Character Table for Point Group C108v

Character table for the C_108v point group

C108v E 2 C108 2 C54 2 C36 2 C27 2 C108^5 2 C18 2 C108^7 2 C27^2 2 C12 2 C54^5 2 C108^11 2 C9 2 C108^13 2 C54^7 2 C36^5 2 C27^4 2 C108^17 2 C6 2 C108^19 2 C27^5 2 C36^7 2 C54^11 2 C108^23 2 C9^2 2 C108^25 2 C54^13 2 C4 2 C27^7 2 C108^29 2 C18^5 2 C108^31 2 C27^8 2 C36^11 2 C54^17 2 C108^35 2 C3 2 C108^37 2 C54^19 2 C36^13 2 C27^10 2 C108^41 2 C18^7 2 C108^43 2 C27^11 2 C12^5 2 C54^23 2 C108^47 2 C9^4 2 C108^49 2 C54^25 2 C36^17 2 C27^13 2 C108^53 C2 54 sv 54 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 ... ..T ....T ......T ........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 ..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 ... ... ..... ....... ......... ........... ............. 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 ... ... ..... ....... ......... ........... ............. E1 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. ....TT. ......TT. ........TT. ..........TT. E2 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... ....TT... ......TT... ........TT... E3 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..... ..TT..... ....TT..... ......TT..... E4 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 ... ... ..... ....... TT....... ..TT....... ....TT....... E5 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 ... ... ..... ....... ......... TT......... ..TT......... E6 2.00000 1.87939 1.53209 1.00000 0.34730 -0.34730 -1.00000 -1.53209 -1.87939 -2.00000 -1.87939 -1.53209 -1.00000 -0.34730 0.34730 1.00000 1.53209 1.87939 2.00000 1.87939 1.53209 1.00000 0.34730 -0.34730 -1.00000 -1.53209 -1.87939 -2.00000 -1.87939 -1.53209 -1.00000 -0.34730 0.34730 1.00000 1.53209 1.87939 2.00000 1.87939 1.53209 1.00000 0.34730 -0.34730 -1.00000 -1.53209 -1.87939 -2.00000 -1.87939 -1.53209 -1.00000 -0.34730 0.34730 1.00000 1.53209 1.87939 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... TT........... E7 2.00000 1.83643 1.37248 0.68404 -0.11629 -0.89760 -1.53209 -1.91598 -1.98648 -1.73205 -1.19432 -0.46123 0.34730 1.09902 1.67098 1.96962 1.94609 1.60425 1.00000 0.23219 -0.57361 -1.28558 -1.78727 -1.99662 -1.87939 -1.45475 -0.79216 0.00000 0.79216 1.45475 1.87939 1.99662 1.78727 1.28558 0.57361 -0.23219 -1.00000 -1.60425 -1.94609 -1.96962 -1.67098 -1.09902 -0.34730 0.46123 1.19432 1.73205 1.98648 1.91598 1.53209 0.89760 0.11629 -0.68404 -1.37248 -1.83643 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E8 2.00000 1.78727 1.19432 0.34730 -0.57361 -1.37248 -1.87939 -1.98648 -1.67098 -1.00000 -0.11629 0.79216 1.53209 1.94609 1.94609 1.53209 0.79216 -0.11629 -1.00000 -1.67098 -1.98648 -1.87939 -1.37248 -0.57361 0.34730 1.19432 1.78727 2.00000 1.78727 1.19432 0.34730 -0.57361 -1.37248 -1.87939 -1.98648 -1.67098 -1.00000 -0.11629 0.79216 1.53209 1.94609 1.94609 1.53209 0.79216 -0.11629 -1.00000 -1.67098 -1.98648 -1.87939 -1.37248 -0.57361 0.34730 1.19432 1.78727 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E9 2.00000 1.73205 1.00000 0.00000 -1.00000 -1.73205 -2.00000 -1.73205 -1.00000 0.00000 1.00000 1.73205 2.00000 1.73205 1.00000 0.00000 -1.00000 -1.73205 -2.00000 -1.73205 -1.00000 0.00000 1.00000 1.73205 2.00000 1.73205 1.00000 0.00000 -1.00000 -1.73205 -2.00000 -1.73205 -1.00000 0.00000 1.00000 1.73205 2.00000 1.73205 1.00000 0.00000 -1.00000 -1.73205 -2.00000 -1.73205 -1.00000 0.00000 1.00000 1.73205 2.00000 1.73205 1.00000 0.00000 -1.00000 -1.73205 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E10 2.00000 1.67098 0.79216 -0.34730 -1.37248 -1.94609 -1.87939 -1.19432 -0.11629 1.00000 1.78727 1.98648 1.53209 0.57361 -0.57361 -1.53209 -1.98648 -1.78727 -1.00000 0.11629 1.19432 1.87939 1.94609 1.37248 0.34730 -0.79216 -1.67098 -2.00000 -1.67098 -0.79216 0.34730 1.37248 1.94609 1.87939 1.19432 0.11629 -1.00000 -1.78727 -1.98648 -1.53209 -0.57361 0.57361 1.53209 1.98648 1.78727 1.00000 -0.11629 -1.19432 -1.87939 -1.94609 -1.37248 -0.34730 0.79216 1.67098 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E11 2.00000 1.60425 0.57361 -0.68404 -1.67098 -1.99662 -1.53209 -0.46123 0.79216 1.73205 1.98648 1.45475 0.34730 -0.89760 -1.78727 -1.96962 -1.37248 -0.23219 1.00000 1.83643 1.94609 1.28558 0.11629 -1.09902 -1.87939 -1.91598 -1.19432 0.00000 1.19432 1.91598 1.87939 1.09902 -0.11629 -1.28558 -1.94609 -1.83643 -1.00000 0.23219 1.37248 1.96962 1.78727 0.89760 -0.34730 -1.45475 -1.98648 -1.73205 -0.79216 0.46123 1.53209 1.99662 1.67098 0.68404 -0.57361 -1.60425 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E12 2.00000 1.53209 0.34730 -1.00000 -1.87939 -1.87939 -1.00000 0.34730 1.53209 2.00000 1.53209 0.34730 -1.00000 -1.87939 -1.87939 -1.00000 0.34730 1.53209 2.00000 1.53209 0.34730 -1.00000 -1.87939 -1.87939 -1.00000 0.34730 1.53209 2.00000 1.53209 0.34730 -1.00000 -1.87939 -1.87939 -1.00000 0.34730 1.53209 2.00000 1.53209 0.34730 -1.00000 -1.87939 -1.87939 -1.00000 0.34730 1.53209 2.00000 1.53209 0.34730 -1.00000 -1.87939 -1.87939 -1.00000 0.34730 1.53209 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E13 2.00000 1.45475 0.11629 -1.28558 -1.98648 -1.60425 -0.34730 1.09902 1.94609 1.73205 0.57361 -0.89760 -1.87939 -1.83643 -0.79216 0.68404 1.78727 1.91598 1.00000 -0.46123 -1.67098 -1.96962 -1.19432 0.23219 1.53209 1.99662 1.37248 0.00000 -1.37248 -1.99662 -1.53209 -0.23219 1.19432 1.96962 1.67098 0.46123 -1.00000 -1.91598 -1.78727 -0.68404 0.79216 1.83643 1.87939 0.89760 -0.57361 -1.73205 -1.94609 -1.09902 0.34730 1.60425 1.98648 1.28558 -0.11629 -1.45475 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E14 2.00000 1.37248 -0.11629 -1.53209 -1.98648 -1.19432 0.34730 1.67098 1.94609 1.00000 -0.57361 -1.78727 -1.87939 -0.79216 0.79216 1.87939 1.78727 0.57361 -1.00000 -1.94609 -1.67098 -0.34730 1.19432 1.98648 1.53209 0.11629 -1.37248 -2.00000 -1.37248 0.11629 1.53209 1.98648 1.19432 -0.34730 -1.67098 -1.94609 -1.00000 0.57361 1.78727 1.87939 0.79216 -0.79216 -1.87939 -1.78727 -0.57361 1.00000 1.94609 1.67098 0.34730 -1.19432 -1.98648 -1.53209 -0.11629 1.37248 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E15 2.00000 1.28558 -0.34730 -1.73205 -1.87939 -0.68404 1.00000 1.96962 1.53209 0.00000 -1.53209 -1.96962 -1.00000 0.68404 1.87939 1.73205 0.34730 -1.28558 -2.00000 -1.28558 0.34730 1.73205 1.87939 0.68404 -1.00000 -1.96962 -1.53209 0.00000 1.53209 1.96962 1.00000 -0.68404 -1.87939 -1.73205 -0.34730 1.28558 2.00000 1.28558 -0.34730 -1.73205 -1.87939 -0.68404 1.00000 1.96962 1.53209 0.00000 -1.53209 -1.96962 -1.00000 0.68404 1.87939 1.73205 0.34730 -1.28558 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E16 2.00000 1.19432 -0.57361 -1.87939 -1.67098 -0.11629 1.53209 1.94609 0.79216 -1.00000 -1.98648 -1.37248 0.34730 1.78727 1.78727 0.34730 -1.37248 -1.98648 -1.00000 0.79216 1.94609 1.53209 -0.11629 -1.67098 -1.87939 -0.57361 1.19432 2.00000 1.19432 -0.57361 -1.87939 -1.67098 -0.11629 1.53209 1.94609 0.79216 -1.00000 -1.98648 -1.37248 0.34730 1.78727 1.78727 0.34730 -1.37248 -1.98648 -1.00000 0.79216 1.94609 1.53209 -0.11629 -1.67098 -1.87939 -0.57361 1.19432 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E17 2.00000 1.09902 -0.79216 -1.96962 -1.37248 0.46123 1.87939 1.60425 -0.11629 -1.73205 -1.78727 -0.23219 1.53209 1.91598 0.57361 -1.28558 -1.98648 -0.89760 1.00000 1.99662 1.19432 -0.68404 -1.94609 -1.45475 0.34730 1.83643 1.67098 0.00000 -1.67098 -1.83643 -0.34730 1.45475 1.94609 0.68404 -1.19432 -1.99662 -1.00000 0.89760 1.98648 1.28558 -0.57361 -1.91598 -1.53209 0.23219 1.78727 1.73205 0.11629 -1.60425 -1.87939 -0.46123 1.37248 1.96962 0.79216 -1.09902 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E18 2.00000 1.00000 -1.00000 -2.00000 -1.00000 1.00000 2.00000 1.00000 -1.00000 -2.00000 -1.00000 1.00000 2.00000 1.00000 -1.00000 -2.00000 -1.00000 1.00000 2.00000 1.00000 -1.00000 -2.00000 -1.00000 1.00000 2.00000 1.00000 -1.00000 -2.00000 -1.00000 1.00000 2.00000 1.00000 -1.00000 -2.00000 -1.00000 1.00000 2.00000 1.00000 -1.00000 -2.00000 -1.00000 1.00000 2.00000 1.00000 -1.00000 -2.00000 -1.00000 1.00000 2.00000 1.00000 -1.00000 -2.00000 -1.00000 1.00000 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E19 2.00000 0.89760 -1.19432 -1.96962 -0.57361 1.45475 1.87939 0.23219 -1.67098 -1.73205 0.11629 1.83643 1.53209 -0.46123 -1.94609 -1.28558 0.79216 1.99662 1.00000 -1.09902 -1.98648 -0.68404 1.37248 1.91598 0.34730 -1.60425 -1.78727 0.00000 1.78727 1.60425 -0.34730 -1.91598 -1.37248 0.68404 1.98648 1.09902 -1.00000 -1.99662 -0.79216 1.28558 1.94609 0.46123 -1.53209 -1.83643 -0.11629 1.73205 1.67098 -0.23219 -1.87939 -1.45475 0.57361 1.96962 1.19432 -0.89760 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E20 2.00000 0.79216 -1.37248 -1.87939 -0.11629 1.78727 1.53209 -0.57361 -1.98648 -1.00000 1.19432 1.94609 0.34730 -1.67098 -1.67098 0.34730 1.94609 1.19432 -1.00000 -1.98648 -0.57361 1.53209 1.78727 -0.11629 -1.87939 -1.37248 0.79216 2.00000 0.79216 -1.37248 -1.87939 -0.11629 1.78727 1.53209 -0.57361 -1.98648 -1.00000 1.19432 1.94609 0.34730 -1.67098 -1.67098 0.34730 1.94609 1.19432 -1.00000 -1.98648 -0.57361 1.53209 1.78727 -0.11629 -1.87939 -1.37248 0.79216 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E21 2.00000 0.68404 -1.53209 -1.73205 0.34730 1.96962 1.00000 -1.28558 -1.87939 0.00000 1.87939 1.28558 -1.00000 -1.96962 -0.34730 1.73205 1.53209 -0.68404 -2.00000 -0.68404 1.53209 1.73205 -0.34730 -1.96962 -1.00000 1.28558 1.87939 0.00000 -1.87939 -1.28558 1.00000 1.96962 0.34730 -1.73205 -1.53209 0.68404 2.00000 0.68404 -1.53209 -1.73205 0.34730 1.96962 1.00000 -1.28558 -1.87939 0.00000 1.87939 1.28558 -1.00000 -1.96962 -0.34730 1.73205 1.53209 -0.68404 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E22 2.00000 0.57361 -1.67098 -1.53209 0.79216 1.98648 0.34730 -1.78727 -1.37248 1.00000 1.94609 0.11629 -1.87939 -1.19432 1.19432 1.87939 -0.11629 -1.94609 -1.00000 1.37248 1.78727 -0.34730 -1.98648 -0.79216 1.53209 1.67098 -0.57361 -2.00000 -0.57361 1.67098 1.53209 -0.79216 -1.98648 -0.34730 1.78727 1.37248 -1.00000 -1.94609 -0.11629 1.87939 1.19432 -1.19432 -1.87939 0.11629 1.94609 1.00000 -1.37248 -1.78727 0.34730 1.98648 0.79216 -1.53209 -1.67098 0.57361 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E23 2.00000 0.46123 -1.78727 -1.28558 1.19432 1.83643 -0.34730 -1.99662 -0.57361 1.73205 1.37248 -1.09902 -1.87939 0.23219 1.98648 0.68404 -1.67098 -1.45475 1.00000 1.91598 -0.11629 -1.96962 -0.79216 1.60425 1.53209 -0.89760 -1.94609 0.00000 1.94609 0.89760 -1.53209 -1.60425 0.79216 1.96962 0.11629 -1.91598 -1.00000 1.45475 1.67098 -0.68404 -1.98648 -0.23219 1.87939 1.09902 -1.37248 -1.73205 0.57361 1.99662 0.34730 -1.83643 -1.19432 1.28558 1.78727 -0.46123 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E24 2.00000 0.34730 -1.87939 -1.00000 1.53209 1.53209 -1.00000 -1.87939 0.34730 2.00000 0.34730 -1.87939 -1.00000 1.53209 1.53209 -1.00000 -1.87939 0.34730 2.00000 0.34730 -1.87939 -1.00000 1.53209 1.53209 -1.00000 -1.87939 0.34730 2.00000 0.34730 -1.87939 -1.00000 1.53209 1.53209 -1.00000 -1.87939 0.34730 2.00000 0.34730 -1.87939 -1.00000 1.53209 1.53209 -1.00000 -1.87939 0.34730 2.00000 0.34730 -1.87939 -1.00000 1.53209 1.53209 -1.00000 -1.87939 0.34730 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E25 2.00000 0.23219 -1.94609 -0.68404 1.78727 1.09902 -1.53209 -1.45475 1.19432 1.73205 -0.79216 -1.91598 0.34730 1.99662 0.11629 -1.96962 -0.57361 1.83643 1.00000 -1.60425 -1.37248 1.28558 1.67098 -0.89760 -1.87939 0.46123 1.98648 0.00000 -1.98648 -0.46123 1.87939 0.89760 -1.67098 -1.28558 1.37248 1.60425 -1.00000 -1.83643 0.57361 1.96962 -0.11629 -1.99662 -0.34730 1.91598 0.79216 -1.73205 -1.19432 1.45475 1.53209 -1.09902 -1.78727 0.68404 1.94609 -0.23219 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. 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E52 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 ... ... ..... ....... ......... ........... ............. 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Irrational character values: 1.996616316543 = 2*cos(2*π/108) = 2*cos(π/54) 1.986476715484 = 2*cos(4*π/108) = 2*cos(π/27) 1.969615506024 = 2*cos(6*π/108) = 2*cos(π/18) 1.946089741160 = 2*cos(8*π/108) = 2*cos(2*π/27) 1.915979024631 = 2*cos(10*π/108) = 2*cos(5*π/54) 1.879385241572 = 2*cos(12*π/108) = 2*cos(π/9) 1.836432213761 = 2*cos(14*π/108) = 2*cos(7*π/54) 1.787265280647 = 2*cos(16*π/108) = 2*cos(4*π/27) 1.732050807569 = 2*cos(18*π/108) = 2*cos(π/6) = √3 1.670975622826 = 2*cos(20*π/108) = 2*cos(5*π/27) 1.604246385510 = 2*cos(22*π/108) = 2*cos(11*π/54) 1.532088886238 = 2*cos(24*π/108) = 2*cos(2*π/9) 1.454747283146 = 2*cos(26*π/108) = 2*cos(13*π/54) 1.372483275737 = 2*cos(28*π/108) = 2*cos(7*π/27) 1.285575219373 = 2*cos(30*π/108) = 2*cos(5*π/18) 1.194317183406 = 2*cos(32*π/108) = 2*cos(8*π/27) 1.099017956142 = 2*cos(34*π/108) = 2*cos(17*π/54) 0.897598360401 = 2*cos(38*π/108) = 2*cos(19*π/54) 0.792159532078 = 2*cos(40*π/108) = 2*cos(10*π/27) 0.684040286651 = 2*cos(42*π/108) = 2*cos(7*π/18) 0.573606465422 = 2*cos(44*π/108) = 2*cos(11*π/27) 0.461231741485 = 2*cos(46*π/108) = 2*cos(23*π/54) 0.347296355334 = 2*cos(48*π/108) = 2*cos(4*π/9) 0.232185828250 = 2*cos(50*π/108) = 2*cos(25*π/54) 0.116289657821 = 2*cos(52*π/108) = 2*cos(13*π/27) Symmetry of Rotations and Cartesian products A1 p+d+f+g+h+i+j+k+l+m z, z², z³, z⁴, z⁵, z⁶ A2 R R_z E1 R+p+d+f+g+h+i+j+k+l+m {R_x, R_y}, {x, y}, {xz, yz}, {xz², yz²}, {xz³, yz³}, {xz⁴, yz⁴}, {xz⁵, yz⁵} E2 d+f+g+h+i+j+k+l+m {x²−y², xy}, {z(x²−y²), xyz}, {z²(x²−y²), xyz²}, {z³(x²−y²), xyz³}, {z⁴(x²−y²), xyz⁴} E3 f+g+h+i+j+k+l+m {x(x²−3y²), y(3x²−y²)}, {xz(x²−3y²), yz(3x²−y²)}, {xz²(x²−3y²), yz²(3x²−y²)}, {xz³(x²−3y²), yz³(3x²−y²)} E4 g+h+i+j+k+l+m {(x²−y²)²−4x²y², xy(x²−y²)}, {z((x²−y²)²−4x²y²), xyz(x²−y²)}, {z²((x²−y²)²−4x²y²), xyz²(x²−y²)} E5 h+i+j+k+l+m {x(x²−(5+2√5)y²)(x²−(5−2√5)y²), y((5+2√5)x²−y²)((5−2√5)x²−y²)}, {xz(x²−(5+2√5)y²)(x²−(5−2√5)y²), yz((5+2√5)x²−y²)((5−2√5)x²−y²)} E6 i+j+k+l+m {x²(x²−3y²)²−y²(3x²−y²)², xy(x²−3y²)(3x²−y²)} E7 j+k+l+m E8 k+l+m E9 l+m E10 m Notes: α The order of the C_108v point group is 216, and the order of the principal axis (C₁₀₈) is 108. The group has 57 irreducible representations. β The C_108v point group is isomorphic to D_54d and D₁₀₈. γ The C_108v point group is generated by two symmetry elements, C₁₀₈ and any σ_v (or, non-canonically, any σ_d). Also, the group may be generated from a σ_v plus a σ_d (some pairs will yield smaller groups, though; choosing a minimum angle is safe). δ There are two different sets of symmetry planes containing the principal axis (z axis in standard orientation). By convention, the set denoted as σ_v contains both the xz and the yz planes. ε The lowest nonvanishing multipole moment in C_108v is 2 (dipole moment). ζ This point group is non-Abelian (some symmetry operations are not commutative). Therefore, the character table contains multi-membered classes and degenerate irreducible representations. η 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. θ 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 C_108v point group was created by Gernot Katzer.

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

	C_106v
	C_107v
C₁₀₈	C_108v	C_108h D₁₀₈ D_108h D_108d S₁₀₈
	C_109v
	C_110v

Character table for the C108v point group

Character table for the C_108v point group