194  1.24698 -0.44504 -0.44504  1.24698 -1.80194  2.00000 -1.80194  1.24698 -0.44504 -0.44504  1.24698 -1.80194  2.00000 -1.80194  1.24698 -0.44504 -0.44504  1.24698 -1.80194  2.00000  0.00000  0.00000  2.00000 -1.80194  1.24698 -0.44504 -0.44504  1.24698 -1.80194  2.00000 -1.80194  1.24698 -0.44504 -0.44504  1.24698 -1.80194  2.00000 -1.80194  1.24698 -0.44504 -0.44504  1.24698 -1.80194  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E19g    2.00000 -1.91115  1.65248 -1.24698  0.73068 -0.14946 -0.44504  1.00000 -1.46610  1.80194 -1.97766  1.97766 -1.80194  1.46610 -1.00000  0.44504  0.14946 -0.73068  1.24698 -1.65248  1.91115 -2.00000  0.00000  0.00000  2.00000  1.91115 -1.65248  1.24698 -0.73068  0.14946  0.44504 -1.00000  1.46610 -1.80194  1.97766 -1.97766  1.80194 -1.46610  1.00000 -0.44504 -0.14946  0.73068 -1.24698  1.65248 -1.91115 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E20g    2.00000 -1.97766  1.91115 -1.80194  1.65248 -1.46610  1.24698 -1.00000  0.73068 -0.44504  0.14946  0.14946 -0.44504  0.73068 -1.00000  1.24698 -1.46610  1.65248 -1.80194  1.91115 -1.97766  2.00000  0.00000  0.00000  2.00000 -1.97766  1.91115 -1.80194  1.65248 -1.46610  1.24698 -1.00000  0.73068 -0.44504  0.14946  0.14946 -0.44504  0.73068 -1.00000  1.24698 -1.46610  1.65248 -1.80194  1.91115 -1.97766  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 -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 -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  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  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000  1.00000 -1.00000  1.00000     ... ... ..... ....... ......... ........... .............
E1u     2.00000  1.97766  1.91115  1.80194  1.65248  1.46610  1.24698  1.00000  0.73068  0.44504  0.14946 -0.14946 -0.44504 -0.73068 -1.00000 -1.24698 -1.46610 -1.65248 -1.80194 -1.91115 -1.97766 -2.00000  0.00000  0.00000 -2.00000  1.97766  1.91115  1.80194  1.65248  1.46610  1.24698  1.00000  0.73068  0.44504  0.14946 -0.14946 -0.44504 -0.73068 -1.00000 -1.24698 -1.46610 -1.65248 -1.80194 -1.91115 -1.97766  2.00000  0.00000  0.00000     ... TT. ..... ....TT. ......... ........TT. .............
E2u     2.00000  1.91115  1.65248  1.24698  0.73068  0.14946 -0.44504 -1.00000 -1.46610 -1.80194 -1.97766 -1.97766 -1.80194 -1.46610 -1.00000 -0.44504  0.14946  0.73068  1.24698  1.65248  1.91115  2.00000  0.00000  0.00000 -2.00000 -1.91115 -1.65248 -1.24698 -0.73068 -0.14946  0.44504  1.00000  1.46610  1.80194  1.97766  1.97766  1.80194  1.46610  1.00000  0.44504 -0.14946 -0.73068 -1.24698 -1.65248 -1.91115 -2.00000  0.00000  0.00000     ... ... ..... ..TT... ......... ......TT... .............
E3u     2.00000  1.80194  1.24698  0.44504 -0.44504 -1.24698 -1.80194 -2.00000 -1.80194 -1.24698 -0.44504  0.44504  1.24698  1.80194  2.00000  1.80194  1.24698  0.44504 -0.44504 -1.24698 -1.80194 -2.00000  0.00000  0.00000 -2.00000  1.80194  1.24698  0.44504 -0.44504 -1.24698 -1.80194 -2.00000 -1.80194 -1.24698 -0.44504  0.44504  1.24698  1.80194  2.00000  1.80194  1.24698  0.44504 -0.44504 -1.24698 -1.80194  2.00000  0.00000  0.00000     ... ... ..... TT..... ......... ....TT..... .............
E4u     2.00000  1.65248  0.73068 -0.44504 -1.46610 -1.97766 -1.80194 -1.00000  0.14946  1.24698  1.91115  1.91115  1.24698  0.14946 -1.00000 -1.80194 -1.97766 -1.46610 -0.44504  0.73068  1.65248  2.00000  0.00000  0.00000 -2.00000 -1.65248 -0.73068  0.44504  1.46610  1.97766  1.80194  1.00000 -0.14946 -1.24698 -1.91115 -1.91115 -1.24698 -0.14946  1.00000  1.80194  1.97766  1.46610  0.44504 -0.73068 -1.65248 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ..TT....... .............
E5u     2.00000  1.46610  0.14946 -1.24698 -1.97766 -1.65248 -0.44504  1.00000  1.91115  1.80194  0.73068 -0.73068 -1.80194 -1.91115 -1.00000  0.44504  1.65248  1.97766  1.24698 -0.14946 -1.46610 -2.00000  0.00000  0.00000 -2.00000  1.46610  0.14946 -1.24698 -1.97766 -1.65248 -0.44504  1.00000  1.91115  1.80194  0.73068 -0.73068 -1.80194 -1.91115 -1.00000  0.44504  1.65248  1.97766  1.24698 -0.14946 -1.46610  2.00000  0.00000  0.00000     ... ... ..... ....... ......... TT......... .............
E6u     2.00000  1.24698 -0.44504 -1.80194 -1.80194 -0.44504  1.24698  2.00000  1.24698 -0.44504 -1.80194 -1.80194 -0.44504  1.24698  2.00000  1.24698 -0.44504 -1.80194 -1.80194 -0.44504  1.24698  2.00000  0.00000  0.00000 -2.00000 -1.24698  0.44504  1.80194  1.80194  0.44504 -1.24698 -2.00000 -1.24698  0.44504  1.80194  1.80194  0.44504 -1.24698 -2.00000 -1.24698  0.44504  1.80194  1.80194  0.44504 -1.24698 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E7u     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 -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     ... ... ..... ....... ......... ........... .............
E8u     2.00000  0.73068 -1.46610 -1.80194  0.14946  1.91115  1.24698 -1.00000 -1.97766 -0.44504  1.65248  1.65248 -0.44504 -1.97766 -1.00000  1.24698  1.91115  0.14946 -1.80194 -1.46610  0.73068  2.00000  0.00000  0.00000 -2.00000 -0.73068  1.46610  1.80194 -0.14946 -1.91115 -1.24698  1.00000  1.97766  0.44504 -1.65248 -1.65248  0.44504  1.97766  1.00000 -1.24698 -1.91115 -0.14946  1.80194  1.46610 -0.73068 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E9u     2.00000  0.44504 -1.80194 -1.24698  1.24698  1.80194 -0.44504 -2.00000 -0.44504  1.80194  1.24698 -1.24698 -1.80194  0.44504  2.00000  0.44504 -1.80194 -1.24698  1.24698  1.80194 -0.44504 -2.00000  0.00000  0.00000 -2.00000  0.44504 -1.80194 -1.24698  1.24698  1.80194 -0.44504 -2.00000 -0.44504  1.80194  1.24698 -1.24698 -1.80194  0.44504  2.00000  0.44504 -1.80194 -1.24698  1.24698  1.80194 -0.44504  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E10u    2.00000  0.14946 -1.97766 -0.44504  1.91115  0.73068 -1.80194 -1.00000  1.65248  1.24698 -1.46610 -1.46610  1.24698  1.65248 -1.00000 -1.80194  0.73068  1.91115 -0.44504 -1.97766  0.14946  2.00000  0.00000  0.00000 -2.00000 -0.14946  1.97766  0.44504 -1.91115 -0.73068  1.80194  1.00000 -1.65248 -1.24698  1.46610  1.46610 -1.24698 -1.65248  1.00000  1.80194 -0.73068 -1.91115  0.44504  1.97766 -0.14946 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E11u    2.00000 -0.14946 -1.97766  0.44504  1.91115 -0.73068 -1.80194  1.00000  1.65248 -1.24698 -1.46610  1.46610  1.24698 -1.65248 -1.00000  1.80194  0.73068 -1.91115 -0.44504  1.97766  0.14946 -2.00000  0.00000  0.00000 -2.00000 -0.14946 -1.97766  0.44504  1.91115 -0.73068 -1.80194  1.00000  1.65248 -1.24698 -1.46610  1.46610  1.24698 -1.65248 -1.00000  1.80194  0.73068 -1.91115 -0.44504  1.97766  0.14946  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E12u    2.00000 -0.44504 -1.80194  1.24698  1.24698 -1.80194 -0.44504  2.00000 -0.44504 -1.80194  1.24698  1.24698 -1.80194 -0.44504  2.00000 -0.44504 -1.80194  1.24698  1.24698 -1.80194 -0.44504  2.00000  0.00000  0.00000 -2.00000  0.44504  1.80194 -1.24698 -1.24698  1.80194  0.44504 -2.00000  0.44504  1.80194 -1.24698 -1.24698  1.80194  0.44504 -2.00000  0.44504  1.80194 -1.24698 -1.24698  1.80194  0.44504 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E13u    2.00000 -0.73068 -1.46610  1.80194  0.14946 -1.91115  1.24698  1.00000 -1.97766  0.44504  1.65248 -1.65248 -0.44504  1.97766 -1.00000 -1.24698  1.91115 -0.14946 -1.80194  1.46610  0.73068 -2.00000  0.00000  0.00000 -2.00000 -0.73068 -1.46610  1.80194  0.14946 -1.91115  1.24698  1.00000 -1.97766  0.44504  1.65248 -1.65248 -0.44504  1.97766 -1.00000 -1.24698  1.91115 -0.14946 -1.80194  1.46610  0.73068  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E14u    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 -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     ... ... ..... ....... ......... ........... .............
E15u    2.00000 -1.24698 -0.44504  1.80194 -1.80194  0.44504  1.24698 -2.00000  1.24698  0.44504 -1.80194  1.80194 -0.44504 -1.24698  2.00000 -1.24698 -0.44504  1.80194 -1.80194  0.44504  1.24698 -2.00000  0.00000  0.00000 -2.00000 -1.24698 -0.44504  1.80194 -1.80194  0.44504  1.24698 -2.00000  1.24698  0.44504 -1.80194  1.80194 -0.44504 -1.24698  2.00000 -1.24698 -0.44504  1.80194 -1.80194  0.44504  1.24698  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E16u    2.00000 -1.46610  0.14946  1.24698 -1.97766  1.65248 -0.44504 -1.00000  1.91115 -1.80194  0.73068  0.73068 -1.80194  1.91115 -1.00000 -0.44504  1.65248 -1.97766  1.24698  0.14946 -1.46610  2.00000  0.00000  0.00000 -2.00000  1.46610 -0.14946 -1.24698  1.97766 -1.65248  0.44504  1.00000 -1.91115  1.80194 -0.73068 -0.73068  1.80194 -1.91115  1.00000  0.44504 -1.65248  1.97766 -1.24698 -0.14946  1.46610 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E17u    2.00000 -1.65248  0.73068  0.44504 -1.46610  1.97766 -1.80194  1.00000  0.14946 -1.24698  1.91115 -1.91115  1.24698 -0.14946 -1.00000  1.80194 -1.97766  1.46610 -0.44504 -0.73068  1.65248 -2.00000  0.00000  0.00000 -2.00000 -1.65248  0.73068  0.44504 -1.46610  1.97766 -1.80194  1.00000  0.14946 -1.24698  1.91115 -1.91115  1.24698 -0.14946 -1.00000  1.80194 -1.97766  1.46610 -0.44504 -0.73068  1.65248  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E18u    2.00000 -1.80194  1.24698 -0.44504 -0.44504  1.24698 -1.80194  2.00000 -1.80194  1.24698 -0.44504 -0.44504  1.24698 -1.80194  2.00000 -1.80194  1.24698 -0.44504 -0.44504  1.24698 -1.80194  2.00000  0.00000  0.00000 -2.00000  1.80194 -1.24698  0.44504  0.44504 -1.24698  1.80194 -2.00000  1.80194 -1.24698  0.44504  0.44504 -1.24698  1.80194 -2.00000  1.80194 -1.24698  0.44504  0.44504 -1.24698  1.80194 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E19u    2.00000 -1.91115  1.65248 -1.24698  0.73068 -0.14946 -0.44504  1.00000 -1.46610  1.80194 -1.97766  1.97766 -1.80194  1.46610 -1.00000  0.44504  0.14946 -0.73068  1.24698 -1.65248  1.91115 -2.00000  0.00000  0.00000 -2.00000 -1.91115  1.65248 -1.24698  0.73068 -0.14946 -0.44504  1.00000 -1.46610  1.80194 -1.97766  1.97766 -1.80194  1.46610 -1.00000  0.44504  0.14946 -0.73068  1.24698 -1.65248  1.91115  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E20u    2.00000 -1.97766  1.91115 -1.80194  1.65248 -1.46610  1.24698 -1.00000  0.73068 -0.44504  0.14946  0.14946 -0.44504  0.73068 -1.00000  1.24698 -1.46610  1.65248 -1.80194  1.91115 -1.97766  2.00000  0.00000  0.00000 -2.00000  1.97766 -1.91115  1.80194 -1.65248  1.46610 -1.24698  1.00000 -0.73068  0.44504 -0.14946 -0.14946  0.44504 -0.73068  1.00000 -1.24698  1.46610 -1.65248  1.80194 -1.91115  1.97766 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............

 Irrational character values:  1.977661652450 = 2*cos(2*π/42) = 2*cos(π/21)
                               1.911145611572 = 2*cos(4*π/42) = 2*cos(2*π/21)
                               1.801937735805 = 2*cos(6*π/42) = 2*cos(π/7)
                               1.652477548632 = 2*cos(8*π/42) = 2*cos(4*π/21)
                               1.466103743660 = 2*cos(10*π/42) = 2*cos(5*π/21)
                               1.246979603717 = 2*cos(12*π/42) = 2*cos(2*π/7)
                               0.730682048733 = 2*cos(16*π/42) = 2*cos(8*π/21)
                               0.445041867913 = 2*cos(18*π/42) = 2*cos(3*π/7)
                               0.149460187173 = 2*cos(20*π/42) = 2*cos(10*π/21)



 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 D42h point group is 168, and the order of the principal axis (C42) is 42. The group has 48 irreducible representations.

    β  The D42h point group is generated by three symmetry elements that are canonically chosen C42, 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 S42 can be chosen, together with σh or C2 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 has the x axis as a member, while the y axis is a member of the C2 set.

    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 has the xz plane as a member, while the yz plane is a member of the σd set.

    ε  The lowest nonvanishing multipole moment in D42h 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 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.
       All characters of this group can be expressed using complex numbers, elementary arithmetic operations, square roots and third roots.

This Character Table for the D42h point group was created by Gernot Katzer.

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