Character table for the D84 point group

D84     E        2 C84    2 C42    2 C28    2 C21    2 C84^5  2 C14    2 C12    2 C21^2  2 C28^3  2 C42^5  2 C84^11 2 C7     2 C84^13 2 C6     2 C28^5  2 C21^4  2 C84^17 2 C14^3  2 C84^19 2 C21^5  2 C4     2 C42^11 2 C84^23 2 C7^2   2 C84^25 2 C42^13 2 C28^9  2 C3     2 C84^29 2 C14^5  2 C84^31 2 C21^8  2 C28^11 2 C42^17 2 C12^5  2 C7^3   2 C84^37 2 C42^19 2 C28^13 2 C21^10 2 C84^41 C2       42 C2'   42 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  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     ..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  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     ... ... ..... ....... ......... ........... .............
E1      2.00000  1.99441  1.97766  1.94986  1.91115  1.86175  1.80194  1.73205  1.65248  1.56366  1.46610  1.36035  1.24698  1.12664  1.00000  0.86777  0.73068  0.58951  0.44504  0.29808  0.14946  0.00000 -0.14946 -0.29808 -0.44504 -0.58951 -0.73068 -0.86777 -1.00000 -1.12664 -1.24698 -1.36035 -1.46610 -1.56366 -1.65248 -1.73205 -1.80194 -1.86175 -1.91115 -1.94986 -1.97766 -1.99441 -2.00000  0.00000  0.00000     TT. TT. ..TT. ....TT. ......TT. ........TT. ..........TT.
E2      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 -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... ........TT...
E3      2.00000  1.94986  1.80194  1.56366  1.24698  0.86777  0.44504  0.00000 -0.44504 -0.86777 -1.24698 -1.56366 -1.80194 -1.94986 -2.00000 -1.94986 -1.80194 -1.56366 -1.24698 -0.86777 -0.44504  0.00000  0.44504  0.86777  1.24698  1.56366  1.80194  1.94986  2.00000  1.94986  1.80194  1.56366  1.24698  0.86777  0.44504  0.00000 -0.44504 -0.86777 -1.24698 -1.56366 -1.80194 -1.94986 -2.00000  0.00000  0.00000     ... ... ..... TT..... ..TT..... ....TT..... ......TT.....
E4      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  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.......
E5      2.00000  1.86175  1.46610  0.86777  0.14946 -0.58951 -1.24698 -1.73205 -1.97766 -1.94986 -1.65248 -1.12664 -0.44504  0.29808  1.00000  1.56366  1.91115  1.99441  1.80194  1.36035  0.73068  0.00000 -0.73068 -1.36035 -1.80194 -1.99441 -1.91115 -1.56366 -1.00000 -0.29808  0.44504  1.12664  1.65248  1.94986  1.97766  1.73205  1.24698  0.58951 -0.14946 -0.86777 -1.46610 -1.86175 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... TT......... ..TT.........
E6      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 -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...........
E7      2.00000  1.73205  1.00000  0.00000 -1.00000 -1.73205 -2.00000 -1.73205 -1.00000  0.00000  1.00000  1.73205  2.00000  1.73205  1.00000  0.00000 -1.00000 -1.73205 -2.00000 -1.73205 -1.00000  0.00000  1.00000  1.73205  2.00000  1.73205  1.00000  0.00000 -1.00000 -1.73205 -2.00000 -1.73205 -1.00000  0.00000  1.00000  1.73205  2.00000  1.73205  1.00000  0.00000 -1.00000 -1.73205 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E8      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  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     ... ... ..... ....... ......... ........... .............
E9      2.00000  1.56366  0.44504 -0.86777 -1.80194 -1.94986 -1.24698  0.00000  1.24698  1.94986  1.80194  0.86777 -0.44504 -1.56366 -2.00000 -1.56366 -0.44504  0.86777  1.80194  1.94986  1.24698  0.00000 -1.24698 -1.94986 -1.80194 -0.86777  0.44504  1.56366  2.00000  1.56366  0.44504 -0.86777 -1.80194 -1.94986 -1.24698  0.00000  1.24698  1.94986  1.80194  0.86777 -0.44504 -1.56366 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E10     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 -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     ... ... ..... ....... ......... ........... .............
E11     2.00000  1.36035 -0.14946 -1.56366 -1.97766 -1.12664  0.44504  1.73205  1.91115  0.86777 -0.73068 -1.86175 -1.80194 -0.58951  1.00000  1.94986  1.65248  0.29808 -1.24698 -1.99441 -1.46610  0.00000  1.46610  1.99441  1.24698 -0.29808 -1.65248 -1.94986 -1.00000  0.58951  1.80194  1.86175  0.73068 -0.86777 -1.91115 -1.73205 -0.44504  1.12664  1.97766  1.56366  0.14946 -1.36035 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E12     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  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     ... ... ..... ....... ......... ........... .............
E13     2.00000  1.12664 -0.73068 -1.94986 -1.46610  0.29808  1.80194  1.73205  0.14946 -1.56366 -1.91115 -0.58951  1.24698  1.99441  1.00000 -0.86777 -1.97766 -1.36035  0.44504  1.86175  1.65248  0.00000 -1.65248 -1.86175 -0.44504  1.36035  1.97766  0.86777 -1.00000 -1.99441 -1.24698  0.58951  1.91115  1.56366 -0.14946 -1.73205 -1.80194 -0.29808  1.46610  1.94986  0.73068 -1.12664 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E14     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 -1.00000  1.00000  2.00000  1.00000 -1.00000 -2.00000 -1.00000  1.00000  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E15     2.00000  0.86777 -1.24698 -1.94986 -0.44504  1.56366  1.80194  0.00000 -1.80194 -1.56366  0.44504  1.94986  1.24698 -0.86777 -2.00000 -0.86777  1.24698  1.94986  0.44504 -1.56366 -1.80194  0.00000  1.80194  1.56366 -0.44504 -1.94986 -1.24698  0.86777  2.00000  0.86777 -1.24698 -1.94986 -0.44504  1.56366  1.80194  0.00000 -1.80194 -1.56366  0.44504  1.94986  1.24698 -0.86777 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E16     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  0.73068 -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     ... ... ..... ....... ......... ........... .............
E17     2.00000  0.58951 -1.65248 -1.56366  0.73068  1.99441  0.44504 -1.73205 -1.46610  0.86777  1.97766  0.29808 -1.80194 -1.36035  1.00000  1.94986  0.14946 -1.86175 -1.24698  1.12664  1.91115  0.00000 -1.91115 -1.12664  1.24698  1.86175 -0.14946 -1.94986 -1.00000  1.36035  1.80194 -0.29808 -1.97766 -0.86777  1.46610  1.73205 -0.44504 -1.99441 -0.73068  1.56366  1.65248 -0.58951 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E18     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 -0.44504  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     ... ... ..... ....... ......... ........... .............
E19     2.00000  0.29808 -1.91115 -0.86777  1.65248  1.36035 -1.24698 -1.73205  0.73068  1.94986 -0.14946 -1.99441 -0.44504  1.86175  1.00000 -1.56366 -1.46610  1.12664  1.80194 -0.58951 -1.97766  0.00000  1.97766  0.58951 -1.80194 -1.12664  1.46610  1.56366 -1.00000 -1.86175  0.44504  1.99441  0.14946 -1.94986 -0.73068  1.73205  1.24698 -1.36035 -1.65248  0.86777  1.91115 -0.29808 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E20     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  0.14946 -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     ... ... ..... ....... ......... ........... .............
E21     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     ... ... ..... ....... ......... ........... .............
E22     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  0.14946  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     ... ... ..... ....... ......... ........... .............
E23     2.00000 -0.29808 -1.91115  0.86777  1.65248 -1.36035 -1.24698  1.73205  0.73068 -1.94986 -0.14946  1.99441 -0.44504 -1.86175  1.00000  1.56366 -1.46610 -1.12664  1.80194  0.58951 -1.97766  0.00000  1.97766 -0.58951 -1.80194  1.12664  1.46610 -1.56366 -1.00000  1.86175  0.44504 -1.99441  0.14946  1.94986 -0.73068 -1.73205  1.24698  1.36035 -1.65248 -0.86777  1.91115  0.29808 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E24     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 -0.44504 -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     ... ... ..... ....... ......... ........... .............
E25     2.00000 -0.58951 -1.65248  1.56366  0.73068 -1.99441  0.44504  1.73205 -1.46610 -0.86777  1.97766 -0.29808 -1.80194  1.36035  1.00000 -1.94986  0.14946  1.86175 -1.24698 -1.12664  1.91115  0.00000 -1.91115  1.12664  1.24698 -1.86175 -0.14946  1.94986 -1.00000 -1.36035  1.80194  0.29808 -1.97766  0.86777  1.46610 -1.73205 -0.44504  1.99441 -0.73068 -1.56366  1.65248  0.58951 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E26     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  0.73068  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     ... ... ..... ....... ......... ........... .............
E27     2.00000 -0.86777 -1.24698  1.94986 -0.44504 -1.56366  1.80194  0.00000 -1.80194  1.56366  0.44504 -1.94986  1.24698  0.86777 -2.00000  0.86777  1.24698 -1.94986  0.44504  1.56366 -1.80194  0.00000  1.80194 -1.56366 -0.44504  1.94986 -1.24698 -0.86777  2.00000 -0.86777 -1.24698  1.94986 -0.44504 -1.56366  1.80194  0.00000 -1.80194  1.56366  0.44504 -1.94986  1.24698  0.86777 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E28     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 -1.00000 -1.00000  2.00000 -1.00000 -1.00000  2.00000 -1.00000 -1.00000  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E29     2.00000 -1.12664 -0.73068  1.94986 -1.46610 -0.29808  1.80194 -1.73205  0.14946  1.56366 -1.91115  0.58951  1.24698 -1.99441  1.00000  0.86777 -1.97766  1.36035  0.44504 -1.86175  1.65248  0.00000 -1.65248  1.86175 -0.44504 -1.36035  1.97766 -0.86777 -1.00000  1.99441 -1.24698 -0.58951  1.91115 -1.56366 -0.14946  1.73205 -1.80194  0.29808  1.46610 -1.94986  0.73068  1.12664 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E30     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  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     ... ... ..... ....... ......... ........... .............
E31     2.00000 -1.36035 -0.14946  1.56366 -1.97766  1.12664  0.44504 -1.73205  1.91115 -0.86777 -0.73068  1.86175 -1.80194  0.58951  1.00000 -1.94986  1.65248 -0.29808 -1.24698  1.99441 -1.46610  0.00000  1.46610 -1.99441  1.24698  0.29808 -1.65248  1.94986 -1.00000 -0.58951  1.80194 -1.86175  0.73068  0.86777 -1.91115  1.73205 -0.44504 -1.12664  1.97766 -1.56366  0.14946  1.36035 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E32     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 -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     ... ... ..... ....... ......... ........... .............
E33     2.00000 -1.56366  0.44504  0.86777 -1.80194  1.94986 -1.24698  0.00000  1.24698 -1.94986  1.80194 -0.86777 -0.44504  1.56366 -2.00000  1.56366 -0.44504 -0.86777  1.80194 -1.94986  1.24698  0.00000 -1.24698  1.94986 -1.80194  0.86777  0.44504 -1.56366  2.00000 -1.56366  0.44504  0.86777 -1.80194  1.94986 -1.24698  0.00000  1.24698 -1.94986  1.80194 -0.86777 -0.44504  1.56366 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E34     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  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     ... ... ..... ....... ......... ........... .............
E35     2.00000 -1.73205  1.00000  0.00000 -1.00000  1.73205 -2.00000  1.73205 -1.00000  0.00000  1.00000 -1.73205  2.00000 -1.73205  1.00000  0.00000 -1.00000  1.73205 -2.00000  1.73205 -1.00000  0.00000  1.00000 -1.73205  2.00000 -1.73205  1.00000  0.00000 -1.00000  1.73205 -2.00000  1.73205 -1.00000  0.00000  1.00000 -1.73205  2.00000 -1.73205  1.00000  0.00000 -1.00000  1.73205 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E36     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 -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     ... ... ..... ....... ......... ........... .............
E37     2.00000 -1.86175  1.46610 -0.86777  0.14946  0.58951 -1.24698  1.73205 -1.97766  1.94986 -1.65248  1.12664 -0.44504 -0.29808  1.00000 -1.56366  1.91115 -1.99441  1.80194 -1.36035  0.73068  0.00000 -0.73068  1.36035 -1.80194  1.99441 -1.91115  1.56366 -1.00000  0.29808  0.44504 -1.12664  1.65248 -1.94986  1.97766 -1.73205  1.24698 -0.58951 -0.14946  0.86777 -1.46610  1.86175 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E38     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  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     ... ... ..... ....... ......... ........... .............
E39     2.00000 -1.94986  1.80194 -1.56366  1.24698 -0.86777  0.44504  0.00000 -0.44504  0.86777 -1.24698  1.56366 -1.80194  1.94986 -2.00000  1.94986 -1.80194  1.56366 -1.24698  0.86777 -0.44504  0.00000  0.44504 -0.86777  1.24698 -1.56366  1.80194 -1.94986  2.00000 -1.94986  1.80194 -1.56366  1.24698 -0.86777  0.44504  0.00000 -0.44504  0.86777 -1.24698  1.56366 -1.80194  1.94986 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E40     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 -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     ... ... ..... ....... ......... ........... .............
E41     2.00000 -1.99441  1.97766 -1.94986  1.91115 -1.86175  1.80194 -1.73205  1.65248 -1.56366  1.46610 -1.36035  1.24698 -1.12664  1.00000 -0.86777  0.73068 -0.58951  0.44504 -0.29808  0.14946  0.00000 -0.14946  0.29808 -0.44504  0.58951 -0.73068  0.86777 -1.00000  1.12664 -1.24698  1.36035 -1.46610  1.56366 -1.65248  1.73205 -1.80194  1.86175 -1.91115  1.94986 -1.97766  1.99441 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............

 Irrational character values:  1.994407594362 = 2*cos(2*π/84) = 2*cos(π/42)
                               1.977661652450 = 2*cos(4*π/84) = 2*cos(π/21)
                               1.949855824364 = 2*cos(6*π/84) = 2*cos(π/14)
                               1.911145611572 = 2*cos(8*π/84) = 2*cos(2*π/21)
                               1.861747497288 = 2*cos(10*π/84) = 2*cos(5*π/42)
                               1.801937735805 = 2*cos(12*π/84) = 2*cos(π/7)
                               1.732050807569 = 2*cos(14*π/84) = 2*cos(π/6) = √3
                               1.652477548632 = 2*cos(16*π/84) = 2*cos(4*π/21)
                               1.563662964936 = 2*cos(18*π/84) = 2*cos(3*π/14)
                               1.466103743660 = 2*cos(20*π/84) = 2*cos(5*π/21)
                               1.360345475542 = 2*cos(22*π/84) = 2*cos(11*π/42)
                               1.246979603717 = 2*cos(24*π/84) = 2*cos(2*π/7)
                               1.126640116127 = 2*cos(26*π/84) = 2*cos(13*π/42)
                               0.867767478235 = 2*cos(30*π/84) = 2*cos(5*π/14)
                               0.730682048733 = 2*cos(32*π/84) = 2*cos(8*π/21)
                               0.589510348822 = 2*cos(34*π/84) = 2*cos(17*π/42)
                               0.445041867913 = 2*cos(36*π/84) = 2*cos(3*π/7)
                               0.298084532352 = 2*cos(38*π/84) = 2*cos(19*π/42)
                               0.149460187173 = 2*cos(40*π/84) = 2*cos(10*π/21)



 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 D84 point group is 168, and the order of the principal axis (C84) is 84. The group has 45 irreducible representations.

    β  The D84 point group is isomorphic to D42d and C84v.

    γ  The D84 point group is generated by two symmetry elements, C84 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 contains both the x and y axes.

    ε  The lowest nonvanishing multipole moment in D84 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.

    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 D84 point group was created by Gernot Katzer.

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