Character Table for Point Group D126d

Character table for the D_126d point group

D126d E 2 S252 2 C126 2 S84 2 C63 2 S252^5 2 C42 2 S36 2 C63^2 2 S28 2 C126^5 2 S252^11 2 C21 2 S252^13 2 C18 2 S84^5 2 C63^4 2 S252^17 2 C14 2 S252^19 2 C63^5 2 S12 2 C126^11 2 S252^23 2 C21^2 2 S252^25 2 C126^13 2 S28^3 2 C9 2 S252^29 2 C42^5 2 S252^31 2 C63^8 2 S84^11 2 C126^17 2 S36^5 2 C7 2 S252^37 2 C126^19 2 S84^13 2 C63^10 2 S252^41 2 C6 2 S252^43 2 C63^11 2 S28^5 2 C126^23 2 S252^47 2 C21^4 2 S36^7 2 C126^25 2 S84^17 2 C63^13 2 S252^53 2 C14^3 2 S252^55 2 C9^2 2 S84^19 2 C126^29 2 S252^59 2 C21^5 2 S252^61 2 C126^31 2 S4 2 C63^16 2 S252^65 2 C42^11 2 S252^67 2 C63^17 2 S84^23 2 C18^5 2 S252^71 2 C7^2 2 S252^73 2 C126^37 2 S84^25 2 C63^19 2 S36^11 2 C42^13 2 S252^79 2 C63^20 2 S28^9 2 C126^41 2 S252^83 2 C3 2 S252^85 2 C126^43 2 S84^29 2 C63^22 2 S252^89 2 C14^5 2 S36^13 2 C63^23 2 S84^31 2 C126^47 2 S252^95 2 C21^8 2 S252^97 2 C18^7 2 S28^11 2 C63^25 2 S252^101 2 C42^17 2 S252^103 2 C63^26 2 S12^5 2 C126^53 2 S252^107 2 C7^3 2 S252^109 2 C126^55 2 S84^37 2 C9^4 2 S252^113 2 C42^19 2 S252^115 2 C63^29 2 S28^13 2 C126^59 2 S36^17 2 C21^10 2 S252^121 2 C126^61 2 S84^41 2 C63^31 2 S252^125 C2 126 C2' 126 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 1.00000 1.00000 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 1.00000 1.00000 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 -1.00000 1.00000 -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 -1.00000 1.00000 -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 1.99938 1.99751 1.99441 1.99006 1.98448 1.97766 1.96962 1.96034 1.94986 1.93815 1.92525 1.91115 1.89585 1.87939 1.86175 1.84295 1.82301 1.80194 1.77974 1.75644 1.73205 1.70658 1.68005 1.65248 1.62388 1.59427 1.56366 1.53209 1.49956 1.46610 1.43173 1.39647 1.36035 1.32337 1.28558 1.24698 1.20761 1.16749 1.12664 1.08509 1.04287 1.00000 0.95651 0.91242 0.86777 0.82257 0.77687 0.73068 0.68404 0.63697 0.58951 0.54168 0.49351 0.44504 0.39629 0.34730 0.29808 0.24869 0.19914 0.14946 0.09969 0.04986 0.00000 -0.04986 -0.09969 -0.14946 -0.19914 -0.24869 -0.29808 -0.34730 -0.39629 -0.44504 -0.49351 -0.54168 -0.58951 -0.63697 -0.68404 -0.73068 -0.77687 -0.82257 -0.86777 -0.91242 -0.95651 -1.00000 -1.04287 -1.08509 -1.12664 -1.16749 -1.20761 -1.24698 -1.28558 -1.32337 -1.36035 -1.39647 -1.43173 -1.46610 -1.49956 -1.53209 -1.56366 -1.59427 -1.62388 -1.65248 -1.68005 -1.70658 -1.73205 -1.75644 -1.77974 -1.80194 -1.82301 -1.84295 -1.86175 -1.87939 -1.89585 -1.91115 -1.92525 -1.93815 -1.94986 -1.96034 -1.96962 -1.97766 -1.98448 -1.99006 -1.99441 -1.99751 -1.99938 -2.00000 0.00000 0.00000 ... TT. ..... ....TT. ......... ........TT. ............. E2 2.00000 1.99751 1.99006 1.97766 1.96034 1.93815 1.91115 1.87939 1.84295 1.80194 1.75644 1.70658 1.65248 1.59427 1.53209 1.46610 1.39647 1.32337 1.24698 1.16749 1.08509 1.00000 0.91242 0.82257 0.73068 0.63697 0.54168 0.44504 0.34730 0.24869 0.14946 0.04986 -0.04986 -0.14946 -0.24869 -0.34730 -0.44504 -0.54168 -0.63697 -0.73068 -0.82257 -0.91242 -1.00000 -1.08509 -1.16749 -1.24698 -1.32337 -1.39647 -1.46610 -1.53209 -1.59427 -1.65248 -1.70658 -1.75644 -1.80194 -1.84295 -1.87939 -1.91115 -1.93815 -1.96034 -1.97766 -1.99006 -1.99751 -2.00000 -1.99751 -1.99006 -1.97766 -1.96034 -1.93815 -1.91115 -1.87939 -1.84295 -1.80194 -1.75644 -1.70658 -1.65248 -1.59427 -1.53209 -1.46610 -1.39647 -1.32337 -1.24698 -1.16749 -1.08509 -1.00000 -0.91242 -0.82257 -0.73068 -0.63697 -0.54168 -0.44504 -0.34730 -0.24869 -0.14946 -0.04986 0.04986 0.14946 0.24869 0.34730 0.44504 0.54168 0.63697 0.73068 0.82257 0.91242 1.00000 1.08509 1.16749 1.24698 1.32337 1.39647 1.46610 1.53209 1.59427 1.65248 1.70658 1.75644 1.80194 1.84295 1.87939 1.91115 1.93815 1.96034 1.97766 1.99006 1.99751 2.00000 0.00000 0.00000 ... ... TT... ....... ....TT... ........... ........TT... E3 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 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..... ............. E4 2.00000 1.99006 1.96034 1.91115 1.84295 1.75644 1.65248 1.53209 1.39647 1.24698 1.08509 0.91242 0.73068 0.54168 0.34730 0.14946 -0.04986 -0.24869 -0.44504 -0.63697 -0.82257 -1.00000 -1.16749 -1.32337 -1.46610 -1.59427 -1.70658 -1.80194 -1.87939 -1.93815 -1.97766 -1.99751 -1.99751 -1.97766 -1.93815 -1.87939 -1.80194 -1.70658 -1.59427 -1.46610 -1.32337 -1.16749 -1.00000 -0.82257 -0.63697 -0.44504 -0.24869 -0.04986 0.14946 0.34730 0.54168 0.73068 0.91242 1.08509 1.24698 1.39647 1.53209 1.65248 1.75644 1.84295 1.91115 1.96034 1.99006 2.00000 1.99006 1.96034 1.91115 1.84295 1.75644 1.65248 1.53209 1.39647 1.24698 1.08509 0.91242 0.73068 0.54168 0.34730 0.14946 -0.04986 -0.24869 -0.44504 -0.63697 -0.82257 -1.00000 -1.16749 -1.32337 -1.46610 -1.59427 -1.70658 -1.80194 -1.87939 -1.93815 -1.97766 -1.99751 -1.99751 -1.97766 -1.93815 -1.87939 -1.80194 -1.70658 -1.59427 -1.46610 -1.32337 -1.16749 -1.00000 -0.82257 -0.63697 -0.44504 -0.24869 -0.04986 0.14946 0.34730 0.54168 0.73068 0.91242 1.08509 1.24698 1.39647 1.53209 1.65248 1.75644 1.84295 1.91115 1.96034 1.99006 2.00000 0.00000 0.00000 ... ... ..... ....... TT....... ........... ....TT....... E5 2.00000 1.98448 1.93815 1.86175 1.75644 1.62388 1.46610 1.28558 1.08509 0.86777 0.63697 0.39629 0.14946 -0.09969 -0.34730 -0.58951 -0.82257 -1.04287 -1.24698 -1.43173 -1.59427 -1.73205 -1.84295 -1.92525 -1.97766 -1.99938 -1.99006 -1.94986 -1.87939 -1.77974 -1.65248 -1.49956 -1.32337 -1.12664 -0.91242 -0.68404 -0.44504 -0.19914 0.04986 0.29808 0.54168 0.77687 1.00000 1.20761 1.39647 1.56366 1.70658 1.82301 1.91115 1.96962 1.99751 1.99441 1.96034 1.89585 1.80194 1.68005 1.53209 1.36035 1.16749 0.95651 0.73068 0.49351 0.24869 0.00000 -0.24869 -0.49351 -0.73068 -0.95651 -1.16749 -1.36035 -1.53209 -1.68005 -1.80194 -1.89585 -1.96034 -1.99441 -1.99751 -1.96962 -1.91115 -1.82301 -1.70658 -1.56366 -1.39647 -1.20761 -1.00000 -0.77687 -0.54168 -0.29808 -0.04986 0.19914 0.44504 0.68404 0.91242 1.12664 1.32337 1.49956 1.65248 1.77974 1.87939 1.94986 1.99006 1.99938 1.97766 1.92525 1.84295 1.73205 1.59427 1.43173 1.24698 1.04287 0.82257 0.58951 0.34730 0.09969 -0.14946 -0.39629 -0.63697 -0.86777 -1.08509 -1.28558 -1.46610 -1.62388 -1.75644 -1.86175 -1.93815 -1.98448 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... TT......... ............. E6 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 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........... E7 2.00000 1.96962 1.87939 1.73205 1.53209 1.28558 1.00000 0.68404 0.34730 0.00000 -0.34730 -0.68404 -1.00000 -1.28558 -1.53209 -1.73205 -1.87939 -1.96962 -2.00000 -1.96962 -1.87939 -1.73205 -1.53209 -1.28558 -1.00000 -0.68404 -0.34730 0.00000 0.34730 0.68404 1.00000 1.28558 1.53209 1.73205 1.87939 1.96962 2.00000 1.96962 1.87939 1.73205 1.53209 1.28558 1.00000 0.68404 0.34730 0.00000 -0.34730 -0.68404 -1.00000 -1.28558 -1.53209 -1.73205 -1.87939 -1.96962 -2.00000 -1.96962 -1.87939 -1.73205 -1.53209 -1.28558 -1.00000 -0.68404 -0.34730 0.00000 0.34730 0.68404 1.00000 1.28558 1.53209 1.73205 1.87939 1.96962 2.00000 1.96962 1.87939 1.73205 1.53209 1.28558 1.00000 0.68404 0.34730 0.00000 -0.34730 -0.68404 -1.00000 -1.28558 -1.53209 -1.73205 -1.87939 -1.96962 -2.00000 -1.96962 -1.87939 -1.73205 -1.53209 -1.28558 -1.00000 -0.68404 -0.34730 0.00000 0.34730 0.68404 1.00000 1.28558 1.53209 1.73205 1.87939 1.96962 2.00000 1.96962 1.87939 1.73205 1.53209 1.28558 1.00000 0.68404 0.34730 0.00000 -0.34730 -0.68404 -1.00000 -1.28558 -1.53209 -1.73205 -1.87939 -1.96962 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. 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E121 2.00000 -1.98448 1.93815 -1.86175 1.75644 -1.62388 1.46610 -1.28558 1.08509 -0.86777 0.63697 -0.39629 0.14946 0.09969 -0.34730 0.58951 -0.82257 1.04287 -1.24698 1.43173 -1.59427 1.73205 -1.84295 1.92525 -1.97766 1.99938 -1.99006 1.94986 -1.87939 1.77974 -1.65248 1.49956 -1.32337 1.12664 -0.91242 0.68404 -0.44504 0.19914 0.04986 -0.29808 0.54168 -0.77687 1.00000 -1.20761 1.39647 -1.56366 1.70658 -1.82301 1.91115 -1.96962 1.99751 -1.99441 1.96034 -1.89585 1.80194 -1.68005 1.53209 -1.36035 1.16749 -0.95651 0.73068 -0.49351 0.24869 0.00000 -0.24869 0.49351 -0.73068 0.95651 -1.16749 1.36035 -1.53209 1.68005 -1.80194 1.89585 -1.96034 1.99441 -1.99751 1.96962 -1.91115 1.82301 -1.70658 1.56366 -1.39647 1.20761 -1.00000 0.77687 -0.54168 0.29808 -0.04986 -0.19914 0.44504 -0.68404 0.91242 -1.12664 1.32337 -1.49956 1.65248 -1.77974 1.87939 -1.94986 1.99006 -1.99938 1.97766 -1.92525 1.84295 -1.73205 1.59427 -1.43173 1.24698 -1.04287 0.82257 -0.58951 0.34730 -0.09969 -0.14946 0.39629 -0.63697 0.86777 -1.08509 1.28558 -1.46610 1.62388 -1.75644 1.86175 -1.93815 1.98448 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ..TT......... E122 2.00000 -1.99006 1.96034 -1.91115 1.84295 -1.75644 1.65248 -1.53209 1.39647 -1.24698 1.08509 -0.91242 0.73068 -0.54168 0.34730 -0.14946 -0.04986 0.24869 -0.44504 0.63697 -0.82257 1.00000 -1.16749 1.32337 -1.46610 1.59427 -1.70658 1.80194 -1.87939 1.93815 -1.97766 1.99751 -1.99751 1.97766 -1.93815 1.87939 -1.80194 1.70658 -1.59427 1.46610 -1.32337 1.16749 -1.00000 0.82257 -0.63697 0.44504 -0.24869 0.04986 0.14946 -0.34730 0.54168 -0.73068 0.91242 -1.08509 1.24698 -1.39647 1.53209 -1.65248 1.75644 -1.84295 1.91115 -1.96034 1.99006 -2.00000 1.99006 -1.96034 1.91115 -1.84295 1.75644 -1.65248 1.53209 -1.39647 1.24698 -1.08509 0.91242 -0.73068 0.54168 -0.34730 0.14946 0.04986 -0.24869 0.44504 -0.63697 0.82257 -1.00000 1.16749 -1.32337 1.46610 -1.59427 1.70658 -1.80194 1.87939 -1.93815 1.97766 -1.99751 1.99751 -1.97766 1.93815 -1.87939 1.80194 -1.70658 1.59427 -1.46610 1.32337 -1.16749 1.00000 -0.82257 0.63697 -0.44504 0.24869 -0.04986 -0.14946 0.34730 -0.54168 0.73068 -0.91242 1.08509 -1.24698 1.39647 -1.53209 1.65248 -1.75644 1.84295 -1.91115 1.96034 -1.99006 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ..TT....... ............. E123 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 -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..... E124 2.00000 -1.99751 1.99006 -1.97766 1.96034 -1.93815 1.91115 -1.87939 1.84295 -1.80194 1.75644 -1.70658 1.65248 -1.59427 1.53209 -1.46610 1.39647 -1.32337 1.24698 -1.16749 1.08509 -1.00000 0.91242 -0.82257 0.73068 -0.63697 0.54168 -0.44504 0.34730 -0.24869 0.14946 -0.04986 -0.04986 0.14946 -0.24869 0.34730 -0.44504 0.54168 -0.63697 0.73068 -0.82257 0.91242 -1.00000 1.08509 -1.16749 1.24698 -1.32337 1.39647 -1.46610 1.53209 -1.59427 1.65248 -1.70658 1.75644 -1.80194 1.84295 -1.87939 1.91115 -1.93815 1.96034 -1.97766 1.99006 -1.99751 2.00000 -1.99751 1.99006 -1.97766 1.96034 -1.93815 1.91115 -1.87939 1.84295 -1.80194 1.75644 -1.70658 1.65248 -1.59427 1.53209 -1.46610 1.39647 -1.32337 1.24698 -1.16749 1.08509 -1.00000 0.91242 -0.82257 0.73068 -0.63697 0.54168 -0.44504 0.34730 -0.24869 0.14946 -0.04986 -0.04986 0.14946 -0.24869 0.34730 -0.44504 0.54168 -0.63697 0.73068 -0.82257 0.91242 -1.00000 1.08509 -1.16749 1.24698 -1.32337 1.39647 -1.46610 1.53209 -1.59427 1.65248 -1.70658 1.75644 -1.80194 1.84295 -1.87939 1.91115 -1.93815 1.96034 -1.97766 1.99006 -1.99751 2.00000 0.00000 0.00000 ... ... ..... ..TT... ......... ......TT... ............. E125 2.00000 -1.99938 1.99751 -1.99441 1.99006 -1.98448 1.97766 -1.96962 1.96034 -1.94986 1.93815 -1.92525 1.91115 -1.89585 1.87939 -1.86175 1.84295 -1.82301 1.80194 -1.77974 1.75644 -1.73205 1.70658 -1.68005 1.65248 -1.62388 1.59427 -1.56366 1.53209 -1.49956 1.46610 -1.43173 1.39647 -1.36035 1.32337 -1.28558 1.24698 -1.20761 1.16749 -1.12664 1.08509 -1.04287 1.00000 -0.95651 0.91242 -0.86777 0.82257 -0.77687 0.73068 -0.68404 0.63697 -0.58951 0.54168 -0.49351 0.44504 -0.39629 0.34730 -0.29808 0.24869 -0.19914 0.14946 -0.09969 0.04986 0.00000 -0.04986 0.09969 -0.14946 0.19914 -0.24869 0.29808 -0.34730 0.39629 -0.44504 0.49351 -0.54168 0.58951 -0.63697 0.68404 -0.73068 0.77687 -0.82257 0.86777 -0.91242 0.95651 -1.00000 1.04287 -1.08509 1.12664 -1.16749 1.20761 -1.24698 1.28558 -1.32337 1.36035 -1.39647 1.43173 -1.46610 1.49956 -1.53209 1.56366 -1.59427 1.62388 -1.65248 1.68005 -1.70658 1.73205 -1.75644 1.77974 -1.80194 1.82301 -1.84295 1.86175 -1.87939 1.89585 -1.91115 1.92525 -1.93815 1.94986 -1.96034 1.96962 -1.97766 1.98448 -1.99006 1.99441 -1.99751 1.99938 -2.00000 0.00000 0.00000 TT. ... ..TT. ....... ......TT. ........... ..........TT. Irrational character values: 1.999378364002 = 2*cos(2*π/252) = 2*cos(π/126) 1.997513842438 = 2*cos(4*π/252) = 2*cos(π/63) 1.994407594362 = 2*cos(6*π/252) = 2*cos(π/42) 1.990061550731 = 2*cos(8*π/252) = 2*cos(2*π/63) 1.984478413200 = 2*cos(10*π/252) = 2*cos(5*π/126) 1.977661652450 = 2*cos(12*π/252) = 2*cos(π/21) 1.969615506024 = 2*cos(14*π/252) = 2*cos(π/18) 1.960344975697 = 2*cos(16*π/252) = 2*cos(4*π/63) 1.949855824364 = 2*cos(18*π/252) = 2*cos(π/14) 1.938154572458 = 2*cos(20*π/252) = 2*cos(5*π/63) 1.925248493900 = 2*cos(22*π/252) = 2*cos(11*π/126) 1.911145611572 = 2*cos(24*π/252) = 2*cos(2*π/21) 1.895854692334 = 2*cos(26*π/252) = 2*cos(13*π/126) 1.879385241572 = 2*cos(28*π/252) = 2*cos(π/9) 1.861747497288 = 2*cos(30*π/252) = 2*cos(5*π/42) 1.842952423741 = 2*cos(32*π/252) = 2*cos(8*π/63) 1.823011704623 = 2*cos(34*π/252) = 2*cos(17*π/126) 1.801937735805 = 2*cos(36*π/252) = 2*cos(π/7) 1.779743617623 = 2*cos(38*π/252) = 2*cos(19*π/126) 1.756443146740 = 2*cos(40*π/252) = 2*cos(10*π/63) 1.732050807569 = 2*cos(42*π/252) = 2*cos(π/6) = √3 1.706581763264 = 2*cos(44*π/252) = 2*cos(11*π/63) 1.680051846302 = 2*cos(46*π/252) = 2*cos(23*π/126) 1.652477548632 = 2*cos(48*π/252) = 2*cos(4*π/21) 1.623876011432 = 2*cos(50*π/252) = 2*cos(25*π/126) 1.594265014446 = 2*cos(52*π/252) = 2*cos(13*π/63) 1.563662964936 = 2*cos(54*π/252) = 2*cos(3*π/14) 1.532088886238 = 2*cos(56*π/252) = 2*cos(2*π/9) 1.499562405935 = 2*cos(58*π/252) = 2*cos(29*π/126) 1.466103743660 = 2*cos(60*π/252) = 2*cos(5*π/21) 1.431733698519 = 2*cos(62*π/252) = 2*cos(31*π/126) 1.396473636172 = 2*cos(64*π/252) = 2*cos(16*π/63) 1.360345475542 = 2*cos(66*π/252) = 2*cos(11*π/42) 1.323371675194 = 2*cos(68*π/252) = 2*cos(17*π/63) 1.285575219373 = 2*cos(70*π/252) = 2*cos(5*π/18) 1.246979603717 = 2*cos(72*π/252) = 2*cos(2*π/7) 1.207608820651 = 2*cos(74*π/252) = 2*cos(37*π/126) 1.167487344470 = 2*cos(76*π/252) = 2*cos(19*π/63) 1.126640116127 = 2*cos(78*π/252) = 2*cos(13*π/42) 1.085092527732 = 2*cos(80*π/252) = 2*cos(20*π/63) 1.042870406759 = 2*cos(82*π/252) = 2*cos(41*π/126) 0.956507957243 = 2*cos(86*π/252) = 2*cos(43*π/126) 0.912421314706 = 2*cos(88*π/252) = 2*cos(22*π/63) 0.867767478235 = 2*cos(90*π/252) = 2*cos(5*π/14) 0.822574206261 = 2*cos(92*π/252) = 2*cos(23*π/63) 0.776869592549 = 2*cos(94*π/252) = 2*cos(47*π/126) 0.730682048733 = 2*cos(96*π/252) = 2*cos(8*π/21) 0.684040286651 = 2*cos(98*π/252) = 2*cos(7*π/18) 0.636973300503 = 2*cos(100*π/252) = 2*cos(25*π/63) 0.589510348822 = 2*cos(102*π/252) = 2*cos(17*π/42) 0.541680936286 = 2*cos(104*π/252) = 2*cos(26*π/63) 0.493514795381 = 2*cos(106*π/252) = 2*cos(53*π/126) 0.445041867913 = 2*cos(108*π/252) = 2*cos(3*π/7) 0.396292286399 = 2*cos(110*π/252) = 2*cos(55*π/126) 0.347296355334 = 2*cos(112*π/252) = 2*cos(4*π/9) 0.298084532352 = 2*cos(114*π/252) = 2*cos(19*π/42) 0.248687409295 = 2*cos(116*π/252) = 2*cos(29*π/63) 0.199135693192 = 2*cos(118*π/252) = 2*cos(59*π/126) 0.149460187173 = 2*cos(120*π/252) = 2*cos(10*π/21) 0.099691771321 = 2*cos(122*π/252) = 2*cos(61*π/126) 0.049861383476 = 2*cos(124*π/252) = 2*cos(31*π/63) 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 E117 m E118 l E119 k+m E120 j+l E121 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)} E122 h+j+l {z((x²−y²)²−4x²y²), xyz(x²−y²)} E123 g+i+k+m {xz(x²−3y²), yz(3x²−y²)}, {xz³(x²−3y²), yz³(3x²−y²)} E124 f+h+j+l {z(x²−y²), xyz}, {z³(x²−y²), xyz³} E125 R+d+g+i+k+m {R_x, R_y}, {xz, yz}, {xz³, yz³}, {xz⁵, yz⁵} Notes: α The order of the D_126d point group is 504, and the order of the principal axis (S₂₅₂) is 252. The group has 129 irreducible representations. β The D_126d point group is isomorphic to C_252v and D₂₅₂. γ The D_126d point group is generated by two symmetry elements, S₂₅₂ 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_126d 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. 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_126d point group was created by Gernot Katzer.

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

	D_124d
	D_125d
C₁₂₆ C_126v C_126h D₁₂₆ D_126h	D_126d	S₁₂₆
	D_127d
	D_128d

Character table for the D126d point group

Character table for the D_126d point group