Character Table for Point Group D56d

Character table for the D_56d point group

D56d E 2 S112 2 C56 2 S112^3 2 C28 2 S112^5 2 C56^3 2 S16 2 C14 2 S112^9 2 C56^5 2 S112^11 2 C28^3 2 S112^13 2 C8 2 S112^15 2 C7 2 S112^17 2 C56^9 2 S112^19 2 C28^5 2 S16^3 2 C56^11 2 S112^23 2 C14^3 2 S112^25 2 C56^13 2 S112^27 2 C4 2 S112^29 2 C56^15 2 S112^31 2 C7^2 2 S112^33 2 C56^17 2 S16^5 2 C28^9 2 S112^37 2 C56^19 2 S112^39 2 C14^5 2 S112^41 2 C8^3 2 S112^43 2 C28^11 2 S112^45 2 C56^23 2 S112^47 2 C7^3 2 S16^7 2 C56^25 2 S112^51 2 C28^13 2 S112^53 2 C56^27 2 S112^55 C2 56 C2' 56 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 ... ... ....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 ..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 ... ... ..... ....... ......... ........... ............. 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 ... ..T ..... ......T ......... ..........T ............. E1 2.00000 1.99685 1.98742 1.97174 1.94986 1.92183 1.88777 1.84776 1.80194 1.75045 1.69345 1.63112 1.56366 1.49128 1.41421 1.33269 1.24698 1.15734 1.06406 0.96744 0.86777 0.76537 0.66056 0.55367 0.44504 0.33501 0.22393 0.11214 0.00000 -0.11214 -0.22393 -0.33501 -0.44504 -0.55367 -0.66056 -0.76537 -0.86777 -0.96744 -1.06406 -1.15734 -1.24698 -1.33269 -1.41421 -1.49128 -1.56366 -1.63112 -1.69345 -1.75045 -1.80194 -1.84776 -1.88777 -1.92183 -1.94986 -1.97174 -1.98742 -1.99685 -2.00000 0.00000 0.00000 ... TT. ..... ....TT. ......... ........TT. ............. E2 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... ........... ........TT... E3 2.00000 1.97174 1.88777 1.75045 1.56366 1.33269 1.06406 0.76537 0.44504 0.11214 -0.22393 -0.55367 -0.86777 -1.15734 -1.41421 -1.63112 -1.80194 -1.92183 -1.98742 -1.99685 -1.94986 -1.84776 -1.69345 -1.49128 -1.24698 -0.96744 -0.66056 -0.33501 0.00000 0.33501 0.66056 0.96744 1.24698 1.49128 1.69345 1.84776 1.94986 1.99685 1.98742 1.92183 1.80194 1.63112 1.41421 1.15734 0.86777 0.55367 0.22393 -0.11214 -0.44504 -0.76537 -1.06406 -1.33269 -1.56366 -1.75045 -1.88777 -1.97174 -2.00000 0.00000 0.00000 ... ... ..... TT..... ......... ....TT..... ............. E4 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....... ........... ....TT....... E5 2.00000 1.92183 1.69345 1.33269 0.86777 0.33501 -0.22393 -0.76537 -1.24698 -1.63112 -1.88777 -1.99685 -1.94986 -1.75045 -1.41421 -0.96744 -0.44504 0.11214 0.66056 1.15734 1.56366 1.84776 1.98742 1.97174 1.80194 1.49128 1.06406 0.55367 0.00000 -0.55367 -1.06406 -1.49128 -1.80194 -1.97174 -1.98742 -1.84776 -1.56366 -1.15734 -0.66056 -0.11214 0.44504 0.96744 1.41421 1.75045 1.94986 1.99685 1.88777 1.63112 1.24698 0.76537 0.22393 -0.33501 -0.86777 -1.33269 -1.69345 -1.92183 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... TT......... ............. E6 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 ... ... ..... ....... ......... ........... TT........... E7 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 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 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 1.84776 1.41421 0.76537 0.00000 -0.76537 -1.41421 -1.84776 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E8 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 ... ... ..... ....... ......... ........... ............. E9 2.00000 1.75045 1.06406 0.11214 -0.86777 -1.63112 -1.98742 -1.84776 -1.24698 -0.33501 0.66056 1.49128 1.94986 1.92183 1.41421 0.55367 -0.44504 -1.33269 -1.88777 -1.97174 -1.56366 -0.76537 0.22393 1.15734 1.80194 1.99685 1.69345 0.96744 0.00000 -0.96744 -1.69345 -1.99685 -1.80194 -1.15734 -0.22393 0.76537 1.56366 1.97174 1.88777 1.33269 0.44504 -0.55367 -1.41421 -1.92183 -1.94986 -1.49128 -0.66056 0.33501 1.24698 1.84776 1.98742 1.63112 0.86777 -0.11214 -1.06406 -1.75045 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E10 2.00000 1.69345 0.86777 -0.22393 -1.24698 -1.88777 -1.94986 -1.41421 -0.44504 0.66056 1.56366 1.98742 1.80194 1.06406 0.00000 -1.06406 -1.80194 -1.98742 -1.56366 -0.66056 0.44504 1.41421 1.94986 1.88777 1.24698 0.22393 -0.86777 -1.69345 -2.00000 -1.69345 -0.86777 0.22393 1.24698 1.88777 1.94986 1.41421 0.44504 -0.66056 -1.56366 -1.98742 -1.80194 -1.06406 0.00000 1.06406 1.80194 1.98742 1.56366 0.66056 -0.44504 -1.41421 -1.94986 -1.88777 -1.24698 -0.22393 0.86777 1.69345 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E11 2.00000 1.63112 0.66056 -0.55367 -1.56366 -1.99685 -1.69345 -0.76537 0.44504 1.49128 1.98742 1.75045 0.86777 -0.33501 -1.41421 -1.97174 -1.80194 -0.96744 0.22393 1.33269 1.94986 1.84776 1.06406 -0.11214 -1.24698 -1.92183 -1.88777 -1.15734 0.00000 1.15734 1.88777 1.92183 1.24698 0.11214 -1.06406 -1.84776 -1.94986 -1.33269 -0.22393 0.96744 1.80194 1.97174 1.41421 0.33501 -0.86777 -1.75045 -1.98742 -1.49128 -0.44504 0.76537 1.69345 1.99685 1.56366 0.55367 -0.66056 -1.63112 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E12 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 -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 ... ... ..... ....... ......... ........... ............. E13 2.00000 1.49128 0.22393 -1.15734 -1.94986 -1.75045 -0.66056 0.76537 1.80194 1.92183 1.06406 -0.33501 -1.56366 -1.99685 -1.41421 -0.11214 1.24698 1.97174 1.69345 0.55367 -0.86777 -1.84776 -1.88777 -0.96744 0.44504 1.63112 1.98742 1.33269 0.00000 -1.33269 -1.98742 -1.63112 -0.44504 0.96744 1.88777 1.84776 0.86777 -0.55367 -1.69345 -1.97174 -1.24698 0.11214 1.41421 1.99685 1.56366 0.33501 -1.06406 -1.92183 -1.80194 -0.76537 0.66056 1.75045 1.94986 1.15734 -0.22393 -1.49128 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E14 2.00000 1.41421 0.00000 -1.41421 -2.00000 -1.41421 0.00000 1.41421 2.00000 1.41421 0.00000 -1.41421 -2.00000 -1.41421 0.00000 1.41421 2.00000 1.41421 0.00000 -1.41421 -2.00000 -1.41421 0.00000 1.41421 2.00000 1.41421 0.00000 -1.41421 -2.00000 -1.41421 0.00000 1.41421 2.00000 1.41421 0.00000 -1.41421 -2.00000 -1.41421 0.00000 1.41421 2.00000 1.41421 0.00000 -1.41421 -2.00000 -1.41421 0.00000 1.41421 2.00000 1.41421 0.00000 -1.41421 -2.00000 -1.41421 0.00000 1.41421 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E15 2.00000 1.33269 -0.22393 -1.63112 -1.94986 -0.96744 0.66056 1.84776 1.80194 0.55367 -1.06406 -1.97174 -1.56366 -0.11214 1.41421 1.99685 1.24698 -0.33501 -1.69345 -1.92183 -0.86777 0.76537 1.88777 1.75045 0.44504 -1.15734 -1.98742 -1.49128 0.00000 1.49128 1.98742 1.15734 -0.44504 -1.75045 -1.88777 -0.76537 0.86777 1.92183 1.69345 0.33501 -1.24698 -1.99685 -1.41421 0.11214 1.56366 1.97174 1.06406 -0.55367 -1.80194 -1.84776 -0.66056 0.96744 1.94986 1.63112 0.22393 -1.33269 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E16 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 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 ... ... ..... ....... ......... ........... ............. E17 2.00000 1.15734 -0.66056 -1.92183 -1.56366 0.11214 1.69345 1.84776 0.44504 -1.33269 -1.98742 -0.96744 0.86777 1.97174 1.41421 -0.33501 -1.80194 -1.75045 -0.22393 1.49128 1.94986 0.76537 -1.06406 -1.99685 -1.24698 0.55367 1.88777 1.63112 0.00000 -1.63112 -1.88777 -0.55367 1.24698 1.99685 1.06406 -0.76537 -1.94986 -1.49128 0.22393 1.75045 1.80194 0.33501 -1.41421 -1.97174 -0.86777 0.96744 1.98742 1.33269 -0.44504 -1.84776 -1.69345 -0.11214 1.56366 1.92183 0.66056 -1.15734 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E18 2.00000 1.06406 -0.86777 -1.98742 -1.24698 0.66056 1.94986 1.41421 -0.44504 -1.88777 -1.56366 0.22393 1.80194 1.69345 0.00000 -1.69345 -1.80194 -0.22393 1.56366 1.88777 0.44504 -1.41421 -1.94986 -0.66056 1.24698 1.98742 0.86777 -1.06406 -2.00000 -1.06406 0.86777 1.98742 1.24698 -0.66056 -1.94986 -1.41421 0.44504 1.88777 1.56366 -0.22393 -1.80194 -1.69345 0.00000 1.69345 1.80194 0.22393 -1.56366 -1.88777 -0.44504 1.41421 1.94986 0.66056 -1.24698 -1.98742 -0.86777 1.06406 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E19 2.00000 0.96744 -1.06406 -1.99685 -0.86777 1.15734 1.98742 0.76537 -1.24698 -1.97174 -0.66056 1.33269 1.94986 0.55367 -1.41421 -1.92183 -0.44504 1.49128 1.88777 0.33501 -1.56366 -1.84776 -0.22393 1.63112 1.80194 0.11214 -1.69345 -1.75045 0.00000 1.75045 1.69345 -0.11214 -1.80194 -1.63112 0.22393 1.84776 1.56366 -0.33501 -1.88777 -1.49128 0.44504 1.92183 1.41421 -0.55367 -1.94986 -1.33269 0.66056 1.97174 1.24698 -0.76537 -1.98742 -1.15734 0.86777 1.99685 1.06406 -0.96744 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E20 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 -0.86777 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 ... ... ..... ....... ......... ........... ............. E21 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 0.76537 -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 0.76537 -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 0.76537 -1.41421 -1.84776 0.00000 1.84776 1.41421 -0.76537 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E22 2.00000 0.66056 -1.56366 -1.69345 0.44504 1.98742 0.86777 -1.41421 -1.80194 0.22393 1.94986 1.06406 -1.24698 -1.88777 0.00000 1.88777 1.24698 -1.06406 -1.94986 -0.22393 1.80194 1.41421 -0.86777 -1.98742 -0.44504 1.69345 1.56366 -0.66056 -2.00000 -0.66056 1.56366 1.69345 -0.44504 -1.98742 -0.86777 1.41421 1.80194 -0.22393 -1.94986 -1.06406 1.24698 1.88777 0.00000 -1.88777 -1.24698 1.06406 1.94986 0.22393 -1.80194 -1.41421 0.86777 1.98742 0.44504 -1.69345 -1.56366 0.66056 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E23 2.00000 0.55367 -1.69345 -1.49128 0.86777 1.97174 0.22393 -1.84776 -1.24698 1.15734 1.88777 -0.11214 -1.94986 -0.96744 1.41421 1.75045 -0.44504 -1.99685 -0.66056 1.63112 1.56366 -0.76537 -1.98742 -0.33501 1.80194 1.33269 -1.06406 -1.92183 0.00000 1.92183 1.06406 -1.33269 -1.80194 0.33501 1.98742 0.76537 -1.56366 -1.63112 0.66056 1.99685 0.44504 -1.75045 -1.41421 0.96744 1.94986 0.11214 -1.88777 -1.15734 1.24698 1.84776 -0.22393 -1.97174 -0.86777 1.49128 1.69345 -0.55367 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. 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E53 2.00000 -1.97174 1.88777 -1.75045 1.56366 -1.33269 1.06406 -0.76537 0.44504 -0.11214 -0.22393 0.55367 -0.86777 1.15734 -1.41421 1.63112 -1.80194 1.92183 -1.98742 1.99685 -1.94986 1.84776 -1.69345 1.49128 -1.24698 0.96744 -0.66056 0.33501 0.00000 -0.33501 0.66056 -0.96744 1.24698 -1.49128 1.69345 -1.84776 1.94986 -1.99685 1.98742 -1.92183 1.80194 -1.63112 1.41421 -1.15734 0.86777 -0.55367 0.22393 0.11214 -0.44504 0.76537 -1.06406 1.33269 -1.56366 1.75045 -1.88777 1.97174 -2.00000 0.00000 0.00000 ... ... ..... ....... ..TT..... ........... ......TT..... E54 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... ............. E55 2.00000 -1.99685 1.98742 -1.97174 1.94986 -1.92183 1.88777 -1.84776 1.80194 -1.75045 1.69345 -1.63112 1.56366 -1.49128 1.41421 -1.33269 1.24698 -1.15734 1.06406 -0.96744 0.86777 -0.76537 0.66056 -0.55367 0.44504 -0.33501 0.22393 -0.11214 0.00000 0.11214 -0.22393 0.33501 -0.44504 0.55367 -0.66056 0.76537 -0.86777 0.96744 -1.06406 1.15734 -1.24698 1.33269 -1.41421 1.49128 -1.56366 1.63112 -1.69345 1.75045 -1.80194 1.84776 -1.88777 1.92183 -1.94986 1.97174 -1.98742 1.99685 -2.00000 0.00000 0.00000 TT. ... ..TT. ....... ......TT. ........... ..........TT. Irrational character values: 1.996853630036 = 2*cos(2*π/112) = 2*cos(π/56) 1.987424419786 = 2*cos(4*π/112) = 2*cos(π/28) 1.971742037036 = 2*cos(6*π/112) = 2*cos(3*π/56) 1.949855824364 = 2*cos(8*π/112) = 2*cos(π/14) 1.921834643890 = 2*cos(10*π/112) = 2*cos(5*π/56) 1.887766660617 = 2*cos(12*π/112) = 2*cos(3*π/28) 1.847759065023 = 2*cos(14*π/112) = 2*cos(π/8) = √2+√2 1.801937735805 = 2*cos(16*π/112) = 2*cos(π/7) 1.750446843818 = 2*cos(18*π/112) = 2*cos(9*π/56) 1.693448398457 = 2*cos(20*π/112) = 2*cos(5*π/28) 1.631121737919 = 2*cos(22*π/112) = 2*cos(11*π/56) 1.563662964936 = 2*cos(24*π/112) = 2*cos(3*π/14) 1.491284329766 = 2*cos(26*π/112) = 2*cos(13*π/56) 1.414213562373 = 2*cos(28*π/112) = 2*cos(π/4) = √2 1.332693155904 = 2*cos(30*π/112) = 2*cos(15*π/56) 1.246979603717 = 2*cos(32*π/112) = 2*cos(2*π/7) 1.157342592360 = 2*cos(34*π/112) = 2*cos(17*π/56) 1.064064153031 = 2*cos(36*π/112) = 2*cos(9*π/28) 0.967437774210 = 2*cos(38*π/112) = 2*cos(19*π/56) 0.867767478235 = 2*cos(40*π/112) = 2*cos(5*π/14) 0.765366864730 = 2*cos(42*π/112) = 2*cos(3*π/8) = √2−√2 0.660558123910 = 2*cos(44*π/112) = 2*cos(11*π/28) 0.553671022850 = 2*cos(46*π/112) = 2*cos(23*π/56) 0.445041867913 = 2*cos(48*π/112) = 2*cos(3*π/7) 0.335012446609 = 2*cos(50*π/112) = 2*cos(25*π/56) 0.223928952207 = 2*cos(52*π/112) = 2*cos(13*π/28) 0.112140894474 = 2*cos(54*π/112) = 2*cos(27*π/56) 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 E47 m E48 l E49 k+m E50 j+l E51 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)} E52 h+j+l {z((x²−y²)²−4x²y²), xyz(x²−y²)} E53 g+i+k+m {xz(x²−3y²), yz(3x²−y²)}, {xz³(x²−3y²), yz³(3x²−y²)} E54 f+h+j+l {z(x²−y²), xyz}, {z³(x²−y²), xyz³} E55 R+d+g+i+k+m {R_x, R_y}, {xz, yz}, {xz³, yz³}, {xz⁵, yz⁵} Notes: α The order of the D_56d point group is 224, and the order of the principal axis (S₁₁₂) is 112. The group has 59 irreducible representations. β The D_56d point group is isomorphic to C_112v and D₁₁₂. γ The D_56d point group is generated by two symmetry elements, S₁₁₂ 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_56d 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. κ The regular heptagon is the lowest regular polygon that cannot be constructed by compass and ruler alone, which has mystified mathematicians since antiquity. The reason for the inconstructibility of the heptagon is that cos(2π/7) has an algebraic degree of 3 and thus can not be represented by square roots and integer numbers. An algebraic representation becomes possible if cubic roots and complex numbers are used: 2*cos(2π/7) = (³√28+i*84*√3 + ³√28−i*84*√3 − 2)/6. This complex expression for a real value is hardly useful. Therefore, regular polygons of order 14,21,28,35,42 etc. are also inconstructible, and their cosines have no representation in real radicals.

This Character Table for the D_56d point group was created by Gernot Katzer.

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

	D_54d
	D_55d
C₅₆ C_56v C_56h D₅₆ D_56h	D_56d	S₅₆
	D_57d
	D_58d

Character table for the D56d point group

Character table for the D_56d point group