Character Table for Point Group C48h

Character table for the C_48h point group

C48h E 2 C48 2 C24 2 C16 2 C12 2 C48^5 2 C8 2 C48^7 2 C6 2 C16^3 2 C24^5 2 C48^11 2 C4 2 C48^13 2 C24^7 2 C16^5 2 C3 2 C48^17 2 C8^3 2 C48^19 2 C12^5 2 C16^7 2 C24^11 2 C48^23 C2 i 2 S48 2 S24 2 S16 2 S12 2 S48^5 2 S8 2 S48^7 2 S6 2 S16^3 2 S24^5 2 S48^11 2 S4 2 S48^13 2 S24^7 2 S16^5 2 S3 2 S48^17 2 S8^3 2 S48^19 2 S12^5 2 S16^7 2 S24^11 2 S48^23 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 ..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 ... ... ..... ....... ......... ........... ............. E1g * 2.00000 1.98289 1.93185 1.84776 1.73205 1.58671 1.41421 1.21752 1.00000 0.76537 0.51764 0.26105 0.00000 -0.26105 -0.51764 -0.76537 -1.00000 -1.21752 -1.41421 -1.58671 -1.73205 -1.84776 -1.93185 -1.98289 -2.00000 2.00000 -1.98289 -1.93185 -1.84776 -1.73205 -1.58671 -1.41421 -1.21752 -1.00000 -0.76537 -0.51764 -0.26105 0.00000 0.26105 0.51764 0.76537 1.00000 1.21752 1.41421 1.58671 1.73205 1.84776 1.93185 1.98289 -2.00000 TT. ... ..TT. ....... ......TT. ........... ..........TT. E2g * 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 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 ... ... TT... ....... ....TT... ........... ........TT... E3g * 2.00000 1.84776 1.41421 0.76537 0.00000 -0.76537 -1.41421 -1.84776 -2.00000 -1.84776 -1.41421 -0.76537 0.00000 0.76537 1.41421 1.84776 2.00000 1.84776 1.41421 0.76537 0.00000 -0.76537 -1.41421 -1.84776 -2.00000 2.00000 -1.84776 -1.41421 -0.76537 0.00000 0.76537 1.41421 1.84776 2.00000 1.84776 1.41421 0.76537 0.00000 -0.76537 -1.41421 -1.84776 -2.00000 -1.84776 -1.41421 -0.76537 0.00000 0.76537 1.41421 1.84776 -2.00000 ... ... ..... ....... ..TT..... ........... ......TT..... E4g * 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 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 ... ... ..... ....... TT....... ........... ....TT....... E5g * 2.00000 1.58671 0.51764 -0.76537 -1.73205 -1.98289 -1.41421 -0.26105 1.00000 1.84776 1.93185 1.21752 0.00000 -1.21752 -1.93185 -1.84776 -1.00000 0.26105 1.41421 1.98289 1.73205 0.76537 -0.51764 -1.58671 -2.00000 2.00000 -1.58671 -0.51764 0.76537 1.73205 1.98289 1.41421 0.26105 -1.00000 -1.84776 -1.93185 -1.21752 0.00000 1.21752 1.93185 1.84776 1.00000 -0.26105 -1.41421 -1.98289 -1.73205 -0.76537 0.51764 1.58671 -2.00000 ... ... ..... ....... ......... ........... ..TT......... E6g * 2.00000 1.41421 0.00000 -1.41421 -2.00000 -1.41421 0.00000 1.41421 2.00000 1.41421 0.00000 -1.41421 -2.00000 -1.41421 0.00000 1.41421 2.00000 1.41421 0.00000 -1.41421 -2.00000 -1.41421 0.00000 1.41421 2.00000 2.00000 1.41421 0.00000 -1.41421 -2.00000 -1.41421 0.00000 1.41421 2.00000 1.41421 0.00000 -1.41421 -2.00000 -1.41421 0.00000 1.41421 2.00000 1.41421 0.00000 -1.41421 -2.00000 -1.41421 0.00000 1.41421 2.00000 ... ... ..... ....... ......... ........... TT........... E7g * 2.00000 1.21752 -0.51764 -1.84776 -1.73205 -0.26105 1.41421 1.98289 1.00000 -0.76537 -1.93185 -1.58671 0.00000 1.58671 1.93185 0.76537 -1.00000 -1.98289 -1.41421 0.26105 1.73205 1.84776 0.51764 -1.21752 -2.00000 2.00000 -1.21752 0.51764 1.84776 1.73205 0.26105 -1.41421 -1.98289 -1.00000 0.76537 1.93185 1.58671 0.00000 -1.58671 -1.93185 -0.76537 1.00000 1.98289 1.41421 -0.26105 -1.73205 -1.84776 -0.51764 1.21752 -2.00000 ... ... ..... ....... ......... ........... ............. E8g * 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 ... ... ..... ....... ......... ........... ............. E9g * 2.00000 0.76537 -1.41421 -1.84776 0.00000 1.84776 1.41421 -0.76537 -2.00000 -0.76537 1.41421 1.84776 0.00000 -1.84776 -1.41421 0.76537 2.00000 0.76537 -1.41421 -1.84776 0.00000 1.84776 1.41421 -0.76537 -2.00000 2.00000 -0.76537 1.41421 1.84776 0.00000 -1.84776 -1.41421 0.76537 2.00000 0.76537 -1.41421 -1.84776 0.00000 1.84776 1.41421 -0.76537 -2.00000 -0.76537 1.41421 1.84776 0.00000 -1.84776 -1.41421 0.76537 -2.00000 ... ... ..... ....... ......... ........... ............. E10g * 2.00000 0.51764 -1.73205 -1.41421 1.00000 1.93185 0.00000 -1.93185 -1.00000 1.41421 1.73205 -0.51764 -2.00000 -0.51764 1.73205 1.41421 -1.00000 -1.93185 0.00000 1.93185 1.00000 -1.41421 -1.73205 0.51764 2.00000 2.00000 0.51764 -1.73205 -1.41421 1.00000 1.93185 0.00000 -1.93185 -1.00000 1.41421 1.73205 -0.51764 -2.00000 -0.51764 1.73205 1.41421 -1.00000 -1.93185 0.00000 1.93185 1.00000 -1.41421 -1.73205 0.51764 2.00000 ... ... ..... ....... ......... ........... ............. E11g * 2.00000 0.26105 -1.93185 -0.76537 1.73205 1.21752 -1.41421 -1.58671 1.00000 1.84776 -0.51764 -1.98289 0.00000 1.98289 0.51764 -1.84776 -1.00000 1.58671 1.41421 -1.21752 -1.73205 0.76537 1.93185 -0.26105 -2.00000 2.00000 -0.26105 1.93185 0.76537 -1.73205 -1.21752 1.41421 1.58671 -1.00000 -1.84776 0.51764 1.98289 0.00000 -1.98289 -0.51764 1.84776 1.00000 -1.58671 -1.41421 1.21752 1.73205 -0.76537 -1.93185 0.26105 -2.00000 ... ... ..... ....... ......... ........... ............. E12g * 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 ... ... ..... ....... ......... ........... ............. E13g * 2.00000 -0.26105 -1.93185 0.76537 1.73205 -1.21752 -1.41421 1.58671 1.00000 -1.84776 -0.51764 1.98289 0.00000 -1.98289 0.51764 1.84776 -1.00000 -1.58671 1.41421 1.21752 -1.73205 -0.76537 1.93185 0.26105 -2.00000 2.00000 0.26105 1.93185 -0.76537 -1.73205 1.21752 1.41421 -1.58671 -1.00000 1.84776 0.51764 -1.98289 0.00000 1.98289 -0.51764 -1.84776 1.00000 1.58671 -1.41421 -1.21752 1.73205 0.76537 -1.93185 -0.26105 -2.00000 ... ... ..... ....... ......... ........... ............. E14g * 2.00000 -0.51764 -1.73205 1.41421 1.00000 -1.93185 0.00000 1.93185 -1.00000 -1.41421 1.73205 0.51764 -2.00000 0.51764 1.73205 -1.41421 -1.00000 1.93185 0.00000 -1.93185 1.00000 1.41421 -1.73205 -0.51764 2.00000 2.00000 -0.51764 -1.73205 1.41421 1.00000 -1.93185 0.00000 1.93185 -1.00000 -1.41421 1.73205 0.51764 -2.00000 0.51764 1.73205 -1.41421 -1.00000 1.93185 0.00000 -1.93185 1.00000 1.41421 -1.73205 -0.51764 2.00000 ... ... ..... ....... ......... ........... ............. E15g * 2.00000 -0.76537 -1.41421 1.84776 0.00000 -1.84776 1.41421 0.76537 -2.00000 0.76537 1.41421 -1.84776 0.00000 1.84776 -1.41421 -0.76537 2.00000 -0.76537 -1.41421 1.84776 0.00000 -1.84776 1.41421 0.76537 -2.00000 2.00000 0.76537 1.41421 -1.84776 0.00000 1.84776 -1.41421 -0.76537 2.00000 -0.76537 -1.41421 1.84776 0.00000 -1.84776 1.41421 0.76537 -2.00000 0.76537 1.41421 -1.84776 0.00000 1.84776 -1.41421 -0.76537 -2.00000 ... ... ..... ....... ......... ........... ............. E16g * 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 ... ... ..... ....... ......... ........... ............. E17g * 2.00000 -1.21752 -0.51764 1.84776 -1.73205 0.26105 1.41421 -1.98289 1.00000 0.76537 -1.93185 1.58671 0.00000 -1.58671 1.93185 -0.76537 -1.00000 1.98289 -1.41421 -0.26105 1.73205 -1.84776 0.51764 1.21752 -2.00000 2.00000 1.21752 0.51764 -1.84776 1.73205 -0.26105 -1.41421 1.98289 -1.00000 -0.76537 1.93185 -1.58671 0.00000 1.58671 -1.93185 0.76537 1.00000 -1.98289 1.41421 0.26105 -1.73205 1.84776 -0.51764 -1.21752 -2.00000 ... ... ..... ....... ......... ........... ............. E18g * 2.00000 -1.41421 0.00000 1.41421 -2.00000 1.41421 0.00000 -1.41421 2.00000 -1.41421 0.00000 1.41421 -2.00000 1.41421 0.00000 -1.41421 2.00000 -1.41421 0.00000 1.41421 -2.00000 1.41421 0.00000 -1.41421 2.00000 2.00000 -1.41421 0.00000 1.41421 -2.00000 1.41421 0.00000 -1.41421 2.00000 -1.41421 0.00000 1.41421 -2.00000 1.41421 0.00000 -1.41421 2.00000 -1.41421 0.00000 1.41421 -2.00000 1.41421 0.00000 -1.41421 2.00000 ... ... ..... ....... ......... ........... ............. E19g * 2.00000 -1.58671 0.51764 0.76537 -1.73205 1.98289 -1.41421 0.26105 1.00000 -1.84776 1.93185 -1.21752 0.00000 1.21752 -1.93185 1.84776 -1.00000 -0.26105 1.41421 -1.98289 1.73205 -0.76537 -0.51764 1.58671 -2.00000 2.00000 1.58671 -0.51764 -0.76537 1.73205 -1.98289 1.41421 -0.26105 -1.00000 1.84776 -1.93185 1.21752 0.00000 -1.21752 1.93185 -1.84776 1.00000 0.26105 -1.41421 1.98289 -1.73205 0.76537 0.51764 -1.58671 -2.00000 ... ... ..... ....... ......... ........... ............. E20g * 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 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 ... ... ..... ....... ......... ........... ............. E21g * 2.00000 -1.84776 1.41421 -0.76537 0.00000 0.76537 -1.41421 1.84776 -2.00000 1.84776 -1.41421 0.76537 0.00000 -0.76537 1.41421 -1.84776 2.00000 -1.84776 1.41421 -0.76537 0.00000 0.76537 -1.41421 1.84776 -2.00000 2.00000 1.84776 -1.41421 0.76537 0.00000 -0.76537 1.41421 -1.84776 2.00000 -1.84776 1.41421 -0.76537 0.00000 0.76537 -1.41421 1.84776 -2.00000 1.84776 -1.41421 0.76537 0.00000 -0.76537 1.41421 -1.84776 -2.00000 ... ... ..... ....... ......... ........... ............. E22g * 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 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 ... ... ..... ....... ......... ........... ............. E23g * 2.00000 -1.98289 1.93185 -1.84776 1.73205 -1.58671 1.41421 -1.21752 1.00000 -0.76537 0.51764 -0.26105 0.00000 0.26105 -0.51764 0.76537 -1.00000 1.21752 -1.41421 1.58671 -1.73205 1.84776 -1.93185 1.98289 -2.00000 2.00000 1.98289 -1.93185 1.84776 -1.73205 1.58671 -1.41421 1.21752 -1.00000 0.76537 -0.51764 0.26105 0.00000 -0.26105 0.51764 -0.76537 1.00000 -1.21752 1.41421 -1.58671 1.73205 -1.84776 1.93185 -1.98289 -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 ... ..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 ... ... ..... ....... ......... ........... ............. E1u * 2.00000 1.98289 1.93185 1.84776 1.73205 1.58671 1.41421 1.21752 1.00000 0.76537 0.51764 0.26105 0.00000 -0.26105 -0.51764 -0.76537 -1.00000 -1.21752 -1.41421 -1.58671 -1.73205 -1.84776 -1.93185 -1.98289 -2.00000 -2.00000 1.98289 1.93185 1.84776 1.73205 1.58671 1.41421 1.21752 1.00000 0.76537 0.51764 0.26105 0.00000 -0.26105 -0.51764 -0.76537 -1.00000 -1.21752 -1.41421 -1.58671 -1.73205 -1.84776 -1.93185 -1.98289 2.00000 ... TT. ..... ....TT. ......... ........TT. ............. E2u * 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 -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 ... ... ..... ..TT... ......... ......TT... ............. E3u * 2.00000 1.84776 1.41421 0.76537 0.00000 -0.76537 -1.41421 -1.84776 -2.00000 -1.84776 -1.41421 -0.76537 0.00000 0.76537 1.41421 1.84776 2.00000 1.84776 1.41421 0.76537 0.00000 -0.76537 -1.41421 -1.84776 -2.00000 -2.00000 1.84776 1.41421 0.76537 0.00000 -0.76537 -1.41421 -1.84776 -2.00000 -1.84776 -1.41421 -0.76537 0.00000 0.76537 1.41421 1.84776 2.00000 1.84776 1.41421 0.76537 0.00000 -0.76537 -1.41421 -1.84776 2.00000 ... ... ..... TT..... ......... ....TT..... ............. E4u * 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 -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 ... ... ..... ....... ......... ..TT....... ............. E5u * 2.00000 1.58671 0.51764 -0.76537 -1.73205 -1.98289 -1.41421 -0.26105 1.00000 1.84776 1.93185 1.21752 0.00000 -1.21752 -1.93185 -1.84776 -1.00000 0.26105 1.41421 1.98289 1.73205 0.76537 -0.51764 -1.58671 -2.00000 -2.00000 1.58671 0.51764 -0.76537 -1.73205 -1.98289 -1.41421 -0.26105 1.00000 1.84776 1.93185 1.21752 0.00000 -1.21752 -1.93185 -1.84776 -1.00000 0.26105 1.41421 1.98289 1.73205 0.76537 -0.51764 -1.58671 2.00000 ... ... ..... ....... ......... TT......... ............. E6u * 2.00000 1.41421 0.00000 -1.41421 -2.00000 -1.41421 0.00000 1.41421 2.00000 1.41421 0.00000 -1.41421 -2.00000 -1.41421 0.00000 1.41421 2.00000 1.41421 0.00000 -1.41421 -2.00000 -1.41421 0.00000 1.41421 2.00000 -2.00000 -1.41421 0.00000 1.41421 2.00000 1.41421 0.00000 -1.41421 -2.00000 -1.41421 0.00000 1.41421 2.00000 1.41421 0.00000 -1.41421 -2.00000 -1.41421 0.00000 1.41421 2.00000 1.41421 0.00000 -1.41421 -2.00000 ... ... ..... ....... ......... ........... ............. E7u * 2.00000 1.21752 -0.51764 -1.84776 -1.73205 -0.26105 1.41421 1.98289 1.00000 -0.76537 -1.93185 -1.58671 0.00000 1.58671 1.93185 0.76537 -1.00000 -1.98289 -1.41421 0.26105 1.73205 1.84776 0.51764 -1.21752 -2.00000 -2.00000 1.21752 -0.51764 -1.84776 -1.73205 -0.26105 1.41421 1.98289 1.00000 -0.76537 -1.93185 -1.58671 0.00000 1.58671 1.93185 0.76537 -1.00000 -1.98289 -1.41421 0.26105 1.73205 1.84776 0.51764 -1.21752 2.00000 ... ... ..... ....... ......... ........... ............. E8u * 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 ... ... ..... ....... ......... ........... ............. E9u * 2.00000 0.76537 -1.41421 -1.84776 0.00000 1.84776 1.41421 -0.76537 -2.00000 -0.76537 1.41421 1.84776 0.00000 -1.84776 -1.41421 0.76537 2.00000 0.76537 -1.41421 -1.84776 0.00000 1.84776 1.41421 -0.76537 -2.00000 -2.00000 0.76537 -1.41421 -1.84776 0.00000 1.84776 1.41421 -0.76537 -2.00000 -0.76537 1.41421 1.84776 0.00000 -1.84776 -1.41421 0.76537 2.00000 0.76537 -1.41421 -1.84776 0.00000 1.84776 1.41421 -0.76537 2.00000 ... ... ..... ....... ......... ........... ............. E10u * 2.00000 0.51764 -1.73205 -1.41421 1.00000 1.93185 0.00000 -1.93185 -1.00000 1.41421 1.73205 -0.51764 -2.00000 -0.51764 1.73205 1.41421 -1.00000 -1.93185 0.00000 1.93185 1.00000 -1.41421 -1.73205 0.51764 2.00000 -2.00000 -0.51764 1.73205 1.41421 -1.00000 -1.93185 0.00000 1.93185 1.00000 -1.41421 -1.73205 0.51764 2.00000 0.51764 -1.73205 -1.41421 1.00000 1.93185 0.00000 -1.93185 -1.00000 1.41421 1.73205 -0.51764 -2.00000 ... ... ..... ....... ......... ........... ............. E11u * 2.00000 0.26105 -1.93185 -0.76537 1.73205 1.21752 -1.41421 -1.58671 1.00000 1.84776 -0.51764 -1.98289 0.00000 1.98289 0.51764 -1.84776 -1.00000 1.58671 1.41421 -1.21752 -1.73205 0.76537 1.93185 -0.26105 -2.00000 -2.00000 0.26105 -1.93185 -0.76537 1.73205 1.21752 -1.41421 -1.58671 1.00000 1.84776 -0.51764 -1.98289 0.00000 1.98289 0.51764 -1.84776 -1.00000 1.58671 1.41421 -1.21752 -1.73205 0.76537 1.93185 -0.26105 2.00000 ... ... ..... ....... ......... ........... ............. E12u * 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 ... ... ..... ....... ......... ........... ............. E13u * 2.00000 -0.26105 -1.93185 0.76537 1.73205 -1.21752 -1.41421 1.58671 1.00000 -1.84776 -0.51764 1.98289 0.00000 -1.98289 0.51764 1.84776 -1.00000 -1.58671 1.41421 1.21752 -1.73205 -0.76537 1.93185 0.26105 -2.00000 -2.00000 -0.26105 -1.93185 0.76537 1.73205 -1.21752 -1.41421 1.58671 1.00000 -1.84776 -0.51764 1.98289 0.00000 -1.98289 0.51764 1.84776 -1.00000 -1.58671 1.41421 1.21752 -1.73205 -0.76537 1.93185 0.26105 2.00000 ... ... ..... ....... ......... ........... ............. E14u * 2.00000 -0.51764 -1.73205 1.41421 1.00000 -1.93185 0.00000 1.93185 -1.00000 -1.41421 1.73205 0.51764 -2.00000 0.51764 1.73205 -1.41421 -1.00000 1.93185 0.00000 -1.93185 1.00000 1.41421 -1.73205 -0.51764 2.00000 -2.00000 0.51764 1.73205 -1.41421 -1.00000 1.93185 0.00000 -1.93185 1.00000 1.41421 -1.73205 -0.51764 2.00000 -0.51764 -1.73205 1.41421 1.00000 -1.93185 0.00000 1.93185 -1.00000 -1.41421 1.73205 0.51764 -2.00000 ... ... ..... ....... ......... ........... ............. E15u * 2.00000 -0.76537 -1.41421 1.84776 0.00000 -1.84776 1.41421 0.76537 -2.00000 0.76537 1.41421 -1.84776 0.00000 1.84776 -1.41421 -0.76537 2.00000 -0.76537 -1.41421 1.84776 0.00000 -1.84776 1.41421 0.76537 -2.00000 -2.00000 -0.76537 -1.41421 1.84776 0.00000 -1.84776 1.41421 0.76537 -2.00000 0.76537 1.41421 -1.84776 0.00000 1.84776 -1.41421 -0.76537 2.00000 -0.76537 -1.41421 1.84776 0.00000 -1.84776 1.41421 0.76537 2.00000 ... ... ..... ....... ......... ........... ............. E16u * 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 ... ... ..... ....... ......... ........... ............. E17u * 2.00000 -1.21752 -0.51764 1.84776 -1.73205 0.26105 1.41421 -1.98289 1.00000 0.76537 -1.93185 1.58671 0.00000 -1.58671 1.93185 -0.76537 -1.00000 1.98289 -1.41421 -0.26105 1.73205 -1.84776 0.51764 1.21752 -2.00000 -2.00000 -1.21752 -0.51764 1.84776 -1.73205 0.26105 1.41421 -1.98289 1.00000 0.76537 -1.93185 1.58671 0.00000 -1.58671 1.93185 -0.76537 -1.00000 1.98289 -1.41421 -0.26105 1.73205 -1.84776 0.51764 1.21752 2.00000 ... ... ..... ....... ......... ........... ............. E18u * 2.00000 -1.41421 0.00000 1.41421 -2.00000 1.41421 0.00000 -1.41421 2.00000 -1.41421 0.00000 1.41421 -2.00000 1.41421 0.00000 -1.41421 2.00000 -1.41421 0.00000 1.41421 -2.00000 1.41421 0.00000 -1.41421 2.00000 -2.00000 1.41421 0.00000 -1.41421 2.00000 -1.41421 0.00000 1.41421 -2.00000 1.41421 0.00000 -1.41421 2.00000 -1.41421 0.00000 1.41421 -2.00000 1.41421 0.00000 -1.41421 2.00000 -1.41421 0.00000 1.41421 -2.00000 ... ... ..... ....... ......... ........... ............. E19u * 2.00000 -1.58671 0.51764 0.76537 -1.73205 1.98289 -1.41421 0.26105 1.00000 -1.84776 1.93185 -1.21752 0.00000 1.21752 -1.93185 1.84776 -1.00000 -0.26105 1.41421 -1.98289 1.73205 -0.76537 -0.51764 1.58671 -2.00000 -2.00000 -1.58671 0.51764 0.76537 -1.73205 1.98289 -1.41421 0.26105 1.00000 -1.84776 1.93185 -1.21752 0.00000 1.21752 -1.93185 1.84776 -1.00000 -0.26105 1.41421 -1.98289 1.73205 -0.76537 -0.51764 1.58671 2.00000 ... ... ..... ....... ......... ........... ............. E20u * 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 -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 ... ... ..... ....... ......... ........... ............. E21u * 2.00000 -1.84776 1.41421 -0.76537 0.00000 0.76537 -1.41421 1.84776 -2.00000 1.84776 -1.41421 0.76537 0.00000 -0.76537 1.41421 -1.84776 2.00000 -1.84776 1.41421 -0.76537 0.00000 0.76537 -1.41421 1.84776 -2.00000 -2.00000 -1.84776 1.41421 -0.76537 0.00000 0.76537 -1.41421 1.84776 -2.00000 1.84776 -1.41421 0.76537 0.00000 -0.76537 1.41421 -1.84776 2.00000 -1.84776 1.41421 -0.76537 0.00000 0.76537 -1.41421 1.84776 2.00000 ... ... ..... ....... ......... ........... ............. E22u * 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 -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 ... ... ..... ....... ......... ........... ............. E23u * 2.00000 -1.98289 1.93185 -1.84776 1.73205 -1.58671 1.41421 -1.21752 1.00000 -0.76537 0.51764 -0.26105 0.00000 0.26105 -0.51764 0.76537 -1.00000 1.21752 -1.41421 1.58671 -1.73205 1.84776 -1.93185 1.98289 -2.00000 -2.00000 -1.98289 1.93185 -1.84776 1.73205 -1.58671 1.41421 -1.21752 1.00000 -0.76537 0.51764 -0.26105 0.00000 0.26105 -0.51764 0.76537 -1.00000 1.21752 -1.41421 1.58671 -1.73205 1.84776 -1.93185 1.98289 2.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.982889722748 = 2*cos(2*π/48) = 2*cos(π/24) = √2+√2+√3 1.931851652578 = 2*cos(4*π/48) = 2*cos(π/12) = √2+√3 1.847759065023 = 2*cos(6*π/48) = 2*cos(π/8) = √2+√2 1.732050807569 = 2*cos(8*π/48) = 2*cos(π/6) = √3 1.586706680582 = 2*cos(10*π/48) = 2*cos(5*π/24) = √2+√2−√3 1.414213562373 = 2*cos(12*π/48) = 2*cos(π/4) = √2 1.217522858017 = 2*cos(14*π/48) = 2*cos(7*π/24) = √2−√2−√3 0.765366864730 = 2*cos(18*π/48) = 2*cos(3*π/8) = √2−√2 0.517638090205 = 2*cos(20*π/48) = 2*cos(5*π/12) = √2−√3 0.261052384440 = 2*cos(22*π/48) = 2*cos(11*π/24) = √2−√2+√3 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_48h point group is 96, and the order of the principal axis (C₄₈) is 48. The group has 50 irreducible representations. β The C_48h point group is generated by two symmetry elements, which are canonically chosen as C₄₈ and i. Other possible choices are C₄₈ and σ_h, or less commonly S₄₈ with either i or σ_h. γ The lowest nonvanishing multipole moment in C_48h is 4 (quadrupole moment). δ This is an Abelian point group (the commutative law holds between all symmetry operations). The C_48h 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 92 cases have been combined into 46 two-dimensional representations that are no longer irreducible but have real-valued characters. Accordingly, 46 pairs of left and right rotations have been combined into one two-membered pseudo-class each. ζ The 46 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 constructible polygon, as the order of the principal axis is a product of any number of different Fermat primes (3,5,17,257,65537) times an arbitrary power of two. Therefore, all characters have an algebraic degree which is a power of two and can be expressed as radicals involving only square roots and integer numbers.

This Character Table for the C_48h point group was created by Gernot Katzer.

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

	C_46h
	C_47h
C₄₈ C_48v	C_48h	D₄₈ D_48h D_48d S₄₈
	C_49h
	C_50h

Character table for the C48h point group

Character table for the C_48h point group