Character Table for Point Group D112d

Character table for the D_112d point group

D112d E 2 S224 2 C112 2 S224^3 2 C56 2 S224^5 2 C112^3 2 S32 2 C28 2 S224^9 2 C112^5 2 S224^11 2 C56^3 2 S224^13 2 C16 2 S224^15 2 C14 2 S224^17 2 C112^9 2 S224^19 2 C56^5 2 S32^3 2 C112^11 2 S224^23 2 C28^3 2 S224^25 2 C112^13 2 S224^27 2 C8 2 S224^29 2 C112^15 2 S224^31 2 C7 2 S224^33 2 C112^17 2 S32^5 2 C56^9 2 S224^37 2 C112^19 2 S224^39 2 C28^5 2 S224^41 2 C16^3 2 S224^43 2 C56^11 2 S224^45 2 C112^23 2 S224^47 2 C14^3 2 S32^7 2 C112^25 2 S224^51 2 C56^13 2 S224^53 2 C112^27 2 S224^55 2 C4 2 S224^57 2 C112^29 2 S224^59 2 C56^15 2 S224^61 2 C112^31 2 S32^9 2 C7^2 2 S224^65 2 C112^33 2 S224^67 2 C56^17 2 S224^69 2 C16^5 2 S224^71 2 C28^9 2 S224^73 2 C112^37 2 S224^75 2 C56^19 2 S32^11 2 C112^39 2 S224^79 2 C14^5 2 S224^81 2 C112^41 2 S224^83 2 C8^3 2 S224^85 2 C112^43 2 S224^87 2 C28^11 2 S224^89 2 C112^45 2 S32^13 2 C56^23 2 S224^93 2 C112^47 2 S224^95 2 C7^3 2 S224^97 2 C16^7 2 S224^99 2 C56^25 2 S224^101 2 C112^51 2 S224^103 2 C28^13 2 S32^15 2 C112^53 2 S224^107 2 C56^27 2 S224^109 2 C112^55 2 S224^111 C2 112 C2' 112 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 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 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 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 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 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -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 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -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 1.99921 1.99685 1.99292 1.98742 1.98036 1.97174 1.96157 1.94986 1.93661 1.92183 1.90555 1.88777 1.86850 1.84776 1.82557 1.80194 1.77689 1.75045 1.72263 1.69345 1.66294 1.63112 1.59802 1.56366 1.52807 1.49128 1.45332 1.41421 1.37399 1.33269 1.29034 1.24698 1.20263 1.15734 1.11114 1.06406 1.01615 0.96744 0.91796 0.86777 0.81689 0.76537 0.71324 0.66056 0.60735 0.55367 0.49955 0.44504 0.39018 0.33501 0.27958 0.22393 0.16810 0.11214 0.05609 0.00000 -0.05609 -0.11214 -0.16810 -0.22393 -0.27958 -0.33501 -0.39018 -0.44504 -0.49955 -0.55367 -0.60735 -0.66056 -0.71324 -0.76537 -0.81689 -0.86777 -0.91796 -0.96744 -1.01615 -1.06406 -1.11114 -1.15734 -1.20263 -1.24698 -1.29034 -1.33269 -1.37399 -1.41421 -1.45332 -1.49128 -1.52807 -1.56366 -1.59802 -1.63112 -1.66294 -1.69345 -1.72263 -1.75045 -1.77689 -1.80194 -1.82557 -1.84776 -1.86850 -1.88777 -1.90555 -1.92183 -1.93661 -1.94986 -1.96157 -1.97174 -1.98036 -1.98742 -1.99292 -1.99685 -1.99921 -2.00000 0.00000 0.00000 ... TT. ..... ....TT. ......... ........TT. ............. E2 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 -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... E3 2.00000 1.99292 1.97174 1.93661 1.88777 1.82557 1.75045 1.66294 1.56366 1.45332 1.33269 1.20263 1.06406 0.91796 0.76537 0.60735 0.44504 0.27958 0.11214 -0.05609 -0.22393 -0.39018 -0.55367 -0.71324 -0.86777 -1.01615 -1.15734 -1.29034 -1.41421 -1.52807 -1.63112 -1.72263 -1.80194 -1.86850 -1.92183 -1.96157 -1.98742 -1.99921 -1.99685 -1.98036 -1.94986 -1.90555 -1.84776 -1.77689 -1.69345 -1.59802 -1.49128 -1.37399 -1.24698 -1.11114 -0.96744 -0.81689 -0.66056 -0.49955 -0.33501 -0.16810 0.00000 0.16810 0.33501 0.49955 0.66056 0.81689 0.96744 1.11114 1.24698 1.37399 1.49128 1.59802 1.69345 1.77689 1.84776 1.90555 1.94986 1.98036 1.99685 1.99921 1.98742 1.96157 1.92183 1.86850 1.80194 1.72263 1.63112 1.52807 1.41421 1.29034 1.15734 1.01615 0.86777 0.71324 0.55367 0.39018 0.22393 0.05609 -0.11214 -0.27958 -0.44504 -0.60735 -0.76537 -0.91796 -1.06406 -1.20263 -1.33269 -1.45332 -1.56366 -1.66294 -1.75045 -1.82557 -1.88777 -1.93661 -1.97174 -1.99292 -2.00000 0.00000 0.00000 ... ... ..... TT..... ......... ....TT..... ............. E4 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 -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....... E5 2.00000 1.98036 1.92183 1.82557 1.69345 1.52807 1.33269 1.11114 0.86777 0.60735 0.33501 0.05609 -0.22393 -0.49955 -0.76537 -1.01615 -1.24698 -1.45332 -1.63112 -1.77689 -1.88777 -1.96157 -1.99685 -1.99292 -1.94986 -1.86850 -1.75045 -1.59802 -1.41421 -1.20263 -0.96744 -0.71324 -0.44504 -0.16810 0.11214 0.39018 0.66056 0.91796 1.15734 1.37399 1.56366 1.72263 1.84776 1.93661 1.98742 1.99921 1.97174 1.90555 1.80194 1.66294 1.49128 1.29034 1.06406 0.81689 0.55367 0.27958 0.00000 -0.27958 -0.55367 -0.81689 -1.06406 -1.29034 -1.49128 -1.66294 -1.80194 -1.90555 -1.97174 -1.99921 -1.98742 -1.93661 -1.84776 -1.72263 -1.56366 -1.37399 -1.15734 -0.91796 -0.66056 -0.39018 -0.11214 0.16810 0.44504 0.71324 0.96744 1.20263 1.41421 1.59802 1.75045 1.86850 1.94986 1.99292 1.99685 1.96157 1.88777 1.77689 1.63112 1.45332 1.24698 1.01615 0.76537 0.49955 0.22393 -0.05609 -0.33501 -0.60735 -0.86777 -1.11114 -1.33269 -1.52807 -1.69345 -1.82557 -1.92183 -1.98036 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... TT......... ............. E6 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 -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........... E7 2.00000 1.96157 1.84776 1.66294 1.41421 1.11114 0.76537 0.39018 0.00000 -0.39018 -0.76537 -1.11114 -1.41421 -1.66294 -1.84776 -1.96157 -2.00000 -1.96157 -1.84776 -1.66294 -1.41421 -1.11114 -0.76537 -0.39018 0.00000 0.39018 0.76537 1.11114 1.41421 1.66294 1.84776 1.96157 2.00000 1.96157 1.84776 1.66294 1.41421 1.11114 0.76537 0.39018 0.00000 -0.39018 -0.76537 -1.11114 -1.41421 -1.66294 -1.84776 -1.96157 -2.00000 -1.96157 -1.84776 -1.66294 -1.41421 -1.11114 -0.76537 -0.39018 0.00000 0.39018 0.76537 1.11114 1.41421 1.66294 1.84776 1.96157 2.00000 1.96157 1.84776 1.66294 1.41421 1.11114 0.76537 0.39018 0.00000 -0.39018 -0.76537 -1.11114 -1.41421 -1.66294 -1.84776 -1.96157 -2.00000 -1.96157 -1.84776 -1.66294 -1.41421 -1.11114 -0.76537 -0.39018 0.00000 0.39018 0.76537 1.11114 1.41421 1.66294 1.84776 1.96157 2.00000 1.96157 1.84776 1.66294 1.41421 1.11114 0.76537 0.39018 0.00000 -0.39018 -0.76537 -1.11114 -1.41421 -1.66294 -1.84776 -1.96157 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E8 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 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 ... ... ..... ....... ......... ........... ............. E9 2.00000 1.93661 1.75045 1.45332 1.06406 0.60735 0.11214 -0.39018 -0.86777 -1.29034 -1.63112 -1.86850 -1.98742 -1.98036 -1.84776 -1.59802 -1.24698 -0.81689 -0.33501 0.16810 0.66056 1.11114 1.49128 1.77689 1.94986 1.99921 1.92183 1.72263 1.41421 1.01615 0.55367 0.05609 -0.44504 -0.91796 -1.33269 -1.66294 -1.88777 -1.99292 -1.97174 -1.82557 -1.56366 -1.20263 -0.76537 -0.27958 0.22393 0.71324 1.15734 1.52807 1.80194 1.96157 1.99685 1.90555 1.69345 1.37399 0.96744 0.49955 0.00000 -0.49955 -0.96744 -1.37399 -1.69345 -1.90555 -1.99685 -1.96157 -1.80194 -1.52807 -1.15734 -0.71324 -0.22393 0.27958 0.76537 1.20263 1.56366 1.82557 1.97174 1.99292 1.88777 1.66294 1.33269 0.91796 0.44504 -0.05609 -0.55367 -1.01615 -1.41421 -1.72263 -1.92183 -1.99921 -1.94986 -1.77689 -1.49128 -1.11114 -0.66056 -0.16810 0.33501 0.81689 1.24698 1.59802 1.84776 1.98036 1.98742 1.86850 1.63112 1.29034 0.86777 0.39018 -0.11214 -0.60735 -1.06406 -1.45332 -1.75045 -1.93661 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. 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E110 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 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... ............. E111 2.00000 -1.99921 1.99685 -1.99292 1.98742 -1.98036 1.97174 -1.96157 1.94986 -1.93661 1.92183 -1.90555 1.88777 -1.86850 1.84776 -1.82557 1.80194 -1.77689 1.75045 -1.72263 1.69345 -1.66294 1.63112 -1.59802 1.56366 -1.52807 1.49128 -1.45332 1.41421 -1.37399 1.33269 -1.29034 1.24698 -1.20263 1.15734 -1.11114 1.06406 -1.01615 0.96744 -0.91796 0.86777 -0.81689 0.76537 -0.71324 0.66056 -0.60735 0.55367 -0.49955 0.44504 -0.39018 0.33501 -0.27958 0.22393 -0.16810 0.11214 -0.05609 0.00000 0.05609 -0.11214 0.16810 -0.22393 0.27958 -0.33501 0.39018 -0.44504 0.49955 -0.55367 0.60735 -0.66056 0.71324 -0.76537 0.81689 -0.86777 0.91796 -0.96744 1.01615 -1.06406 1.11114 -1.15734 1.20263 -1.24698 1.29034 -1.33269 1.37399 -1.41421 1.45332 -1.49128 1.52807 -1.56366 1.59802 -1.63112 1.66294 -1.69345 1.72263 -1.75045 1.77689 -1.80194 1.82557 -1.84776 1.86850 -1.88777 1.90555 -1.92183 1.93661 -1.94986 1.96157 -1.97174 1.98036 -1.98742 1.99292 -1.99685 1.99921 -2.00000 0.00000 0.00000 TT. ... ..TT. ....... ......TT. ........... ..........TT. Irrational character values: 1.999213252766 = 2*cos(2*π/224) = 2*cos(π/112) 1.996853630036 = 2*cos(4*π/224) = 2*cos(π/56) 1.992922988235 = 2*cos(6*π/224) = 2*cos(3*π/112) 1.987424419786 = 2*cos(8*π/224) = 2*cos(π/28) 1.980362250673 = 2*cos(10*π/224) = 2*cos(5*π/112) 1.971742037036 = 2*cos(12*π/224) = 2*cos(3*π/56) 1.961570560806 = 2*cos(14*π/224) = 2*cos(π/16) = √2+√2+√2 1.949855824364 = 2*cos(16*π/224) = 2*cos(π/14) 1.936607044245 = 2*cos(18*π/224) = 2*cos(9*π/112) 1.921834643890 = 2*cos(20*π/224) = 2*cos(5*π/56) 1.905550245446 = 2*cos(22*π/224) = 2*cos(11*π/112) 1.887766660617 = 2*cos(24*π/224) = 2*cos(3*π/28) 1.868497880589 = 2*cos(26*π/224) = 2*cos(13*π/112) 1.847759065023 = 2*cos(28*π/224) = 2*cos(π/8) = √2+√2 1.825566530123 = 2*cos(30*π/224) = 2*cos(15*π/112) 1.801937735805 = 2*cos(32*π/224) = 2*cos(π/7) 1.776891271958 = 2*cos(34*π/224) = 2*cos(17*π/112) 1.750446843818 = 2*cos(36*π/224) = 2*cos(9*π/56) 1.722625256465 = 2*cos(38*π/224) = 2*cos(19*π/112) 1.693448398457 = 2*cos(40*π/224) = 2*cos(5*π/28) 1.662939224605 = 2*cos(42*π/224) = 2*cos(3*π/16) = √2+√2−√2 1.631121737919 = 2*cos(44*π/224) = 2*cos(11*π/56) 1.598020970716 = 2*cos(46*π/224) = 2*cos(23*π/112) 1.563662964936 = 2*cos(48*π/224) = 2*cos(3*π/14) 1.528074751643 = 2*cos(50*π/224) = 2*cos(25*π/112) 1.491284329766 = 2*cos(52*π/224) = 2*cos(13*π/56) 1.453320644068 = 2*cos(54*π/224) = 2*cos(27*π/112) 1.414213562373 = 2*cos(56*π/224) = 2*cos(π/4) = √2 1.373993852070 = 2*cos(58*π/224) = 2*cos(29*π/112) 1.332693155904 = 2*cos(60*π/224) = 2*cos(15*π/56) 1.290343967084 = 2*cos(62*π/224) = 2*cos(31*π/112) 1.246979603717 = 2*cos(64*π/224) = 2*cos(2*π/7) 1.202634182597 = 2*cos(66*π/224) = 2*cos(33*π/112) 1.157342592360 = 2*cos(68*π/224) = 2*cos(17*π/56) 1.111140466039 = 2*cos(70*π/224) = 2*cos(5*π/16) = √2−√2−√2 1.064064153031 = 2*cos(72*π/224) = 2*cos(9*π/28) 1.016150690493 = 2*cos(74*π/224) = 2*cos(37*π/112) 0.967437774210 = 2*cos(76*π/224) = 2*cos(19*π/56) 0.917963728935 = 2*cos(78*π/224) = 2*cos(39*π/112) 0.867767478235 = 2*cos(80*π/224) = 2*cos(5*π/14) 0.816888513872 = 2*cos(82*π/224) = 2*cos(41*π/112) 0.765366864730 = 2*cos(84*π/224) = 2*cos(3*π/8) = √2−√2 0.713243065325 = 2*cos(86*π/224) = 2*cos(43*π/112) 0.660558123910 = 2*cos(88*π/224) = 2*cos(11*π/28) 0.607353490219 = 2*cos(90*π/224) = 2*cos(45*π/112) 0.553671022850 = 2*cos(92*π/224) = 2*cos(23*π/56) 0.499552956334 = 2*cos(94*π/224) = 2*cos(47*π/112) 0.445041867913 = 2*cos(96*π/224) = 2*cos(3*π/7) 0.390180644032 = 2*cos(98*π/224) = 2*cos(7*π/16) = √2−√2+√2 0.335012446609 = 2*cos(100*π/224) = 2*cos(25*π/56) 0.279580679071 = 2*cos(102*π/224) = 2*cos(51*π/112) 0.223928952207 = 2*cos(104*π/224) = 2*cos(13*π/28) 0.168101049858 = 2*cos(106*π/224) = 2*cos(53*π/112) 0.112140894474 = 2*cos(108*π/224) = 2*cos(27*π/56) 0.056092512552 = 2*cos(110*π/224) = 2*cos(55*π/112) 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 E103 m E104 l E105 k+m E106 j+l E107 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)} E108 h+j+l {z((x²−y²)²−4x²y²), xyz(x²−y²)} E109 g+i+k+m {xz(x²−3y²), yz(3x²−y²)}, {xz³(x²−3y²), yz³(3x²−y²)} E110 f+h+j+l {z(x²−y²), xyz}, {z³(x²−y²), xyz³} E111 R+d+g+i+k+m {R_x, R_y}, {xz, yz}, {xz³, yz³}, {xz⁵, yz⁵} Notes: α The order of the D_112d point group is 448, and the order of the principal axis (S₂₂₄) is 224. The group has 115 irreducible representations. β The D_112d point group is isomorphic to C_224v and D₂₂₄. γ The D_112d point group is generated by two symmetry elements, S₂₂₄ 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 lowest nonvanishing multipole moment in D_112d is 4 (quadrupole moment). ε This point group is non-Abelian (some symmetry operations are not commutative). Therefore, the character table contains multi-membered classes and degenerate irreducible representations. ζ 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. η 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. θ 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_112d point group was created by Gernot Katzer.

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

	D_110d
	D_111d
C₁₁₂ C_112v C_112h D₁₁₂ D_112h	D_112d	S₁₁₂
	D_113d
	D_114d

Character table for the D112d point group

Character table for the D_112d point group