Character Table for Point Group D108d

Character table for the D_108d point group

D108d E 2 S216 2 C108 2 S72 2 C54 2 S216^5 2 C36 2 S216^7 2 C27 2 S24 2 C108^5 2 S216^11 2 C18 2 S216^13 2 C108^7 2 S72^5 2 C27^2 2 S216^17 2 C12 2 S216^19 2 C54^5 2 S72^7 2 C108^11 2 S216^23 2 C9 2 S216^25 2 C108^13 2 S8 2 C54^7 2 S216^29 2 C36^5 2 S216^31 2 C27^4 2 S72^11 2 C108^17 2 S216^35 2 C6 2 S216^37 2 C108^19 2 S72^13 2 C27^5 2 S216^41 2 C36^7 2 S216^43 2 C54^11 2 S24^5 2 C108^23 2 S216^47 2 C9^2 2 S216^49 2 C108^25 2 S72^17 2 C54^13 2 S216^53 2 C4 2 S216^55 2 C27^7 2 S72^19 2 C108^29 2 S216^59 2 C18^5 2 S216^61 2 C108^31 2 S24^7 2 C27^8 2 S216^65 2 C36^11 2 S216^67 2 C54^17 2 S72^23 2 C108^35 2 S216^71 2 C3 2 S216^73 2 C108^37 2 S72^25 2 C54^19 2 S216^77 2 C36^13 2 S216^79 2 C27^10 2 S8^3 2 C108^41 2 S216^83 2 C18^7 2 S216^85 2 C108^43 2 S72^29 2 C27^11 2 S216^89 2 C12^5 2 S216^91 2 C54^23 2 S72^31 2 C108^47 2 S216^95 2 C9^4 2 S216^97 2 C108^49 2 S24^11 2 C54^25 2 S216^101 2 C36^17 2 S216^103 2 C27^13 2 S72^35 2 C108^53 2 S216^107 C2 108 C2' 108 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 ... ... ....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 ..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 ... ... ..... ....... ......... ........... ............. 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 ... ..T ..... ......T ......... ..........T ............. E1 2.00000 1.99915 1.99662 1.99239 1.98648 1.97888 1.96962 1.95868 1.94609 1.93185 1.91598 1.89849 1.87939 1.85870 1.83643 1.81262 1.78727 1.76040 1.73205 1.70223 1.67098 1.63830 1.60425 1.56883 1.53209 1.49405 1.45475 1.41421 1.37248 1.32959 1.28558 1.24047 1.19432 1.14715 1.09902 1.04995 1.00000 0.94920 0.89760 0.84524 0.79216 0.73841 0.68404 0.62909 0.57361 0.51764 0.46123 0.40444 0.34730 0.28986 0.23219 0.17431 0.11629 0.05817 0.00000 -0.05817 -0.11629 -0.17431 -0.23219 -0.28986 -0.34730 -0.40444 -0.46123 -0.51764 -0.57361 -0.62909 -0.68404 -0.73841 -0.79216 -0.84524 -0.89760 -0.94920 -1.00000 -1.04995 -1.09902 -1.14715 -1.19432 -1.24047 -1.28558 -1.32959 -1.37248 -1.41421 -1.45475 -1.49405 -1.53209 -1.56883 -1.60425 -1.63830 -1.67098 -1.70223 -1.73205 -1.76040 -1.78727 -1.81262 -1.83643 -1.85870 -1.87939 -1.89849 -1.91598 -1.93185 -1.94609 -1.95868 -1.96962 -1.97888 -1.98648 -1.99239 -1.99662 -1.99915 -2.00000 0.00000 0.00000 ... TT. ..... ....TT. ......... ........TT. ............. E2 2.00000 1.99662 1.98648 1.96962 1.94609 1.91598 1.87939 1.83643 1.78727 1.73205 1.67098 1.60425 1.53209 1.45475 1.37248 1.28558 1.19432 1.09902 1.00000 0.89760 0.79216 0.68404 0.57361 0.46123 0.34730 0.23219 0.11629 0.00000 -0.11629 -0.23219 -0.34730 -0.46123 -0.57361 -0.68404 -0.79216 -0.89760 -1.00000 -1.09902 -1.19432 -1.28558 -1.37248 -1.45475 -1.53209 -1.60425 -1.67098 -1.73205 -1.78727 -1.83643 -1.87939 -1.91598 -1.94609 -1.96962 -1.98648 -1.99662 -2.00000 -1.99662 -1.98648 -1.96962 -1.94609 -1.91598 -1.87939 -1.83643 -1.78727 -1.73205 -1.67098 -1.60425 -1.53209 -1.45475 -1.37248 -1.28558 -1.19432 -1.09902 -1.00000 -0.89760 -0.79216 -0.68404 -0.57361 -0.46123 -0.34730 -0.23219 -0.11629 0.00000 0.11629 0.23219 0.34730 0.46123 0.57361 0.68404 0.79216 0.89760 1.00000 1.09902 1.19432 1.28558 1.37248 1.45475 1.53209 1.60425 1.67098 1.73205 1.78727 1.83643 1.87939 1.91598 1.94609 1.96962 1.98648 1.99662 2.00000 0.00000 0.00000 ... ... TT... ....... ....TT... ........... ........TT... E3 2.00000 1.99239 1.96962 1.93185 1.87939 1.81262 1.73205 1.63830 1.53209 1.41421 1.28558 1.14715 1.00000 0.84524 0.68404 0.51764 0.34730 0.17431 0.00000 -0.17431 -0.34730 -0.51764 -0.68404 -0.84524 -1.00000 -1.14715 -1.28558 -1.41421 -1.53209 -1.63830 -1.73205 -1.81262 -1.87939 -1.93185 -1.96962 -1.99239 -2.00000 -1.99239 -1.96962 -1.93185 -1.87939 -1.81262 -1.73205 -1.63830 -1.53209 -1.41421 -1.28558 -1.14715 -1.00000 -0.84524 -0.68404 -0.51764 -0.34730 -0.17431 0.00000 0.17431 0.34730 0.51764 0.68404 0.84524 1.00000 1.14715 1.28558 1.41421 1.53209 1.63830 1.73205 1.81262 1.87939 1.93185 1.96962 1.99239 2.00000 1.99239 1.96962 1.93185 1.87939 1.81262 1.73205 1.63830 1.53209 1.41421 1.28558 1.14715 1.00000 0.84524 0.68404 0.51764 0.34730 0.17431 0.00000 -0.17431 -0.34730 -0.51764 -0.68404 -0.84524 -1.00000 -1.14715 -1.28558 -1.41421 -1.53209 -1.63830 -1.73205 -1.81262 -1.87939 -1.93185 -1.96962 -1.99239 -2.00000 0.00000 0.00000 ... ... ..... TT..... ......... ....TT..... ............. E4 2.00000 1.98648 1.94609 1.87939 1.78727 1.67098 1.53209 1.37248 1.19432 1.00000 0.79216 0.57361 0.34730 0.11629 -0.11629 -0.34730 -0.57361 -0.79216 -1.00000 -1.19432 -1.37248 -1.53209 -1.67098 -1.78727 -1.87939 -1.94609 -1.98648 -2.00000 -1.98648 -1.94609 -1.87939 -1.78727 -1.67098 -1.53209 -1.37248 -1.19432 -1.00000 -0.79216 -0.57361 -0.34730 -0.11629 0.11629 0.34730 0.57361 0.79216 1.00000 1.19432 1.37248 1.53209 1.67098 1.78727 1.87939 1.94609 1.98648 2.00000 1.98648 1.94609 1.87939 1.78727 1.67098 1.53209 1.37248 1.19432 1.00000 0.79216 0.57361 0.34730 0.11629 -0.11629 -0.34730 -0.57361 -0.79216 -1.00000 -1.19432 -1.37248 -1.53209 -1.67098 -1.78727 -1.87939 -1.94609 -1.98648 -2.00000 -1.98648 -1.94609 -1.87939 -1.78727 -1.67098 -1.53209 -1.37248 -1.19432 -1.00000 -0.79216 -0.57361 -0.34730 -0.11629 0.11629 0.34730 0.57361 0.79216 1.00000 1.19432 1.37248 1.53209 1.67098 1.78727 1.87939 1.94609 1.98648 2.00000 0.00000 0.00000 ... ... ..... ....... TT....... ........... ....TT....... E5 2.00000 1.97888 1.91598 1.81262 1.67098 1.49405 1.28558 1.04995 0.79216 0.51764 0.23219 -0.05817 -0.34730 -0.62909 -0.89760 -1.14715 -1.37248 -1.56883 -1.73205 -1.85870 -1.94609 -1.99239 -1.99662 -1.95868 -1.87939 -1.76040 -1.60425 -1.41421 -1.19432 -0.94920 -0.68404 -0.40444 -0.11629 0.17431 0.46123 0.73841 1.00000 1.24047 1.45475 1.63830 1.78727 1.89849 1.96962 1.99915 1.98648 1.93185 1.83643 1.70223 1.53209 1.32959 1.09902 0.84524 0.57361 0.28986 0.00000 -0.28986 -0.57361 -0.84524 -1.09902 -1.32959 -1.53209 -1.70223 -1.83643 -1.93185 -1.98648 -1.99915 -1.96962 -1.89849 -1.78727 -1.63830 -1.45475 -1.24047 -1.00000 -0.73841 -0.46123 -0.17431 0.11629 0.40444 0.68404 0.94920 1.19432 1.41421 1.60425 1.76040 1.87939 1.95868 1.99662 1.99239 1.94609 1.85870 1.73205 1.56883 1.37248 1.14715 0.89760 0.62909 0.34730 0.05817 -0.23219 -0.51764 -0.79216 -1.04995 -1.28558 -1.49405 -1.67098 -1.81262 -1.91598 -1.97888 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... TT......... ............. E6 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 ... ... ..... ....... ......... ........... TT........... E7 2.00000 1.95868 1.83643 1.63830 1.37248 1.04995 0.68404 0.28986 -0.11629 -0.51764 -0.89760 -1.24047 -1.53209 -1.76040 -1.91598 -1.99239 -1.98648 -1.89849 -1.73205 -1.49405 -1.19432 -0.84524 -0.46123 -0.05817 0.34730 0.73841 1.09902 1.41421 1.67098 1.85870 1.96962 1.99915 1.94609 1.81262 1.60425 1.32959 1.00000 0.62909 0.23219 -0.17431 -0.57361 -0.94920 -1.28558 -1.56883 -1.78727 -1.93185 -1.99662 -1.97888 -1.87939 -1.70223 -1.45475 -1.14715 -0.79216 -0.40444 0.00000 0.40444 0.79216 1.14715 1.45475 1.70223 1.87939 1.97888 1.99662 1.93185 1.78727 1.56883 1.28558 0.94920 0.57361 0.17431 -0.23219 -0.62909 -1.00000 -1.32959 -1.60425 -1.81262 -1.94609 -1.99915 -1.96962 -1.85870 -1.67098 -1.41421 -1.09902 -0.73841 -0.34730 0.05817 0.46123 0.84524 1.19432 1.49405 1.73205 1.89849 1.98648 1.99239 1.91598 1.76040 1.53209 1.24047 0.89760 0.51764 0.11629 -0.28986 -0.68404 -1.04995 -1.37248 -1.63830 -1.83643 -1.95868 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E8 2.00000 1.94609 1.78727 1.53209 1.19432 0.79216 0.34730 -0.11629 -0.57361 -1.00000 -1.37248 -1.67098 -1.87939 -1.98648 -1.98648 -1.87939 -1.67098 -1.37248 -1.00000 -0.57361 -0.11629 0.34730 0.79216 1.19432 1.53209 1.78727 1.94609 2.00000 1.94609 1.78727 1.53209 1.19432 0.79216 0.34730 -0.11629 -0.57361 -1.00000 -1.37248 -1.67098 -1.87939 -1.98648 -1.98648 -1.87939 -1.67098 -1.37248 -1.00000 -0.57361 -0.11629 0.34730 0.79216 1.19432 1.53209 1.78727 1.94609 2.00000 1.94609 1.78727 1.53209 1.19432 0.79216 0.34730 -0.11629 -0.57361 -1.00000 -1.37248 -1.67098 -1.87939 -1.98648 -1.98648 -1.87939 -1.67098 -1.37248 -1.00000 -0.57361 -0.11629 0.34730 0.79216 1.19432 1.53209 1.78727 1.94609 2.00000 1.94609 1.78727 1.53209 1.19432 0.79216 0.34730 -0.11629 -0.57361 -1.00000 -1.37248 -1.67098 -1.87939 -1.98648 -1.98648 -1.87939 -1.67098 -1.37248 -1.00000 -0.57361 -0.11629 0.34730 0.79216 1.19432 1.53209 1.78727 1.94609 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E9 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 -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 ... ... ..... ....... ......... ........... ............. E10 2.00000 1.91598 1.67098 1.28558 0.79216 0.23219 -0.34730 -0.89760 -1.37248 -1.73205 -1.94609 -1.99662 -1.87939 -1.60425 -1.19432 -0.68404 -0.11629 0.46123 1.00000 1.45475 1.78727 1.96962 1.98648 1.83643 1.53209 1.09902 0.57361 0.00000 -0.57361 -1.09902 -1.53209 -1.83643 -1.98648 -1.96962 -1.78727 -1.45475 -1.00000 -0.46123 0.11629 0.68404 1.19432 1.60425 1.87939 1.99662 1.94609 1.73205 1.37248 0.89760 0.34730 -0.23219 -0.79216 -1.28558 -1.67098 -1.91598 -2.00000 -1.91598 -1.67098 -1.28558 -0.79216 -0.23219 0.34730 0.89760 1.37248 1.73205 1.94609 1.99662 1.87939 1.60425 1.19432 0.68404 0.11629 -0.46123 -1.00000 -1.45475 -1.78727 -1.96962 -1.98648 -1.83643 -1.53209 -1.09902 -0.57361 0.00000 0.57361 1.09902 1.53209 1.83643 1.98648 1.96962 1.78727 1.45475 1.00000 0.46123 -0.11629 -0.68404 -1.19432 -1.60425 -1.87939 -1.99662 -1.94609 -1.73205 -1.37248 -0.89760 -0.34730 0.23219 0.79216 1.28558 1.67098 1.91598 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. 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E106 2.00000 -1.99662 1.98648 -1.96962 1.94609 -1.91598 1.87939 -1.83643 1.78727 -1.73205 1.67098 -1.60425 1.53209 -1.45475 1.37248 -1.28558 1.19432 -1.09902 1.00000 -0.89760 0.79216 -0.68404 0.57361 -0.46123 0.34730 -0.23219 0.11629 0.00000 -0.11629 0.23219 -0.34730 0.46123 -0.57361 0.68404 -0.79216 0.89760 -1.00000 1.09902 -1.19432 1.28558 -1.37248 1.45475 -1.53209 1.60425 -1.67098 1.73205 -1.78727 1.83643 -1.87939 1.91598 -1.94609 1.96962 -1.98648 1.99662 -2.00000 1.99662 -1.98648 1.96962 -1.94609 1.91598 -1.87939 1.83643 -1.78727 1.73205 -1.67098 1.60425 -1.53209 1.45475 -1.37248 1.28558 -1.19432 1.09902 -1.00000 0.89760 -0.79216 0.68404 -0.57361 0.46123 -0.34730 0.23219 -0.11629 0.00000 0.11629 -0.23219 0.34730 -0.46123 0.57361 -0.68404 0.79216 -0.89760 1.00000 -1.09902 1.19432 -1.28558 1.37248 -1.45475 1.53209 -1.60425 1.67098 -1.73205 1.78727 -1.83643 1.87939 -1.91598 1.94609 -1.96962 1.98648 -1.99662 2.00000 0.00000 0.00000 ... ... ..... ..TT... ......... ......TT... ............. E107 2.00000 -1.99915 1.99662 -1.99239 1.98648 -1.97888 1.96962 -1.95868 1.94609 -1.93185 1.91598 -1.89849 1.87939 -1.85870 1.83643 -1.81262 1.78727 -1.76040 1.73205 -1.70223 1.67098 -1.63830 1.60425 -1.56883 1.53209 -1.49405 1.45475 -1.41421 1.37248 -1.32959 1.28558 -1.24047 1.19432 -1.14715 1.09902 -1.04995 1.00000 -0.94920 0.89760 -0.84524 0.79216 -0.73841 0.68404 -0.62909 0.57361 -0.51764 0.46123 -0.40444 0.34730 -0.28986 0.23219 -0.17431 0.11629 -0.05817 0.00000 0.05817 -0.11629 0.17431 -0.23219 0.28986 -0.34730 0.40444 -0.46123 0.51764 -0.57361 0.62909 -0.68404 0.73841 -0.79216 0.84524 -0.89760 0.94920 -1.00000 1.04995 -1.09902 1.14715 -1.19432 1.24047 -1.28558 1.32959 -1.37248 1.41421 -1.45475 1.49405 -1.53209 1.56883 -1.60425 1.63830 -1.67098 1.70223 -1.73205 1.76040 -1.78727 1.81262 -1.83643 1.85870 -1.87939 1.89849 -1.91598 1.93185 -1.94609 1.95868 -1.96962 1.97888 -1.98648 1.99239 -1.99662 1.99915 -2.00000 0.00000 0.00000 TT. ... ..TT. ....... ......TT. ........... ..........TT. Irrational character values: 1.999153900164 = 2*cos(2*π/216) = 2*cos(π/108) 1.996616316543 = 2*cos(4*π/216) = 2*cos(π/54) 1.992389396183 = 2*cos(6*π/216) = 2*cos(π/36) 1.986476715484 = 2*cos(8*π/216) = 2*cos(π/27) 1.978883277162 = 2*cos(10*π/216) = 2*cos(5*π/108) 1.969615506024 = 2*cos(12*π/216) = 2*cos(π/18) 1.958681243531 = 2*cos(14*π/216) = 2*cos(7*π/108) 1.946089741160 = 2*cos(16*π/216) = 2*cos(2*π/27) 1.931851652578 = 2*cos(18*π/216) = 2*cos(π/12) = √2+√3 1.915979024631 = 2*cos(20*π/216) = 2*cos(5*π/54) 1.898485287146 = 2*cos(22*π/216) = 2*cos(11*π/108) 1.879385241572 = 2*cos(24*π/216) = 2*cos(π/9) 1.858695048454 = 2*cos(26*π/216) = 2*cos(13*π/108) 1.836432213761 = 2*cos(28*π/216) = 2*cos(7*π/54) 1.812615574073 = 2*cos(30*π/216) = 2*cos(5*π/36) 1.787265280647 = 2*cos(32*π/216) = 2*cos(4*π/27) 1.760402782360 = 2*cos(34*π/216) = 2*cos(17*π/108) 1.732050807569 = 2*cos(36*π/216) = 2*cos(π/6) = √3 1.702233344874 = 2*cos(38*π/216) = 2*cos(19*π/108) 1.670975622826 = 2*cos(40*π/216) = 2*cos(5*π/27) 1.638304088578 = 2*cos(42*π/216) = 2*cos(7*π/36) 1.604246385510 = 2*cos(44*π/216) = 2*cos(11*π/54) 1.568831329839 = 2*cos(46*π/216) = 2*cos(23*π/108) 1.532088886238 = 2*cos(48*π/216) = 2*cos(2*π/9) 1.494050142482 = 2*cos(50*π/216) = 2*cos(25*π/108) 1.454747283146 = 2*cos(52*π/216) = 2*cos(13*π/54) 1.414213562373 = 2*cos(54*π/216) = 2*cos(π/4) = √2 1.372483275737 = 2*cos(56*π/216) = 2*cos(7*π/27) 1.329591731228 = 2*cos(58*π/216) = 2*cos(29*π/108) 1.285575219373 = 2*cos(60*π/216) = 2*cos(5*π/18) 1.240470982537 = 2*cos(62*π/216) = 2*cos(31*π/108) 1.194317183406 = 2*cos(64*π/216) = 2*cos(8*π/27) 1.147152872702 = 2*cos(66*π/216) = 2*cos(11*π/36) 1.099017956142 = 2*cos(68*π/216) = 2*cos(17*π/54) 1.049953160669 = 2*cos(70*π/216) = 2*cos(35*π/108) 0.949200739495 = 2*cos(74*π/216) = 2*cos(37*π/108) 0.897598360401 = 2*cos(76*π/216) = 2*cos(19*π/54) 0.845236523481 = 2*cos(78*π/216) = 2*cos(13*π/36) 0.792159532078 = 2*cos(80*π/216) = 2*cos(10*π/27) 0.738412294625 = 2*cos(82*π/216) = 2*cos(41*π/108) 0.684040286651 = 2*cos(84*π/216) = 2*cos(7*π/18) 0.629089512303 = 2*cos(86*π/216) = 2*cos(43*π/108) 0.573606465422 = 2*cos(88*π/216) = 2*cos(11*π/27) 0.517638090205 = 2*cos(90*π/216) = 2*cos(5*π/12) = √2−√3 0.461231741485 = 2*cos(92*π/216) = 2*cos(23*π/54) 0.404435144664 = 2*cos(94*π/216) = 2*cos(47*π/108) 0.347296355334 = 2*cos(96*π/216) = 2*cos(4*π/9) 0.289863718614 = 2*cos(98*π/216) = 2*cos(49*π/108) 0.232185828250 = 2*cos(100*π/216) = 2*cos(25*π/54) 0.174311485495 = 2*cos(102*π/216) = 2*cos(17*π/36) 0.116289657821 = 2*cos(104*π/216) = 2*cos(13*π/27) 0.058169437486 = 2*cos(106*π/216) = 2*cos(53*π/108) 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 E99 m E100 l E101 k+m E102 j+l E103 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)} E104 h+j+l {z((x²−y²)²−4x²y²), xyz(x²−y²)} E105 g+i+k+m {xz(x²−3y²), yz(3x²−y²)}, {xz³(x²−3y²), yz³(3x²−y²)} E106 f+h+j+l {z(x²−y²), xyz}, {z³(x²−y²), xyz³} E107 R+d+g+i+k+m {R_x, R_y}, {xz, yz}, {xz³, yz³}, {xz⁵, yz⁵} Notes: α The order of the D_108d point group is 432, and the order of the principal axis (S₂₁₆) is 216. The group has 111 irreducible representations. β The D_108d point group is isomorphic to C_216v and D₂₁₆. γ The D_108d point group is generated by two symmetry elements, S₂₁₆ 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_108d 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_108d point group was created by Gernot Katzer.

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

	D_106d
	D_107d
C₁₀₈ C_108v C_108h D₁₀₈ D_108h	D_108d	S₁₀₈
	D_109d
	D_110d

Character table for the D108d point group

Character table for the D_108d point group