Character Table for Point Group C54h

Character table for the C_54h point group

C54h E 2 C54 2 C27 2 C18 2 C27^2 2 C54^5 2 C9 2 C54^7 2 C27^4 2 C6 2 C27^5 2 C54^11 2 C9^2 2 C54^13 2 C27^7 2 C18^5 2 C27^8 2 C54^17 2 C3 2 C54^19 2 C27^10 2 C18^7 2 C27^11 2 C54^23 2 C9^4 2 C54^25 2 C27^13 C2 i 2 S54 2 S27 2 S18 2 S27^2 2 S54^5 2 S9 2 S54^7 2 S27^4 2 S6 2 S27^5 2 S54^11 2 S9^2 2 S54^13 2 S27^7 2 S18^5 2 S27^8 2 S54^17 2 S3 2 S54^19 2 S27^10 2 S18^7 2 S27^11 2 S54^23 2 S9^4 2 S54^25 2 S27^13 sh <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> Ag 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 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 Bg 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 -1.00000 ... ... ..... ....... ......... ........... ............. E1g * 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 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 TT. ... ..TT. ....... ......TT. ........... ..........TT. E2g * 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 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 ... ... TT... ....... ....TT... ........... ........TT... E3g * 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 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 ... ... ..... ....... ..TT..... ........... ......TT..... E4g * 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 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 ... ... ..... ....... TT....... ........... ....TT....... E5g * 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 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 ... ... ..... ....... ......... ........... ..TT......... E6g * 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 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 ... ... ..... ....... ......... ........... TT........... E7g * 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 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 ... ... ..... ....... ......... ........... ............. E8g * 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 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 ... ... ..... ....... ......... ........... ............. E9g * 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 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 ... ... ..... ....... ......... ........... ............. E10g * 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 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 ... ... ..... ....... ......... ........... ............. E11g * 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 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 ... ... ..... ....... ......... ........... ............. E12g * 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 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 ... ... ..... ....... ......... ........... ............. E13g * 2.00000 0.11629 -1.98648 -0.34730 1.94609 0.57361 -1.87939 -0.79216 1.78727 1.00000 -1.67098 -1.19432 1.53209 1.37248 -1.37248 -1.53209 1.19432 1.67098 -1.00000 -1.78727 0.79216 1.87939 -0.57361 -1.94609 0.34730 1.98648 -0.11629 -2.00000 2.00000 -0.11629 1.98648 0.34730 -1.94609 -0.57361 1.87939 0.79216 -1.78727 -1.00000 1.67098 1.19432 -1.53209 -1.37248 1.37248 1.53209 -1.19432 -1.67098 1.00000 1.78727 -0.79216 -1.87939 0.57361 1.94609 -0.34730 -1.98648 0.11629 -2.00000 ... ... ..... ....... ......... ........... ............. E14g * 2.00000 -0.11629 -1.98648 0.34730 1.94609 -0.57361 -1.87939 0.79216 1.78727 -1.00000 -1.67098 1.19432 1.53209 -1.37248 -1.37248 1.53209 1.19432 -1.67098 -1.00000 1.78727 0.79216 -1.87939 -0.57361 1.94609 0.34730 -1.98648 -0.11629 2.00000 2.00000 -0.11629 -1.98648 0.34730 1.94609 -0.57361 -1.87939 0.79216 1.78727 -1.00000 -1.67098 1.19432 1.53209 -1.37248 -1.37248 1.53209 1.19432 -1.67098 -1.00000 1.78727 0.79216 -1.87939 -0.57361 1.94609 0.34730 -1.98648 -0.11629 2.00000 ... ... ..... ....... ......... ........... ............. E15g * 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 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 ... ... ..... ....... ......... ........... ............. E16g * 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 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 ... ... ..... ....... ......... ........... ............. E17g * 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 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 ... ... ..... ....... ......... ........... ............. E18g * 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 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 ... ... ..... ....... ......... ........... ............. E19g * 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 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 ... ... ..... ....... ......... ........... ............. E20g * 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 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 ... ... ..... ....... ......... ........... ............. E21g * 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 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 ... ... ..... ....... ......... ........... ............. E22g * 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 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 ... ... ..... ....... ......... ........... ............. E23g * 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 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 ... ... ..... ....... ......... ........... ............. E24g * 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 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 ... ... ..... ....... ......... ........... ............. E25g * 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 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 ... ... ..... ....... ......... ........... ............. E26g * 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 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 ... ... ..... ....... ......... ........... ............. Au 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -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 ............. Bu 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 -1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 1.00000 ... ... ..... ....... ......... ........... ............. E1u * 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 -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 ... TT. ..... ....TT. ......... ........TT. ............. E2u * 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 -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 ... ... ..... ..TT... ......... ......TT... ............. E3u * 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 -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 ... ... ..... TT..... ......... ....TT..... ............. E4u * 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 -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 ... ... ..... ....... ......... ..TT....... ............. E5u * 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 -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 ... ... ..... ....... ......... TT......... ............. E6u * 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 -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 ... ... ..... ....... ......... ........... ............. E7u * 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 -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 ... ... ..... ....... ......... ........... ............. E8u * 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 -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 ... ... ..... ....... ......... ........... ............. E9u * 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 -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 ... ... ..... ....... ......... ........... ............. E10u * 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 -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 ... ... ..... ....... ......... ........... ............. E11u * 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 -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 ... ... ..... ....... ......... ........... ............. E12u * 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 -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 ... ... ..... ....... ......... ........... ............. E13u * 2.00000 0.11629 -1.98648 -0.34730 1.94609 0.57361 -1.87939 -0.79216 1.78727 1.00000 -1.67098 -1.19432 1.53209 1.37248 -1.37248 -1.53209 1.19432 1.67098 -1.00000 -1.78727 0.79216 1.87939 -0.57361 -1.94609 0.34730 1.98648 -0.11629 -2.00000 -2.00000 0.11629 -1.98648 -0.34730 1.94609 0.57361 -1.87939 -0.79216 1.78727 1.00000 -1.67098 -1.19432 1.53209 1.37248 -1.37248 -1.53209 1.19432 1.67098 -1.00000 -1.78727 0.79216 1.87939 -0.57361 -1.94609 0.34730 1.98648 -0.11629 2.00000 ... ... ..... ....... ......... ........... ............. E14u * 2.00000 -0.11629 -1.98648 0.34730 1.94609 -0.57361 -1.87939 0.79216 1.78727 -1.00000 -1.67098 1.19432 1.53209 -1.37248 -1.37248 1.53209 1.19432 -1.67098 -1.00000 1.78727 0.79216 -1.87939 -0.57361 1.94609 0.34730 -1.98648 -0.11629 2.00000 -2.00000 0.11629 1.98648 -0.34730 -1.94609 0.57361 1.87939 -0.79216 -1.78727 1.00000 1.67098 -1.19432 -1.53209 1.37248 1.37248 -1.53209 -1.19432 1.67098 1.00000 -1.78727 -0.79216 1.87939 0.57361 -1.94609 -0.34730 1.98648 0.11629 -2.00000 ... ... ..... ....... ......... ........... ............. E15u * 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 -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 ... ... ..... ....... ......... ........... ............. E16u * 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 -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 ... ... ..... ....... ......... ........... ............. E17u * 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 -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 ... ... ..... ....... ......... ........... ............. E18u * 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 -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 ... ... ..... ....... ......... ........... ............. E19u * 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 -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 ... ... ..... ....... ......... ........... ............. E20u * 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 -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 ... ... ..... ....... ......... ........... ............. E21u * 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 -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 ... ... ..... ....... ......... ........... ............. E22u * 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 -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 ... ... ..... ....... ......... ........... ............. E23u * 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 -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 ... ... ..... ....... ......... ........... ............. E24u * 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 -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 ... ... ..... ....... ......... ........... ............. E25u * 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 -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 ... ... ..... ....... ......... ........... ............. E26u * 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 -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 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.986476715484 = 2*cos(2*π/54) = 2*cos(π/27) 1.946089741160 = 2*cos(4*π/54) = 2*cos(2*π/27) 1.879385241572 = 2*cos(6*π/54) = 2*cos(π/9) 1.787265280647 = 2*cos(8*π/54) = 2*cos(4*π/27) 1.670975622826 = 2*cos(10*π/54) = 2*cos(5*π/27) 1.532088886238 = 2*cos(12*π/54) = 2*cos(2*π/9) 1.372483275737 = 2*cos(14*π/54) = 2*cos(7*π/27) 1.194317183406 = 2*cos(16*π/54) = 2*cos(8*π/27) 0.792159532078 = 2*cos(20*π/54) = 2*cos(10*π/27) 0.573606465422 = 2*cos(22*π/54) = 2*cos(11*π/27) 0.347296355334 = 2*cos(24*π/54) = 2*cos(4*π/9) 0.116289657821 = 2*cos(26*π/54) = 2*cos(13*π/27) Symmetry of Rotations and Cartesian products Ag R+d+g+i+k+m R_z, z², z⁴, z⁶ E1g R+d+g+i+k+m {R_x, R_y}, {xz, yz}, {xz³, yz³}, {xz⁵, yz⁵} E2g d+g+i+k+m {x²−y², xy}, {z²(x²−y²), xyz²}, {z⁴(x²−y²), xyz⁴} E3g g+i+k+m {xz(x²−3y²), yz(3x²−y²)}, {xz³(x²−3y²), yz³(3x²−y²)} E4g g+i+k+m {(x²−y²)²−4x²y², xy(x²−y²)}, {z²((x²−y²)²−4x²y²), xyz²(x²−y²)} E5g 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)} E6g i+k+m {x²(x²−3y²)²−y²(3x²−y²)², xy(x²−3y²)(3x²−y²)} E7g k+m E8g k+m E9g m E10g m Au p+f+h+j+l z, z³, z⁵ E1u p+f+h+j+l {x, y}, {xz², yz²}, {xz⁴, yz⁴} E2u f+h+j+l {z(x²−y²), xyz}, {z³(x²−y²), xyz³} E3u f+h+j+l {x(x²−3y²), y(3x²−y²)}, {xz²(x²−3y²), yz²(3x²−y²)} E4u h+j+l {z((x²−y²)²−4x²y²), xyz(x²−y²)} E5u 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)} E6u j+l E7u j+l E8u l E9u l Notes: α The order of the C_54h point group is 108, and the order of the principal axis (C₅₄) is 54. The group has 56 irreducible representations. β The C_54h point group is generated by two symmetry elements, which are canonically chosen as C₅₄ and i. Other possible choices are C₅₄ and σ_h, or less commonly S₅₄ with either C₂ or σ_h. γ The lowest nonvanishing multipole moment in C_54h is 4 (quadrupole moment). δ This is an Abelian point group (the commutative law holds between all symmetry operations). The C_54h group is Abelian because all its symmetry operations are coaxial. This is a sufficient condition. In Abelian groups, all symmetry operations form a class of their own, and all irreducible representations are one-dimensional. ε Because the group is Abelian and the maximum order of rotation is >2, some irreducible representations have complex characters. These 104 cases have been combined into 52 two-dimensional representations that are no longer irreducible but have real-valued characters. Accordingly, 52 pairs of left and right rotations have been combined into one two-membered pseudo-class each. ζ The 52 reducible “E” representations almost behave like true irreducible representations. Their norm, however, is twice the group order. Therefore, they have been marked with an asterisk in the table. This is essential when trying to decompose a reducible representation into “irreducible” ones using the familiar projection formula. 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. All characters of this group can be expressed using complex numbers, elementary arithmetic operations, square roots and third roots.

This Character Table for the C_54h point group was created by Gernot Katzer.

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

	C_52h
	C_53h
C₅₄ C_54v	C_54h	D₅₄ D_54h D_54d S₅₄
	C_55h
	C_56h

Character table for the C54h point group

Character table for the C_54h point group