Character Table for Point Group C50h

Character table for the C_50h point group

C50h E 2 C50 2 C25 2 C50^3 2 C25^2 2 C10 2 C25^3 2 C50^7 2 C25^4 2 C50^9 2 C5 2 C50^11 2 C25^6 2 C50^13 2 C25^7 2 C10^3 2 C25^8 2 C50^17 2 C25^9 2 C50^19 2 C5^2 2 C50^21 2 C25^11 2 C50^23 2 C25^12 C2 i 2 S50 2 S25 2 S50^3 2 S25^2 2 S10 2 S25^3 2 S50^7 2 S25^4 2 S50^9 2 S5 2 S50^11 2 S25^6 2 S50^13 2 S25^7 2 S10^3 2 S25^8 2 S50^17 2 S25^9 2 S50^19 2 S5^2 2 S50^21 2 S25^11 2 S50^23 2 S25^12 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 ..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 ... ... ..... ....... ......... ........... ............. E1g * 2.00000 1.98423 1.93717 1.85955 1.75261 1.61803 1.45794 1.27485 1.07165 0.85156 0.61803 0.37476 0.12558 -0.12558 -0.37476 -0.61803 -0.85156 -1.07165 -1.27485 -1.45794 -1.61803 -1.75261 -1.85955 -1.93717 -1.98423 -2.00000 2.00000 -1.98423 -1.93717 -1.85955 -1.75261 -1.61803 -1.45794 -1.27485 -1.07165 -0.85156 -0.61803 -0.37476 -0.12558 0.12558 0.37476 0.61803 0.85156 1.07165 1.27485 1.45794 1.61803 1.75261 1.85955 1.93717 1.98423 -2.00000 TT. ... ..TT. ....... ......TT. ........... ..........TT. E2g * 2.00000 1.93717 1.75261 1.45794 1.07165 0.61803 0.12558 -0.37476 -0.85156 -1.27485 -1.61803 -1.85955 -1.98423 -1.98423 -1.85955 -1.61803 -1.27485 -0.85156 -0.37476 0.12558 0.61803 1.07165 1.45794 1.75261 1.93717 2.00000 2.00000 1.93717 1.75261 1.45794 1.07165 0.61803 0.12558 -0.37476 -0.85156 -1.27485 -1.61803 -1.85955 -1.98423 -1.98423 -1.85955 -1.61803 -1.27485 -0.85156 -0.37476 0.12558 0.61803 1.07165 1.45794 1.75261 1.93717 2.00000 ... ... TT... ....... ....TT... ........... ........TT... E3g * 2.00000 1.85955 1.45794 0.85156 0.12558 -0.61803 -1.27485 -1.75261 -1.98423 -1.93717 -1.61803 -1.07165 -0.37476 0.37476 1.07165 1.61803 1.93717 1.98423 1.75261 1.27485 0.61803 -0.12558 -0.85156 -1.45794 -1.85955 -2.00000 2.00000 -1.85955 -1.45794 -0.85156 -0.12558 0.61803 1.27485 1.75261 1.98423 1.93717 1.61803 1.07165 0.37476 -0.37476 -1.07165 -1.61803 -1.93717 -1.98423 -1.75261 -1.27485 -0.61803 0.12558 0.85156 1.45794 1.85955 -2.00000 ... ... ..... ....... ..TT..... ........... ......TT..... E4g * 2.00000 1.75261 1.07165 0.12558 -0.85156 -1.61803 -1.98423 -1.85955 -1.27485 -0.37476 0.61803 1.45794 1.93717 1.93717 1.45794 0.61803 -0.37476 -1.27485 -1.85955 -1.98423 -1.61803 -0.85156 0.12558 1.07165 1.75261 2.00000 2.00000 1.75261 1.07165 0.12558 -0.85156 -1.61803 -1.98423 -1.85955 -1.27485 -0.37476 0.61803 1.45794 1.93717 1.93717 1.45794 0.61803 -0.37476 -1.27485 -1.85955 -1.98423 -1.61803 -0.85156 0.12558 1.07165 1.75261 2.00000 ... ... ..... ....... TT....... ........... ....TT....... E5g * 2.00000 1.61803 0.61803 -0.61803 -1.61803 -2.00000 -1.61803 -0.61803 0.61803 1.61803 2.00000 1.61803 0.61803 -0.61803 -1.61803 -2.00000 -1.61803 -0.61803 0.61803 1.61803 2.00000 1.61803 0.61803 -0.61803 -1.61803 -2.00000 2.00000 -1.61803 -0.61803 0.61803 1.61803 2.00000 1.61803 0.61803 -0.61803 -1.61803 -2.00000 -1.61803 -0.61803 0.61803 1.61803 2.00000 1.61803 0.61803 -0.61803 -1.61803 -2.00000 -1.61803 -0.61803 0.61803 1.61803 -2.00000 ... ... ..... ....... ......... ........... ..TT......... E6g * 2.00000 1.45794 0.12558 -1.27485 -1.98423 -1.61803 -0.37476 1.07165 1.93717 1.75261 0.61803 -0.85156 -1.85955 -1.85955 -0.85156 0.61803 1.75261 1.93717 1.07165 -0.37476 -1.61803 -1.98423 -1.27485 0.12558 1.45794 2.00000 2.00000 1.45794 0.12558 -1.27485 -1.98423 -1.61803 -0.37476 1.07165 1.93717 1.75261 0.61803 -0.85156 -1.85955 -1.85955 -0.85156 0.61803 1.75261 1.93717 1.07165 -0.37476 -1.61803 -1.98423 -1.27485 0.12558 1.45794 2.00000 ... ... ..... ....... ......... ........... TT........... E7g * 2.00000 1.27485 -0.37476 -1.75261 -1.85955 -0.61803 1.07165 1.98423 1.45794 -0.12558 -1.61803 -1.93717 -0.85156 0.85156 1.93717 1.61803 0.12558 -1.45794 -1.98423 -1.07165 0.61803 1.85955 1.75261 0.37476 -1.27485 -2.00000 2.00000 -1.27485 0.37476 1.75261 1.85955 0.61803 -1.07165 -1.98423 -1.45794 0.12558 1.61803 1.93717 0.85156 -0.85156 -1.93717 -1.61803 -0.12558 1.45794 1.98423 1.07165 -0.61803 -1.85955 -1.75261 -0.37476 1.27485 -2.00000 ... ... ..... ....... ......... ........... ............. E8g * 2.00000 1.07165 -0.85156 -1.98423 -1.27485 0.61803 1.93717 1.45794 -0.37476 -1.85955 -1.61803 0.12558 1.75261 1.75261 0.12558 -1.61803 -1.85955 -0.37476 1.45794 1.93717 0.61803 -1.27485 -1.98423 -0.85156 1.07165 2.00000 2.00000 1.07165 -0.85156 -1.98423 -1.27485 0.61803 1.93717 1.45794 -0.37476 -1.85955 -1.61803 0.12558 1.75261 1.75261 0.12558 -1.61803 -1.85955 -0.37476 1.45794 1.93717 0.61803 -1.27485 -1.98423 -0.85156 1.07165 2.00000 ... ... ..... ....... ......... ........... ............. E9g * 2.00000 0.85156 -1.27485 -1.93717 -0.37476 1.61803 1.75261 -0.12558 -1.85955 -1.45794 0.61803 1.98423 1.07165 -1.07165 -1.98423 -0.61803 1.45794 1.85955 0.12558 -1.75261 -1.61803 0.37476 1.93717 1.27485 -0.85156 -2.00000 2.00000 -0.85156 1.27485 1.93717 0.37476 -1.61803 -1.75261 0.12558 1.85955 1.45794 -0.61803 -1.98423 -1.07165 1.07165 1.98423 0.61803 -1.45794 -1.85955 -0.12558 1.75261 1.61803 -0.37476 -1.93717 -1.27485 0.85156 -2.00000 ... ... ..... ....... ......... ........... ............. E10g * 2.00000 0.61803 -1.61803 -1.61803 0.61803 2.00000 0.61803 -1.61803 -1.61803 0.61803 2.00000 0.61803 -1.61803 -1.61803 0.61803 2.00000 0.61803 -1.61803 -1.61803 0.61803 2.00000 0.61803 -1.61803 -1.61803 0.61803 2.00000 2.00000 0.61803 -1.61803 -1.61803 0.61803 2.00000 0.61803 -1.61803 -1.61803 0.61803 2.00000 0.61803 -1.61803 -1.61803 0.61803 2.00000 0.61803 -1.61803 -1.61803 0.61803 2.00000 0.61803 -1.61803 -1.61803 0.61803 2.00000 ... ... ..... ....... ......... ........... ............. E11g * 2.00000 0.37476 -1.85955 -1.07165 1.45794 1.61803 -0.85156 -1.93717 0.12558 1.98423 0.61803 -1.75261 -1.27485 1.27485 1.75261 -0.61803 -1.98423 -0.12558 1.93717 0.85156 -1.61803 -1.45794 1.07165 1.85955 -0.37476 -2.00000 2.00000 -0.37476 1.85955 1.07165 -1.45794 -1.61803 0.85156 1.93717 -0.12558 -1.98423 -0.61803 1.75261 1.27485 -1.27485 -1.75261 0.61803 1.98423 0.12558 -1.93717 -0.85156 1.61803 1.45794 -1.07165 -1.85955 0.37476 -2.00000 ... ... ..... ....... ......... ........... ............. E12g * 2.00000 0.12558 -1.98423 -0.37476 1.93717 0.61803 -1.85955 -0.85156 1.75261 1.07165 -1.61803 -1.27485 1.45794 1.45794 -1.27485 -1.61803 1.07165 1.75261 -0.85156 -1.85955 0.61803 1.93717 -0.37476 -1.98423 0.12558 2.00000 2.00000 0.12558 -1.98423 -0.37476 1.93717 0.61803 -1.85955 -0.85156 1.75261 1.07165 -1.61803 -1.27485 1.45794 1.45794 -1.27485 -1.61803 1.07165 1.75261 -0.85156 -1.85955 0.61803 1.93717 -0.37476 -1.98423 0.12558 2.00000 ... ... ..... ....... ......... ........... ............. E13g * 2.00000 -0.12558 -1.98423 0.37476 1.93717 -0.61803 -1.85955 0.85156 1.75261 -1.07165 -1.61803 1.27485 1.45794 -1.45794 -1.27485 1.61803 1.07165 -1.75261 -0.85156 1.85955 0.61803 -1.93717 -0.37476 1.98423 0.12558 -2.00000 2.00000 0.12558 1.98423 -0.37476 -1.93717 0.61803 1.85955 -0.85156 -1.75261 1.07165 1.61803 -1.27485 -1.45794 1.45794 1.27485 -1.61803 -1.07165 1.75261 0.85156 -1.85955 -0.61803 1.93717 0.37476 -1.98423 -0.12558 -2.00000 ... ... ..... ....... ......... ........... ............. E14g * 2.00000 -0.37476 -1.85955 1.07165 1.45794 -1.61803 -0.85156 1.93717 0.12558 -1.98423 0.61803 1.75261 -1.27485 -1.27485 1.75261 0.61803 -1.98423 0.12558 1.93717 -0.85156 -1.61803 1.45794 1.07165 -1.85955 -0.37476 2.00000 2.00000 -0.37476 -1.85955 1.07165 1.45794 -1.61803 -0.85156 1.93717 0.12558 -1.98423 0.61803 1.75261 -1.27485 -1.27485 1.75261 0.61803 -1.98423 0.12558 1.93717 -0.85156 -1.61803 1.45794 1.07165 -1.85955 -0.37476 2.00000 ... ... ..... ....... ......... ........... ............. E15g * 2.00000 -0.61803 -1.61803 1.61803 0.61803 -2.00000 0.61803 1.61803 -1.61803 -0.61803 2.00000 -0.61803 -1.61803 1.61803 0.61803 -2.00000 0.61803 1.61803 -1.61803 -0.61803 2.00000 -0.61803 -1.61803 1.61803 0.61803 -2.00000 2.00000 0.61803 1.61803 -1.61803 -0.61803 2.00000 -0.61803 -1.61803 1.61803 0.61803 -2.00000 0.61803 1.61803 -1.61803 -0.61803 2.00000 -0.61803 -1.61803 1.61803 0.61803 -2.00000 0.61803 1.61803 -1.61803 -0.61803 -2.00000 ... ... ..... ....... ......... ........... ............. E16g * 2.00000 -0.85156 -1.27485 1.93717 -0.37476 -1.61803 1.75261 0.12558 -1.85955 1.45794 0.61803 -1.98423 1.07165 1.07165 -1.98423 0.61803 1.45794 -1.85955 0.12558 1.75261 -1.61803 -0.37476 1.93717 -1.27485 -0.85156 2.00000 2.00000 -0.85156 -1.27485 1.93717 -0.37476 -1.61803 1.75261 0.12558 -1.85955 1.45794 0.61803 -1.98423 1.07165 1.07165 -1.98423 0.61803 1.45794 -1.85955 0.12558 1.75261 -1.61803 -0.37476 1.93717 -1.27485 -0.85156 2.00000 ... ... ..... ....... ......... ........... ............. E17g * 2.00000 -1.07165 -0.85156 1.98423 -1.27485 -0.61803 1.93717 -1.45794 -0.37476 1.85955 -1.61803 -0.12558 1.75261 -1.75261 0.12558 1.61803 -1.85955 0.37476 1.45794 -1.93717 0.61803 1.27485 -1.98423 0.85156 1.07165 -2.00000 2.00000 1.07165 0.85156 -1.98423 1.27485 0.61803 -1.93717 1.45794 0.37476 -1.85955 1.61803 0.12558 -1.75261 1.75261 -0.12558 -1.61803 1.85955 -0.37476 -1.45794 1.93717 -0.61803 -1.27485 1.98423 -0.85156 -1.07165 -2.00000 ... ... ..... ....... ......... ........... ............. E18g * 2.00000 -1.27485 -0.37476 1.75261 -1.85955 0.61803 1.07165 -1.98423 1.45794 0.12558 -1.61803 1.93717 -0.85156 -0.85156 1.93717 -1.61803 0.12558 1.45794 -1.98423 1.07165 0.61803 -1.85955 1.75261 -0.37476 -1.27485 2.00000 2.00000 -1.27485 -0.37476 1.75261 -1.85955 0.61803 1.07165 -1.98423 1.45794 0.12558 -1.61803 1.93717 -0.85156 -0.85156 1.93717 -1.61803 0.12558 1.45794 -1.98423 1.07165 0.61803 -1.85955 1.75261 -0.37476 -1.27485 2.00000 ... ... ..... ....... ......... ........... ............. E19g * 2.00000 -1.45794 0.12558 1.27485 -1.98423 1.61803 -0.37476 -1.07165 1.93717 -1.75261 0.61803 0.85156 -1.85955 1.85955 -0.85156 -0.61803 1.75261 -1.93717 1.07165 0.37476 -1.61803 1.98423 -1.27485 -0.12558 1.45794 -2.00000 2.00000 1.45794 -0.12558 -1.27485 1.98423 -1.61803 0.37476 1.07165 -1.93717 1.75261 -0.61803 -0.85156 1.85955 -1.85955 0.85156 0.61803 -1.75261 1.93717 -1.07165 -0.37476 1.61803 -1.98423 1.27485 0.12558 -1.45794 -2.00000 ... ... ..... ....... ......... ........... ............. E20g * 2.00000 -1.61803 0.61803 0.61803 -1.61803 2.00000 -1.61803 0.61803 0.61803 -1.61803 2.00000 -1.61803 0.61803 0.61803 -1.61803 2.00000 -1.61803 0.61803 0.61803 -1.61803 2.00000 -1.61803 0.61803 0.61803 -1.61803 2.00000 2.00000 -1.61803 0.61803 0.61803 -1.61803 2.00000 -1.61803 0.61803 0.61803 -1.61803 2.00000 -1.61803 0.61803 0.61803 -1.61803 2.00000 -1.61803 0.61803 0.61803 -1.61803 2.00000 -1.61803 0.61803 0.61803 -1.61803 2.00000 ... ... ..... ....... ......... ........... ............. E21g * 2.00000 -1.75261 1.07165 -0.12558 -0.85156 1.61803 -1.98423 1.85955 -1.27485 0.37476 0.61803 -1.45794 1.93717 -1.93717 1.45794 -0.61803 -0.37476 1.27485 -1.85955 1.98423 -1.61803 0.85156 0.12558 -1.07165 1.75261 -2.00000 2.00000 1.75261 -1.07165 0.12558 0.85156 -1.61803 1.98423 -1.85955 1.27485 -0.37476 -0.61803 1.45794 -1.93717 1.93717 -1.45794 0.61803 0.37476 -1.27485 1.85955 -1.98423 1.61803 -0.85156 -0.12558 1.07165 -1.75261 -2.00000 ... ... ..... ....... ......... ........... ............. E22g * 2.00000 -1.85955 1.45794 -0.85156 0.12558 0.61803 -1.27485 1.75261 -1.98423 1.93717 -1.61803 1.07165 -0.37476 -0.37476 1.07165 -1.61803 1.93717 -1.98423 1.75261 -1.27485 0.61803 0.12558 -0.85156 1.45794 -1.85955 2.00000 2.00000 -1.85955 1.45794 -0.85156 0.12558 0.61803 -1.27485 1.75261 -1.98423 1.93717 -1.61803 1.07165 -0.37476 -0.37476 1.07165 -1.61803 1.93717 -1.98423 1.75261 -1.27485 0.61803 0.12558 -0.85156 1.45794 -1.85955 2.00000 ... ... ..... ....... ......... ........... ............. E23g * 2.00000 -1.93717 1.75261 -1.45794 1.07165 -0.61803 0.12558 0.37476 -0.85156 1.27485 -1.61803 1.85955 -1.98423 1.98423 -1.85955 1.61803 -1.27485 0.85156 -0.37476 -0.12558 0.61803 -1.07165 1.45794 -1.75261 1.93717 -2.00000 2.00000 1.93717 -1.75261 1.45794 -1.07165 0.61803 -0.12558 -0.37476 0.85156 -1.27485 1.61803 -1.85955 1.98423 -1.98423 1.85955 -1.61803 1.27485 -0.85156 0.37476 0.12558 -0.61803 1.07165 -1.45794 1.75261 -1.93717 -2.00000 ... ... ..... ....... ......... ........... ............. E24g * 2.00000 -1.98423 1.93717 -1.85955 1.75261 -1.61803 1.45794 -1.27485 1.07165 -0.85156 0.61803 -0.37476 0.12558 0.12558 -0.37476 0.61803 -0.85156 1.07165 -1.27485 1.45794 -1.61803 1.75261 -1.85955 1.93717 -1.98423 2.00000 2.00000 -1.98423 1.93717 -1.85955 1.75261 -1.61803 1.45794 -1.27485 1.07165 -0.85156 0.61803 -0.37476 0.12558 0.12558 -0.37476 0.61803 -0.85156 1.07165 -1.27485 1.45794 -1.61803 1.75261 -1.85955 1.93717 -1.98423 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 ... ..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 ... ... ..... ....... ......... ........... ............. E1u * 2.00000 1.98423 1.93717 1.85955 1.75261 1.61803 1.45794 1.27485 1.07165 0.85156 0.61803 0.37476 0.12558 -0.12558 -0.37476 -0.61803 -0.85156 -1.07165 -1.27485 -1.45794 -1.61803 -1.75261 -1.85955 -1.93717 -1.98423 -2.00000 -2.00000 1.98423 1.93717 1.85955 1.75261 1.61803 1.45794 1.27485 1.07165 0.85156 0.61803 0.37476 0.12558 -0.12558 -0.37476 -0.61803 -0.85156 -1.07165 -1.27485 -1.45794 -1.61803 -1.75261 -1.85955 -1.93717 -1.98423 2.00000 ... TT. ..... ....TT. ......... ........TT. ............. E2u * 2.00000 1.93717 1.75261 1.45794 1.07165 0.61803 0.12558 -0.37476 -0.85156 -1.27485 -1.61803 -1.85955 -1.98423 -1.98423 -1.85955 -1.61803 -1.27485 -0.85156 -0.37476 0.12558 0.61803 1.07165 1.45794 1.75261 1.93717 2.00000 -2.00000 -1.93717 -1.75261 -1.45794 -1.07165 -0.61803 -0.12558 0.37476 0.85156 1.27485 1.61803 1.85955 1.98423 1.98423 1.85955 1.61803 1.27485 0.85156 0.37476 -0.12558 -0.61803 -1.07165 -1.45794 -1.75261 -1.93717 -2.00000 ... ... ..... ..TT... ......... ......TT... ............. E3u * 2.00000 1.85955 1.45794 0.85156 0.12558 -0.61803 -1.27485 -1.75261 -1.98423 -1.93717 -1.61803 -1.07165 -0.37476 0.37476 1.07165 1.61803 1.93717 1.98423 1.75261 1.27485 0.61803 -0.12558 -0.85156 -1.45794 -1.85955 -2.00000 -2.00000 1.85955 1.45794 0.85156 0.12558 -0.61803 -1.27485 -1.75261 -1.98423 -1.93717 -1.61803 -1.07165 -0.37476 0.37476 1.07165 1.61803 1.93717 1.98423 1.75261 1.27485 0.61803 -0.12558 -0.85156 -1.45794 -1.85955 2.00000 ... ... ..... TT..... ......... ....TT..... ............. E4u * 2.00000 1.75261 1.07165 0.12558 -0.85156 -1.61803 -1.98423 -1.85955 -1.27485 -0.37476 0.61803 1.45794 1.93717 1.93717 1.45794 0.61803 -0.37476 -1.27485 -1.85955 -1.98423 -1.61803 -0.85156 0.12558 1.07165 1.75261 2.00000 -2.00000 -1.75261 -1.07165 -0.12558 0.85156 1.61803 1.98423 1.85955 1.27485 0.37476 -0.61803 -1.45794 -1.93717 -1.93717 -1.45794 -0.61803 0.37476 1.27485 1.85955 1.98423 1.61803 0.85156 -0.12558 -1.07165 -1.75261 -2.00000 ... ... ..... ....... ......... ..TT....... ............. E5u * 2.00000 1.61803 0.61803 -0.61803 -1.61803 -2.00000 -1.61803 -0.61803 0.61803 1.61803 2.00000 1.61803 0.61803 -0.61803 -1.61803 -2.00000 -1.61803 -0.61803 0.61803 1.61803 2.00000 1.61803 0.61803 -0.61803 -1.61803 -2.00000 -2.00000 1.61803 0.61803 -0.61803 -1.61803 -2.00000 -1.61803 -0.61803 0.61803 1.61803 2.00000 1.61803 0.61803 -0.61803 -1.61803 -2.00000 -1.61803 -0.61803 0.61803 1.61803 2.00000 1.61803 0.61803 -0.61803 -1.61803 2.00000 ... ... ..... ....... ......... TT......... ............. E6u * 2.00000 1.45794 0.12558 -1.27485 -1.98423 -1.61803 -0.37476 1.07165 1.93717 1.75261 0.61803 -0.85156 -1.85955 -1.85955 -0.85156 0.61803 1.75261 1.93717 1.07165 -0.37476 -1.61803 -1.98423 -1.27485 0.12558 1.45794 2.00000 -2.00000 -1.45794 -0.12558 1.27485 1.98423 1.61803 0.37476 -1.07165 -1.93717 -1.75261 -0.61803 0.85156 1.85955 1.85955 0.85156 -0.61803 -1.75261 -1.93717 -1.07165 0.37476 1.61803 1.98423 1.27485 -0.12558 -1.45794 -2.00000 ... ... ..... ....... ......... ........... ............. E7u * 2.00000 1.27485 -0.37476 -1.75261 -1.85955 -0.61803 1.07165 1.98423 1.45794 -0.12558 -1.61803 -1.93717 -0.85156 0.85156 1.93717 1.61803 0.12558 -1.45794 -1.98423 -1.07165 0.61803 1.85955 1.75261 0.37476 -1.27485 -2.00000 -2.00000 1.27485 -0.37476 -1.75261 -1.85955 -0.61803 1.07165 1.98423 1.45794 -0.12558 -1.61803 -1.93717 -0.85156 0.85156 1.93717 1.61803 0.12558 -1.45794 -1.98423 -1.07165 0.61803 1.85955 1.75261 0.37476 -1.27485 2.00000 ... ... ..... ....... ......... ........... ............. E8u * 2.00000 1.07165 -0.85156 -1.98423 -1.27485 0.61803 1.93717 1.45794 -0.37476 -1.85955 -1.61803 0.12558 1.75261 1.75261 0.12558 -1.61803 -1.85955 -0.37476 1.45794 1.93717 0.61803 -1.27485 -1.98423 -0.85156 1.07165 2.00000 -2.00000 -1.07165 0.85156 1.98423 1.27485 -0.61803 -1.93717 -1.45794 0.37476 1.85955 1.61803 -0.12558 -1.75261 -1.75261 -0.12558 1.61803 1.85955 0.37476 -1.45794 -1.93717 -0.61803 1.27485 1.98423 0.85156 -1.07165 -2.00000 ... ... ..... ....... ......... ........... ............. E9u * 2.00000 0.85156 -1.27485 -1.93717 -0.37476 1.61803 1.75261 -0.12558 -1.85955 -1.45794 0.61803 1.98423 1.07165 -1.07165 -1.98423 -0.61803 1.45794 1.85955 0.12558 -1.75261 -1.61803 0.37476 1.93717 1.27485 -0.85156 -2.00000 -2.00000 0.85156 -1.27485 -1.93717 -0.37476 1.61803 1.75261 -0.12558 -1.85955 -1.45794 0.61803 1.98423 1.07165 -1.07165 -1.98423 -0.61803 1.45794 1.85955 0.12558 -1.75261 -1.61803 0.37476 1.93717 1.27485 -0.85156 2.00000 ... ... ..... ....... ......... ........... ............. E10u * 2.00000 0.61803 -1.61803 -1.61803 0.61803 2.00000 0.61803 -1.61803 -1.61803 0.61803 2.00000 0.61803 -1.61803 -1.61803 0.61803 2.00000 0.61803 -1.61803 -1.61803 0.61803 2.00000 0.61803 -1.61803 -1.61803 0.61803 2.00000 -2.00000 -0.61803 1.61803 1.61803 -0.61803 -2.00000 -0.61803 1.61803 1.61803 -0.61803 -2.00000 -0.61803 1.61803 1.61803 -0.61803 -2.00000 -0.61803 1.61803 1.61803 -0.61803 -2.00000 -0.61803 1.61803 1.61803 -0.61803 -2.00000 ... ... ..... ....... ......... ........... ............. E11u * 2.00000 0.37476 -1.85955 -1.07165 1.45794 1.61803 -0.85156 -1.93717 0.12558 1.98423 0.61803 -1.75261 -1.27485 1.27485 1.75261 -0.61803 -1.98423 -0.12558 1.93717 0.85156 -1.61803 -1.45794 1.07165 1.85955 -0.37476 -2.00000 -2.00000 0.37476 -1.85955 -1.07165 1.45794 1.61803 -0.85156 -1.93717 0.12558 1.98423 0.61803 -1.75261 -1.27485 1.27485 1.75261 -0.61803 -1.98423 -0.12558 1.93717 0.85156 -1.61803 -1.45794 1.07165 1.85955 -0.37476 2.00000 ... ... ..... ....... ......... ........... ............. E12u * 2.00000 0.12558 -1.98423 -0.37476 1.93717 0.61803 -1.85955 -0.85156 1.75261 1.07165 -1.61803 -1.27485 1.45794 1.45794 -1.27485 -1.61803 1.07165 1.75261 -0.85156 -1.85955 0.61803 1.93717 -0.37476 -1.98423 0.12558 2.00000 -2.00000 -0.12558 1.98423 0.37476 -1.93717 -0.61803 1.85955 0.85156 -1.75261 -1.07165 1.61803 1.27485 -1.45794 -1.45794 1.27485 1.61803 -1.07165 -1.75261 0.85156 1.85955 -0.61803 -1.93717 0.37476 1.98423 -0.12558 -2.00000 ... ... ..... ....... ......... ........... ............. E13u * 2.00000 -0.12558 -1.98423 0.37476 1.93717 -0.61803 -1.85955 0.85156 1.75261 -1.07165 -1.61803 1.27485 1.45794 -1.45794 -1.27485 1.61803 1.07165 -1.75261 -0.85156 1.85955 0.61803 -1.93717 -0.37476 1.98423 0.12558 -2.00000 -2.00000 -0.12558 -1.98423 0.37476 1.93717 -0.61803 -1.85955 0.85156 1.75261 -1.07165 -1.61803 1.27485 1.45794 -1.45794 -1.27485 1.61803 1.07165 -1.75261 -0.85156 1.85955 0.61803 -1.93717 -0.37476 1.98423 0.12558 2.00000 ... ... ..... ....... ......... ........... ............. E14u * 2.00000 -0.37476 -1.85955 1.07165 1.45794 -1.61803 -0.85156 1.93717 0.12558 -1.98423 0.61803 1.75261 -1.27485 -1.27485 1.75261 0.61803 -1.98423 0.12558 1.93717 -0.85156 -1.61803 1.45794 1.07165 -1.85955 -0.37476 2.00000 -2.00000 0.37476 1.85955 -1.07165 -1.45794 1.61803 0.85156 -1.93717 -0.12558 1.98423 -0.61803 -1.75261 1.27485 1.27485 -1.75261 -0.61803 1.98423 -0.12558 -1.93717 0.85156 1.61803 -1.45794 -1.07165 1.85955 0.37476 -2.00000 ... ... ..... ....... ......... ........... ............. E15u * 2.00000 -0.61803 -1.61803 1.61803 0.61803 -2.00000 0.61803 1.61803 -1.61803 -0.61803 2.00000 -0.61803 -1.61803 1.61803 0.61803 -2.00000 0.61803 1.61803 -1.61803 -0.61803 2.00000 -0.61803 -1.61803 1.61803 0.61803 -2.00000 -2.00000 -0.61803 -1.61803 1.61803 0.61803 -2.00000 0.61803 1.61803 -1.61803 -0.61803 2.00000 -0.61803 -1.61803 1.61803 0.61803 -2.00000 0.61803 1.61803 -1.61803 -0.61803 2.00000 -0.61803 -1.61803 1.61803 0.61803 2.00000 ... ... ..... ....... ......... ........... ............. E16u * 2.00000 -0.85156 -1.27485 1.93717 -0.37476 -1.61803 1.75261 0.12558 -1.85955 1.45794 0.61803 -1.98423 1.07165 1.07165 -1.98423 0.61803 1.45794 -1.85955 0.12558 1.75261 -1.61803 -0.37476 1.93717 -1.27485 -0.85156 2.00000 -2.00000 0.85156 1.27485 -1.93717 0.37476 1.61803 -1.75261 -0.12558 1.85955 -1.45794 -0.61803 1.98423 -1.07165 -1.07165 1.98423 -0.61803 -1.45794 1.85955 -0.12558 -1.75261 1.61803 0.37476 -1.93717 1.27485 0.85156 -2.00000 ... ... ..... ....... ......... ........... ............. E17u * 2.00000 -1.07165 -0.85156 1.98423 -1.27485 -0.61803 1.93717 -1.45794 -0.37476 1.85955 -1.61803 -0.12558 1.75261 -1.75261 0.12558 1.61803 -1.85955 0.37476 1.45794 -1.93717 0.61803 1.27485 -1.98423 0.85156 1.07165 -2.00000 -2.00000 -1.07165 -0.85156 1.98423 -1.27485 -0.61803 1.93717 -1.45794 -0.37476 1.85955 -1.61803 -0.12558 1.75261 -1.75261 0.12558 1.61803 -1.85955 0.37476 1.45794 -1.93717 0.61803 1.27485 -1.98423 0.85156 1.07165 2.00000 ... ... ..... ....... ......... ........... ............. E18u * 2.00000 -1.27485 -0.37476 1.75261 -1.85955 0.61803 1.07165 -1.98423 1.45794 0.12558 -1.61803 1.93717 -0.85156 -0.85156 1.93717 -1.61803 0.12558 1.45794 -1.98423 1.07165 0.61803 -1.85955 1.75261 -0.37476 -1.27485 2.00000 -2.00000 1.27485 0.37476 -1.75261 1.85955 -0.61803 -1.07165 1.98423 -1.45794 -0.12558 1.61803 -1.93717 0.85156 0.85156 -1.93717 1.61803 -0.12558 -1.45794 1.98423 -1.07165 -0.61803 1.85955 -1.75261 0.37476 1.27485 -2.00000 ... ... ..... ....... ......... ........... ............. E19u * 2.00000 -1.45794 0.12558 1.27485 -1.98423 1.61803 -0.37476 -1.07165 1.93717 -1.75261 0.61803 0.85156 -1.85955 1.85955 -0.85156 -0.61803 1.75261 -1.93717 1.07165 0.37476 -1.61803 1.98423 -1.27485 -0.12558 1.45794 -2.00000 -2.00000 -1.45794 0.12558 1.27485 -1.98423 1.61803 -0.37476 -1.07165 1.93717 -1.75261 0.61803 0.85156 -1.85955 1.85955 -0.85156 -0.61803 1.75261 -1.93717 1.07165 0.37476 -1.61803 1.98423 -1.27485 -0.12558 1.45794 2.00000 ... ... ..... ....... ......... ........... ............. E20u * 2.00000 -1.61803 0.61803 0.61803 -1.61803 2.00000 -1.61803 0.61803 0.61803 -1.61803 2.00000 -1.61803 0.61803 0.61803 -1.61803 2.00000 -1.61803 0.61803 0.61803 -1.61803 2.00000 -1.61803 0.61803 0.61803 -1.61803 2.00000 -2.00000 1.61803 -0.61803 -0.61803 1.61803 -2.00000 1.61803 -0.61803 -0.61803 1.61803 -2.00000 1.61803 -0.61803 -0.61803 1.61803 -2.00000 1.61803 -0.61803 -0.61803 1.61803 -2.00000 1.61803 -0.61803 -0.61803 1.61803 -2.00000 ... ... ..... ....... ......... ........... ............. E21u * 2.00000 -1.75261 1.07165 -0.12558 -0.85156 1.61803 -1.98423 1.85955 -1.27485 0.37476 0.61803 -1.45794 1.93717 -1.93717 1.45794 -0.61803 -0.37476 1.27485 -1.85955 1.98423 -1.61803 0.85156 0.12558 -1.07165 1.75261 -2.00000 -2.00000 -1.75261 1.07165 -0.12558 -0.85156 1.61803 -1.98423 1.85955 -1.27485 0.37476 0.61803 -1.45794 1.93717 -1.93717 1.45794 -0.61803 -0.37476 1.27485 -1.85955 1.98423 -1.61803 0.85156 0.12558 -1.07165 1.75261 2.00000 ... ... ..... ....... ......... ........... ............. E22u * 2.00000 -1.85955 1.45794 -0.85156 0.12558 0.61803 -1.27485 1.75261 -1.98423 1.93717 -1.61803 1.07165 -0.37476 -0.37476 1.07165 -1.61803 1.93717 -1.98423 1.75261 -1.27485 0.61803 0.12558 -0.85156 1.45794 -1.85955 2.00000 -2.00000 1.85955 -1.45794 0.85156 -0.12558 -0.61803 1.27485 -1.75261 1.98423 -1.93717 1.61803 -1.07165 0.37476 0.37476 -1.07165 1.61803 -1.93717 1.98423 -1.75261 1.27485 -0.61803 -0.12558 0.85156 -1.45794 1.85955 -2.00000 ... ... ..... ....... ......... ........... ............. E23u * 2.00000 -1.93717 1.75261 -1.45794 1.07165 -0.61803 0.12558 0.37476 -0.85156 1.27485 -1.61803 1.85955 -1.98423 1.98423 -1.85955 1.61803 -1.27485 0.85156 -0.37476 -0.12558 0.61803 -1.07165 1.45794 -1.75261 1.93717 -2.00000 -2.00000 -1.93717 1.75261 -1.45794 1.07165 -0.61803 0.12558 0.37476 -0.85156 1.27485 -1.61803 1.85955 -1.98423 1.98423 -1.85955 1.61803 -1.27485 0.85156 -0.37476 -0.12558 0.61803 -1.07165 1.45794 -1.75261 1.93717 2.00000 ... ... ..... ....... ......... ........... ............. E24u * 2.00000 -1.98423 1.93717 -1.85955 1.75261 -1.61803 1.45794 -1.27485 1.07165 -0.85156 0.61803 -0.37476 0.12558 0.12558 -0.37476 0.61803 -0.85156 1.07165 -1.27485 1.45794 -1.61803 1.75261 -1.85955 1.93717 -1.98423 2.00000 -2.00000 1.98423 -1.93717 1.85955 -1.75261 1.61803 -1.45794 1.27485 -1.07165 0.85156 -0.61803 0.37476 -0.12558 -0.12558 0.37476 -0.61803 0.85156 -1.07165 1.27485 -1.45794 1.61803 -1.75261 1.85955 -1.93717 1.98423 -2.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.984229402629 = 2*cos(2*π/50) = 2*cos(π/25) 1.937166322257 = 2*cos(4*π/50) = 2*cos(2*π/25) 1.859552971777 = 2*cos(6*π/50) = 2*cos(3*π/25) 1.752613360088 = 2*cos(8*π/50) = 2*cos(4*π/25) 1.618033988750 = 2*cos(10*π/50) = 2*cos(π/5) = (√5+1)/2 1.457937254843 = 2*cos(12*π/50) = 2*cos(6*π/25) 1.274847979497 = 2*cos(14*π/50) = 2*cos(7*π/25) 1.071653589958 = 2*cos(16*π/50) = 2*cos(8*π/25) 0.851558583130 = 2*cos(18*π/50) = 2*cos(9*π/25) 0.618033988750 = 2*cos(20*π/50) = 2*cos(2*π/5) = (√5−1)/2 0.374762629171 = 2*cos(22*π/50) = 2*cos(11*π/25) 0.125581039059 = 2*cos(24*π/50) = 2*cos(12*π/25) 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_50h point group is 100, and the order of the principal axis (C₅₀) is 50. The group has 52 irreducible representations. β The C_50h point group is generated by two symmetry elements, which are canonically chosen as C₅₀ and i. Other possible choices are C₅₀ and σ_h, or less commonly S₅₀ with either C₂ or σ_h. γ The lowest nonvanishing multipole moment in C_50h is 4 (quadrupole moment). δ This is an Abelian point group (the commutative law holds between all symmetry operations). The C_50h 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 96 cases have been combined into 48 two-dimensional representations that are no longer irreducible but have real-valued characters. Accordingly, 48 pairs of left and right rotations have been combined into one two-membered pseudo-class each. ζ The 48 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. For this group, however, some of the irrational characters cannot be expressed by a closed algebraic form using real numbers only.

This Character Table for the C_50h point group was created by Gernot Katzer.

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

	C_48h
	C_49h
C₅₀ C_50v	C_50h	D₅₀ D_50h D_50d S₅₀
	C_51h
	C_52h

Character table for the C50h point group

Character table for the C_50h point group