Character Table for Point Group D84d

Character table for the D_84d point group

D84d E 2 S168 2 C84 2 S56 2 C42 2 S168^5 2 C28 2 S24 2 C21 2 S56^3 2 C84^5 2 S168^11 2 C14 2 S168^13 2 C12 2 S56^5 2 C21^2 2 S168^17 2 C28^3 2 S168^19 2 C42^5 2 S8 2 C84^11 2 S168^23 2 C7 2 S168^25 2 C84^13 2 S56^9 2 C6 2 S168^29 2 C28^5 2 S168^31 2 C21^4 2 S56^11 2 C84^17 2 S24^5 2 C14^3 2 S168^37 2 C84^19 2 S56^13 2 C21^5 2 S168^41 2 C4 2 S168^43 2 C42^11 2 S56^15 2 C84^23 2 S168^47 2 C7^2 2 S24^7 2 C84^25 2 S56^17 2 C42^13 2 S168^53 2 C28^9 2 S168^55 2 C3 2 S56^19 2 C84^29 2 S168^59 2 C14^5 2 S168^61 2 C84^31 2 S8^3 2 C21^8 2 S168^65 2 C28^11 2 S168^67 2 C42^17 2 S56^23 2 C12^5 2 S168^71 2 C7^3 2 S168^73 2 C84^37 2 S56^25 2 C42^19 2 S24^11 2 C28^13 2 S168^79 2 C21^10 2 S56^27 2 C84^41 2 S168^83 C2 84 C2' 84 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 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 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 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 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 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -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 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -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 1.99860 1.99441 1.98742 1.97766 1.96513 1.94986 1.93185 1.91115 1.88777 1.86175 1.83312 1.80194 1.76823 1.73205 1.69345 1.65248 1.60920 1.56366 1.51594 1.46610 1.41421 1.36035 1.30457 1.24698 1.18764 1.12664 1.06406 1.00000 0.93454 0.86777 0.79978 0.73068 0.66056 0.58951 0.51764 0.44504 0.37182 0.29808 0.22393 0.14946 0.07478 0.00000 -0.07478 -0.14946 -0.22393 -0.29808 -0.37182 -0.44504 -0.51764 -0.58951 -0.66056 -0.73068 -0.79978 -0.86777 -0.93454 -1.00000 -1.06406 -1.12664 -1.18764 -1.24698 -1.30457 -1.36035 -1.41421 -1.46610 -1.51594 -1.56366 -1.60920 -1.65248 -1.69345 -1.73205 -1.76823 -1.80194 -1.83312 -1.86175 -1.88777 -1.91115 -1.93185 -1.94986 -1.96513 -1.97766 -1.98742 -1.99441 -1.99860 -2.00000 0.00000 0.00000 ... TT. ..... ....TT. ......... ........TT. ............. E2 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 -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... E3 2.00000 1.98742 1.94986 1.88777 1.80194 1.69345 1.56366 1.41421 1.24698 1.06406 0.86777 0.66056 0.44504 0.22393 0.00000 -0.22393 -0.44504 -0.66056 -0.86777 -1.06406 -1.24698 -1.41421 -1.56366 -1.69345 -1.80194 -1.88777 -1.94986 -1.98742 -2.00000 -1.98742 -1.94986 -1.88777 -1.80194 -1.69345 -1.56366 -1.41421 -1.24698 -1.06406 -0.86777 -0.66056 -0.44504 -0.22393 0.00000 0.22393 0.44504 0.66056 0.86777 1.06406 1.24698 1.41421 1.56366 1.69345 1.80194 1.88777 1.94986 1.98742 2.00000 1.98742 1.94986 1.88777 1.80194 1.69345 1.56366 1.41421 1.24698 1.06406 0.86777 0.66056 0.44504 0.22393 0.00000 -0.22393 -0.44504 -0.66056 -0.86777 -1.06406 -1.24698 -1.41421 -1.56366 -1.69345 -1.80194 -1.88777 -1.94986 -1.98742 -2.00000 0.00000 0.00000 ... ... ..... TT..... ......... ....TT..... ............. E4 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 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....... E5 2.00000 1.96513 1.86175 1.69345 1.46610 1.18764 0.86777 0.51764 0.14946 -0.22393 -0.58951 -0.93454 -1.24698 -1.51594 -1.73205 -1.88777 -1.97766 -1.99860 -1.94986 -1.83312 -1.65248 -1.41421 -1.12664 -0.79978 -0.44504 -0.07478 0.29808 0.66056 1.00000 1.30457 1.56366 1.76823 1.91115 1.98742 1.99441 1.93185 1.80194 1.60920 1.36035 1.06406 0.73068 0.37182 0.00000 -0.37182 -0.73068 -1.06406 -1.36035 -1.60920 -1.80194 -1.93185 -1.99441 -1.98742 -1.91115 -1.76823 -1.56366 -1.30457 -1.00000 -0.66056 -0.29808 0.07478 0.44504 0.79978 1.12664 1.41421 1.65248 1.83312 1.94986 1.99860 1.97766 1.88777 1.73205 1.51594 1.24698 0.93454 0.58951 0.22393 -0.14946 -0.51764 -0.86777 -1.18764 -1.46610 -1.69345 -1.86175 -1.96513 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... TT......... ............. E6 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 -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........... E7 2.00000 1.93185 1.73205 1.41421 1.00000 0.51764 0.00000 -0.51764 -1.00000 -1.41421 -1.73205 -1.93185 -2.00000 -1.93185 -1.73205 -1.41421 -1.00000 -0.51764 0.00000 0.51764 1.00000 1.41421 1.73205 1.93185 2.00000 1.93185 1.73205 1.41421 1.00000 0.51764 0.00000 -0.51764 -1.00000 -1.41421 -1.73205 -1.93185 -2.00000 -1.93185 -1.73205 -1.41421 -1.00000 -0.51764 0.00000 0.51764 1.00000 1.41421 1.73205 1.93185 2.00000 1.93185 1.73205 1.41421 1.00000 0.51764 0.00000 -0.51764 -1.00000 -1.41421 -1.73205 -1.93185 -2.00000 -1.93185 -1.73205 -1.41421 -1.00000 -0.51764 0.00000 0.51764 1.00000 1.41421 1.73205 1.93185 2.00000 1.93185 1.73205 1.41421 1.00000 0.51764 0.00000 -0.51764 -1.00000 -1.41421 -1.73205 -1.93185 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E8 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 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 ... ... ..... ....... ......... ........... ............. E9 2.00000 1.88777 1.56366 1.06406 0.44504 -0.22393 -0.86777 -1.41421 -1.80194 -1.98742 -1.94986 -1.69345 -1.24698 -0.66056 0.00000 0.66056 1.24698 1.69345 1.94986 1.98742 1.80194 1.41421 0.86777 0.22393 -0.44504 -1.06406 -1.56366 -1.88777 -2.00000 -1.88777 -1.56366 -1.06406 -0.44504 0.22393 0.86777 1.41421 1.80194 1.98742 1.94986 1.69345 1.24698 0.66056 0.00000 -0.66056 -1.24698 -1.69345 -1.94986 -1.98742 -1.80194 -1.41421 -0.86777 -0.22393 0.44504 1.06406 1.56366 1.88777 2.00000 1.88777 1.56366 1.06406 0.44504 -0.22393 -0.86777 -1.41421 -1.80194 -1.98742 -1.94986 -1.69345 -1.24698 -0.66056 0.00000 0.66056 1.24698 1.69345 1.94986 1.98742 1.80194 1.41421 0.86777 0.22393 -0.44504 -1.06406 -1.56366 -1.88777 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E10 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 -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 ... ... ..... ....... ......... ........... ............. E11 2.00000 1.83312 1.36035 0.66056 -0.14946 -0.93454 -1.56366 -1.93185 -1.97766 -1.69345 -1.12664 -0.37182 0.44504 1.18764 1.73205 1.98742 1.91115 1.51594 0.86777 0.07478 -0.73068 -1.41421 -1.86175 -1.99860 -1.80194 -1.30457 -0.58951 0.22393 1.00000 1.60920 1.94986 1.96513 1.65248 1.06406 0.29808 -0.51764 -1.24698 -1.76823 -1.99441 -1.88777 -1.46610 -0.79978 0.00000 0.79978 1.46610 1.88777 1.99441 1.76823 1.24698 0.51764 -0.29808 -1.06406 -1.65248 -1.96513 -1.94986 -1.60920 -1.00000 -0.22393 0.58951 1.30457 1.80194 1.99860 1.86175 1.41421 0.73068 -0.07478 -0.86777 -1.51594 -1.91115 -1.98742 -1.73205 -1.18764 -0.44504 0.37182 1.12664 1.69345 1.97766 1.93185 1.56366 0.93454 0.14946 -0.66056 -1.36035 -1.83312 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E12 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 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 ... ... ..... ....... ......... ........... ............. E13 2.00000 1.76823 1.12664 0.22393 -0.73068 -1.51594 -1.94986 -1.93185 -1.46610 -0.66056 0.29808 1.18764 1.80194 1.99860 1.73205 1.06406 0.14946 -0.79978 -1.56366 -1.96513 -1.91115 -1.41421 -0.58951 0.37182 1.24698 1.83312 1.99441 1.69345 1.00000 0.07478 -0.86777 -1.60920 -1.97766 -1.88777 -1.36035 -0.51764 0.44504 1.30457 1.86175 1.98742 1.65248 0.93454 0.00000 -0.93454 -1.65248 -1.98742 -1.86175 -1.30457 -0.44504 0.51764 1.36035 1.88777 1.97766 1.60920 0.86777 -0.07478 -1.00000 -1.69345 -1.99441 -1.83312 -1.24698 -0.37182 0.58951 1.41421 1.91115 1.96513 1.56366 0.79978 -0.14946 -1.06406 -1.73205 -1.99860 -1.80194 -1.18764 -0.29808 0.66056 1.46610 1.93185 1.94986 1.51594 0.73068 -0.22393 -1.12664 -1.76823 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E14 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 -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 ... ... ..... ....... ......... ........... ............. 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E71 2.00000 -1.76823 1.12664 -0.22393 -0.73068 1.51594 -1.94986 1.93185 -1.46610 0.66056 0.29808 -1.18764 1.80194 -1.99860 1.73205 -1.06406 0.14946 0.79978 -1.56366 1.96513 -1.91115 1.41421 -0.58951 -0.37182 1.24698 -1.83312 1.99441 -1.69345 1.00000 -0.07478 -0.86777 1.60920 -1.97766 1.88777 -1.36035 0.51764 0.44504 -1.30457 1.86175 -1.98742 1.65248 -0.93454 0.00000 0.93454 -1.65248 1.98742 -1.86175 1.30457 -0.44504 -0.51764 1.36035 -1.88777 1.97766 -1.60920 0.86777 0.07478 -1.00000 1.69345 -1.99441 1.83312 -1.24698 0.37182 0.58951 -1.41421 1.91115 -1.96513 1.56366 -0.79978 -0.14946 1.06406 -1.73205 1.99860 -1.80194 1.18764 -0.29808 -0.66056 1.46610 -1.93185 1.94986 -1.51594 0.73068 0.22393 -1.12664 1.76823 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E72 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 -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 ... ... ..... ....... ......... ........... ............. E73 2.00000 -1.83312 1.36035 -0.66056 -0.14946 0.93454 -1.56366 1.93185 -1.97766 1.69345 -1.12664 0.37182 0.44504 -1.18764 1.73205 -1.98742 1.91115 -1.51594 0.86777 -0.07478 -0.73068 1.41421 -1.86175 1.99860 -1.80194 1.30457 -0.58951 -0.22393 1.00000 -1.60920 1.94986 -1.96513 1.65248 -1.06406 0.29808 0.51764 -1.24698 1.76823 -1.99441 1.88777 -1.46610 0.79978 0.00000 -0.79978 1.46610 -1.88777 1.99441 -1.76823 1.24698 -0.51764 -0.29808 1.06406 -1.65248 1.96513 -1.94986 1.60920 -1.00000 0.22393 0.58951 -1.30457 1.80194 -1.99860 1.86175 -1.41421 0.73068 0.07478 -0.86777 1.51594 -1.91115 1.98742 -1.73205 1.18764 -0.44504 -0.37182 1.12664 -1.69345 1.97766 -1.93185 1.56366 -0.93454 0.14946 0.66056 -1.36035 1.83312 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E74 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 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 ... ... ..... ....... ......... ........... ............. 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E76 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 -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 ... ... ..... ....... ......... ........... ............. E77 2.00000 -1.93185 1.73205 -1.41421 1.00000 -0.51764 0.00000 0.51764 -1.00000 1.41421 -1.73205 1.93185 -2.00000 1.93185 -1.73205 1.41421 -1.00000 0.51764 0.00000 -0.51764 1.00000 -1.41421 1.73205 -1.93185 2.00000 -1.93185 1.73205 -1.41421 1.00000 -0.51764 0.00000 0.51764 -1.00000 1.41421 -1.73205 1.93185 -2.00000 1.93185 -1.73205 1.41421 -1.00000 0.51764 0.00000 -0.51764 1.00000 -1.41421 1.73205 -1.93185 2.00000 -1.93185 1.73205 -1.41421 1.00000 -0.51764 0.00000 0.51764 -1.00000 1.41421 -1.73205 1.93185 -2.00000 1.93185 -1.73205 1.41421 -1.00000 0.51764 0.00000 -0.51764 1.00000 -1.41421 1.73205 -1.93185 2.00000 -1.93185 1.73205 -1.41421 1.00000 -0.51764 0.00000 0.51764 -1.00000 1.41421 -1.73205 1.93185 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E78 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 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 ... ... ..... ....... ......... ........... ............. E79 2.00000 -1.96513 1.86175 -1.69345 1.46610 -1.18764 0.86777 -0.51764 0.14946 0.22393 -0.58951 0.93454 -1.24698 1.51594 -1.73205 1.88777 -1.97766 1.99860 -1.94986 1.83312 -1.65248 1.41421 -1.12664 0.79978 -0.44504 0.07478 0.29808 -0.66056 1.00000 -1.30457 1.56366 -1.76823 1.91115 -1.98742 1.99441 -1.93185 1.80194 -1.60920 1.36035 -1.06406 0.73068 -0.37182 0.00000 0.37182 -0.73068 1.06406 -1.36035 1.60920 -1.80194 1.93185 -1.99441 1.98742 -1.91115 1.76823 -1.56366 1.30457 -1.00000 0.66056 -0.29808 -0.07478 0.44504 -0.79978 1.12664 -1.41421 1.65248 -1.83312 1.94986 -1.99860 1.97766 -1.88777 1.73205 -1.51594 1.24698 -0.93454 0.58951 -0.22393 -0.14946 0.51764 -0.86777 1.18764 -1.46610 1.69345 -1.86175 1.96513 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ..TT......... E80 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 -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....... ............. E81 2.00000 -1.98742 1.94986 -1.88777 1.80194 -1.69345 1.56366 -1.41421 1.24698 -1.06406 0.86777 -0.66056 0.44504 -0.22393 0.00000 0.22393 -0.44504 0.66056 -0.86777 1.06406 -1.24698 1.41421 -1.56366 1.69345 -1.80194 1.88777 -1.94986 1.98742 -2.00000 1.98742 -1.94986 1.88777 -1.80194 1.69345 -1.56366 1.41421 -1.24698 1.06406 -0.86777 0.66056 -0.44504 0.22393 0.00000 -0.22393 0.44504 -0.66056 0.86777 -1.06406 1.24698 -1.41421 1.56366 -1.69345 1.80194 -1.88777 1.94986 -1.98742 2.00000 -1.98742 1.94986 -1.88777 1.80194 -1.69345 1.56366 -1.41421 1.24698 -1.06406 0.86777 -0.66056 0.44504 -0.22393 0.00000 0.22393 -0.44504 0.66056 -0.86777 1.06406 -1.24698 1.41421 -1.56366 1.69345 -1.80194 1.88777 -1.94986 1.98742 -2.00000 0.00000 0.00000 ... ... ..... ....... ..TT..... ........... ......TT..... E82 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 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... ............. E83 2.00000 -1.99860 1.99441 -1.98742 1.97766 -1.96513 1.94986 -1.93185 1.91115 -1.88777 1.86175 -1.83312 1.80194 -1.76823 1.73205 -1.69345 1.65248 -1.60920 1.56366 -1.51594 1.46610 -1.41421 1.36035 -1.30457 1.24698 -1.18764 1.12664 -1.06406 1.00000 -0.93454 0.86777 -0.79978 0.73068 -0.66056 0.58951 -0.51764 0.44504 -0.37182 0.29808 -0.22393 0.14946 -0.07478 0.00000 0.07478 -0.14946 0.22393 -0.29808 0.37182 -0.44504 0.51764 -0.58951 0.66056 -0.73068 0.79978 -0.86777 0.93454 -1.00000 1.06406 -1.12664 1.18764 -1.24698 1.30457 -1.36035 1.41421 -1.46610 1.51594 -1.56366 1.60920 -1.65248 1.69345 -1.73205 1.76823 -1.80194 1.83312 -1.86175 1.88777 -1.91115 1.93185 -1.94986 1.96513 -1.97766 1.98742 -1.99441 1.99860 -2.00000 0.00000 0.00000 TT. ... ..TT. ....... ......TT. ........... ..........TT. Irrational character values: 1.998601409577 = 2*cos(2*π/168) = 2*cos(π/84) 1.994407594362 = 2*cos(4*π/168) = 2*cos(π/42) 1.987424419786 = 2*cos(6*π/168) = 2*cos(π/28) 1.977661652450 = 2*cos(8*π/168) = 2*cos(π/21) 1.965132946467 = 2*cos(10*π/168) = 2*cos(5*π/84) 1.949855824364 = 2*cos(12*π/168) = 2*cos(π/14) 1.931851652578 = 2*cos(14*π/168) = 2*cos(π/12) = √2+√3 1.911145611572 = 2*cos(16*π/168) = 2*cos(2*π/21) 1.887766660617 = 2*cos(18*π/168) = 2*cos(3*π/28) 1.861747497288 = 2*cos(20*π/168) = 2*cos(5*π/42) 1.833124511740 = 2*cos(22*π/168) = 2*cos(11*π/84) 1.801937735805 = 2*cos(24*π/168) = 2*cos(π/7) 1.768230787009 = 2*cos(26*π/168) = 2*cos(13*π/84) 1.732050807569 = 2*cos(28*π/168) = 2*cos(π/6) = √3 1.693448398457 = 2*cos(30*π/168) = 2*cos(5*π/28) 1.652477548632 = 2*cos(32*π/168) = 2*cos(4*π/21) 1.609195559533 = 2*cos(34*π/168) = 2*cos(17*π/84) 1.563662964936 = 2*cos(36*π/168) = 2*cos(3*π/14) 1.515943446291 = 2*cos(38*π/168) = 2*cos(19*π/84) 1.466103743660 = 2*cos(40*π/168) = 2*cos(5*π/21) 1.414213562373 = 2*cos(42*π/168) = 2*cos(π/4) = √2 1.360345475542 = 2*cos(44*π/168) = 2*cos(11*π/42) 1.304574822556 = 2*cos(46*π/168) = 2*cos(23*π/84) 1.246979603717 = 2*cos(48*π/168) = 2*cos(2*π/7) 1.187640371147 = 2*cos(50*π/168) = 2*cos(25*π/84) 1.126640116127 = 2*cos(52*π/168) = 2*cos(13*π/42) 1.064064153031 = 2*cos(54*π/168) = 2*cos(9*π/28) 0.934537256546 = 2*cos(58*π/168) = 2*cos(29*π/84) 0.867767478235 = 2*cos(60*π/168) = 2*cos(5*π/14) 0.799784048639 = 2*cos(62*π/168) = 2*cos(31*π/84) 0.730682048733 = 2*cos(64*π/168) = 2*cos(8*π/21) 0.660558123910 = 2*cos(66*π/168) = 2*cos(11*π/28) 0.589510348822 = 2*cos(68*π/168) = 2*cos(17*π/42) 0.517638090205 = 2*cos(70*π/168) = 2*cos(5*π/12) = √2−√3 0.445041867913 = 2*cos(72*π/168) = 2*cos(3*π/7) 0.371823214326 = 2*cos(74*π/168) = 2*cos(37*π/84) 0.298084532352 = 2*cos(76*π/168) = 2*cos(19*π/42) 0.223928952207 = 2*cos(78*π/168) = 2*cos(13*π/28) 0.149460187173 = 2*cos(80*π/168) = 2*cos(10*π/21) 0.074782388553 = 2*cos(82*π/168) = 2*cos(41*π/84) Symmetry of Rotations and Cartesian products A1 d+g+i+k+m z², z⁴, z⁶ A2 R R_z 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 {x²−y², xy}, {z²(x²−y²), xyz²}, {z⁴(x²−y²), xyz⁴} E3 f+h+j+l {x(x²−3y²), y(3x²−y²)}, {xz²(x²−3y²), yz²(3x²−y²)} E4 g+i+k+m {(x²−y²)²−4x²y², xy(x²−y²)}, {z²((x²−y²)²−4x²y²), xyz²(x²−y²)} E5 h+j+l {x(x²−(5+2√5)y²)(x²−(5−2√5)y²), y((5+2√5)x²−y²)((5−2√5)x²−y²)} E6 i+k+m {x²(x²−3y²)²−y²(3x²−y²)², xy(x²−3y²)(3x²−y²)} E7 j+l E8 k+m E9 l E10 m E75 m E76 l E77 k+m E78 j+l E79 i+k+m {xz(x²−(5+2√5)y²)(x²−(5−2√5)y²), yz((5+2√5)x²−y²)((5−2√5)x²−y²)} E80 h+j+l {z((x²−y²)²−4x²y²), xyz(x²−y²)} E81 g+i+k+m {xz(x²−3y²), yz(3x²−y²)}, {xz³(x²−3y²), yz³(3x²−y²)} E82 f+h+j+l {z(x²−y²), xyz}, {z³(x²−y²), xyz³} E83 R+d+g+i+k+m {R_x, R_y}, {xz, yz}, {xz³, yz³}, {xz⁵, yz⁵} Notes: α The order of the D_84d point group is 336, and the order of the principal axis (S₁₆₈) is 168. The group has 87 irreducible representations. β The D_84d point group is isomorphic to C_168v and D₁₆₈. γ The D_84d point group is generated by two symmetry elements, S₁₆₈ and either a perpendicular C₂^′ or a vertical σ_d. Also, the group may be generated from a C₂^′ plus a σ_d (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 σ_d 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 D_84d 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 D_84d point group was created by Gernot Katzer.

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

	D_82d
	D_83d
C₈₄ C_84v C_84h D₈₄ D_84h	D_84d	S₈₄
	D_85d
	D_86d

Character table for the D84d point group

Character table for the D_84d point group