Character Table for Point Group C56h

Character table for the C_56h point group

C56h E 2 C56 2 C28 2 C56^3 2 C14 2 C56^5 2 C28^3 2 C8 2 C7 2 C56^9 2 C28^5 2 C56^11 2 C14^3 2 C56^13 2 C4 2 C56^15 2 C7^2 2 C56^17 2 C28^9 2 C56^19 2 C14^5 2 C8^3 2 C28^11 2 C56^23 2 C7^3 2 C56^25 2 C28^13 2 C56^27 C2 i 2 S56 2 S28 2 S56^3 2 S14 2 S56^5 2 S28^3 2 S8 2 S7 2 S56^9 2 S28^5 2 S56^11 2 S14^3 2 S56^13 2 S4 2 S56^15 2 S7^2 2 S56^17 2 S28^9 2 S56^19 2 S14^5 2 S8^3 2 S28^11 2 S56^23 2 S7^3 2 S56^25 2 S28^13 2 S56^27 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 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 -1.00000 1.00000 ... ... ..... ....... ......... ........... ............. E1g * 2.00000 1.98742 1.94986 1.88777 1.80194 1.69345 1.56366 1.41421 1.24698 1.06406 0.86777 0.66056 0.44504 0.22393 0.00000 -0.22393 -0.44504 -0.66056 -0.86777 -1.06406 -1.24698 -1.41421 -1.56366 -1.69345 -1.80194 -1.88777 -1.94986 -1.98742 -2.00000 2.00000 -1.98742 -1.94986 -1.88777 -1.80194 -1.69345 -1.56366 -1.41421 -1.24698 -1.06406 -0.86777 -0.66056 -0.44504 -0.22393 0.00000 0.22393 0.44504 0.66056 0.86777 1.06406 1.24698 1.41421 1.56366 1.69345 1.80194 1.88777 1.94986 1.98742 -2.00000 TT. ... ..TT. ....... ......TT. ........... ..........TT. E2g * 2.00000 1.94986 1.80194 1.56366 1.24698 0.86777 0.44504 0.00000 -0.44504 -0.86777 -1.24698 -1.56366 -1.80194 -1.94986 -2.00000 -1.94986 -1.80194 -1.56366 -1.24698 -0.86777 -0.44504 0.00000 0.44504 0.86777 1.24698 1.56366 1.80194 1.94986 2.00000 2.00000 1.94986 1.80194 1.56366 1.24698 0.86777 0.44504 0.00000 -0.44504 -0.86777 -1.24698 -1.56366 -1.80194 -1.94986 -2.00000 -1.94986 -1.80194 -1.56366 -1.24698 -0.86777 -0.44504 0.00000 0.44504 0.86777 1.24698 1.56366 1.80194 1.94986 2.00000 ... ... TT... ....... ....TT... ........... ........TT... E3g * 2.00000 1.88777 1.56366 1.06406 0.44504 -0.22393 -0.86777 -1.41421 -1.80194 -1.98742 -1.94986 -1.69345 -1.24698 -0.66056 0.00000 0.66056 1.24698 1.69345 1.94986 1.98742 1.80194 1.41421 0.86777 0.22393 -0.44504 -1.06406 -1.56366 -1.88777 -2.00000 2.00000 -1.88777 -1.56366 -1.06406 -0.44504 0.22393 0.86777 1.41421 1.80194 1.98742 1.94986 1.69345 1.24698 0.66056 0.00000 -0.66056 -1.24698 -1.69345 -1.94986 -1.98742 -1.80194 -1.41421 -0.86777 -0.22393 0.44504 1.06406 1.56366 1.88777 -2.00000 ... ... ..... ....... ..TT..... ........... ......TT..... E4g * 2.00000 1.80194 1.24698 0.44504 -0.44504 -1.24698 -1.80194 -2.00000 -1.80194 -1.24698 -0.44504 0.44504 1.24698 1.80194 2.00000 1.80194 1.24698 0.44504 -0.44504 -1.24698 -1.80194 -2.00000 -1.80194 -1.24698 -0.44504 0.44504 1.24698 1.80194 2.00000 2.00000 1.80194 1.24698 0.44504 -0.44504 -1.24698 -1.80194 -2.00000 -1.80194 -1.24698 -0.44504 0.44504 1.24698 1.80194 2.00000 1.80194 1.24698 0.44504 -0.44504 -1.24698 -1.80194 -2.00000 -1.80194 -1.24698 -0.44504 0.44504 1.24698 1.80194 2.00000 ... ... ..... ....... TT....... ........... ....TT....... E5g * 2.00000 1.69345 0.86777 -0.22393 -1.24698 -1.88777 -1.94986 -1.41421 -0.44504 0.66056 1.56366 1.98742 1.80194 1.06406 0.00000 -1.06406 -1.80194 -1.98742 -1.56366 -0.66056 0.44504 1.41421 1.94986 1.88777 1.24698 0.22393 -0.86777 -1.69345 -2.00000 2.00000 -1.69345 -0.86777 0.22393 1.24698 1.88777 1.94986 1.41421 0.44504 -0.66056 -1.56366 -1.98742 -1.80194 -1.06406 0.00000 1.06406 1.80194 1.98742 1.56366 0.66056 -0.44504 -1.41421 -1.94986 -1.88777 -1.24698 -0.22393 0.86777 1.69345 -2.00000 ... ... ..... ....... ......... ........... ..TT......... E6g * 2.00000 1.56366 0.44504 -0.86777 -1.80194 -1.94986 -1.24698 0.00000 1.24698 1.94986 1.80194 0.86777 -0.44504 -1.56366 -2.00000 -1.56366 -0.44504 0.86777 1.80194 1.94986 1.24698 0.00000 -1.24698 -1.94986 -1.80194 -0.86777 0.44504 1.56366 2.00000 2.00000 1.56366 0.44504 -0.86777 -1.80194 -1.94986 -1.24698 0.00000 1.24698 1.94986 1.80194 0.86777 -0.44504 -1.56366 -2.00000 -1.56366 -0.44504 0.86777 1.80194 1.94986 1.24698 0.00000 -1.24698 -1.94986 -1.80194 -0.86777 0.44504 1.56366 2.00000 ... ... ..... ....... ......... ........... TT........... E7g * 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 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 -1.41421 0.00000 1.41421 -2.00000 ... ... ..... ....... ......... ........... ............. E8g * 2.00000 1.24698 -0.44504 -1.80194 -1.80194 -0.44504 1.24698 2.00000 1.24698 -0.44504 -1.80194 -1.80194 -0.44504 1.24698 2.00000 1.24698 -0.44504 -1.80194 -1.80194 -0.44504 1.24698 2.00000 1.24698 -0.44504 -1.80194 -1.80194 -0.44504 1.24698 2.00000 2.00000 1.24698 -0.44504 -1.80194 -1.80194 -0.44504 1.24698 2.00000 1.24698 -0.44504 -1.80194 -1.80194 -0.44504 1.24698 2.00000 1.24698 -0.44504 -1.80194 -1.80194 -0.44504 1.24698 2.00000 1.24698 -0.44504 -1.80194 -1.80194 -0.44504 1.24698 2.00000 ... ... ..... ....... ......... ........... ............. E9g * 2.00000 1.06406 -0.86777 -1.98742 -1.24698 0.66056 1.94986 1.41421 -0.44504 -1.88777 -1.56366 0.22393 1.80194 1.69345 0.00000 -1.69345 -1.80194 -0.22393 1.56366 1.88777 0.44504 -1.41421 -1.94986 -0.66056 1.24698 1.98742 0.86777 -1.06406 -2.00000 2.00000 -1.06406 0.86777 1.98742 1.24698 -0.66056 -1.94986 -1.41421 0.44504 1.88777 1.56366 -0.22393 -1.80194 -1.69345 0.00000 1.69345 1.80194 0.22393 -1.56366 -1.88777 -0.44504 1.41421 1.94986 0.66056 -1.24698 -1.98742 -0.86777 1.06406 -2.00000 ... ... ..... ....... ......... ........... ............. E10g * 2.00000 0.86777 -1.24698 -1.94986 -0.44504 1.56366 1.80194 0.00000 -1.80194 -1.56366 0.44504 1.94986 1.24698 -0.86777 -2.00000 -0.86777 1.24698 1.94986 0.44504 -1.56366 -1.80194 0.00000 1.80194 1.56366 -0.44504 -1.94986 -1.24698 0.86777 2.00000 2.00000 0.86777 -1.24698 -1.94986 -0.44504 1.56366 1.80194 0.00000 -1.80194 -1.56366 0.44504 1.94986 1.24698 -0.86777 -2.00000 -0.86777 1.24698 1.94986 0.44504 -1.56366 -1.80194 0.00000 1.80194 1.56366 -0.44504 -1.94986 -1.24698 0.86777 2.00000 ... ... ..... ....... ......... ........... ............. E11g * 2.00000 0.66056 -1.56366 -1.69345 0.44504 1.98742 0.86777 -1.41421 -1.80194 0.22393 1.94986 1.06406 -1.24698 -1.88777 0.00000 1.88777 1.24698 -1.06406 -1.94986 -0.22393 1.80194 1.41421 -0.86777 -1.98742 -0.44504 1.69345 1.56366 -0.66056 -2.00000 2.00000 -0.66056 1.56366 1.69345 -0.44504 -1.98742 -0.86777 1.41421 1.80194 -0.22393 -1.94986 -1.06406 1.24698 1.88777 0.00000 -1.88777 -1.24698 1.06406 1.94986 0.22393 -1.80194 -1.41421 0.86777 1.98742 0.44504 -1.69345 -1.56366 0.66056 -2.00000 ... ... ..... ....... ......... ........... ............. E12g * 2.00000 0.44504 -1.80194 -1.24698 1.24698 1.80194 -0.44504 -2.00000 -0.44504 1.80194 1.24698 -1.24698 -1.80194 0.44504 2.00000 0.44504 -1.80194 -1.24698 1.24698 1.80194 -0.44504 -2.00000 -0.44504 1.80194 1.24698 -1.24698 -1.80194 0.44504 2.00000 2.00000 0.44504 -1.80194 -1.24698 1.24698 1.80194 -0.44504 -2.00000 -0.44504 1.80194 1.24698 -1.24698 -1.80194 0.44504 2.00000 0.44504 -1.80194 -1.24698 1.24698 1.80194 -0.44504 -2.00000 -0.44504 1.80194 1.24698 -1.24698 -1.80194 0.44504 2.00000 ... ... ..... ....... ......... ........... ............. E13g * 2.00000 0.22393 -1.94986 -0.66056 1.80194 1.06406 -1.56366 -1.41421 1.24698 1.69345 -0.86777 -1.88777 0.44504 1.98742 0.00000 -1.98742 -0.44504 1.88777 0.86777 -1.69345 -1.24698 1.41421 1.56366 -1.06406 -1.80194 0.66056 1.94986 -0.22393 -2.00000 2.00000 -0.22393 1.94986 0.66056 -1.80194 -1.06406 1.56366 1.41421 -1.24698 -1.69345 0.86777 1.88777 -0.44504 -1.98742 0.00000 1.98742 0.44504 -1.88777 -0.86777 1.69345 1.24698 -1.41421 -1.56366 1.06406 1.80194 -0.66056 -1.94986 0.22393 -2.00000 ... ... ..... ....... ......... ........... ............. E14g * 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 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 0.00000 -2.00000 0.00000 2.00000 ... ... ..... ....... ......... ........... ............. E15g * 2.00000 -0.22393 -1.94986 0.66056 1.80194 -1.06406 -1.56366 1.41421 1.24698 -1.69345 -0.86777 1.88777 0.44504 -1.98742 0.00000 1.98742 -0.44504 -1.88777 0.86777 1.69345 -1.24698 -1.41421 1.56366 1.06406 -1.80194 -0.66056 1.94986 0.22393 -2.00000 2.00000 0.22393 1.94986 -0.66056 -1.80194 1.06406 1.56366 -1.41421 -1.24698 1.69345 0.86777 -1.88777 -0.44504 1.98742 0.00000 -1.98742 0.44504 1.88777 -0.86777 -1.69345 1.24698 1.41421 -1.56366 -1.06406 1.80194 0.66056 -1.94986 -0.22393 -2.00000 ... ... ..... ....... ......... ........... ............. E16g * 2.00000 -0.44504 -1.80194 1.24698 1.24698 -1.80194 -0.44504 2.00000 -0.44504 -1.80194 1.24698 1.24698 -1.80194 -0.44504 2.00000 -0.44504 -1.80194 1.24698 1.24698 -1.80194 -0.44504 2.00000 -0.44504 -1.80194 1.24698 1.24698 -1.80194 -0.44504 2.00000 2.00000 -0.44504 -1.80194 1.24698 1.24698 -1.80194 -0.44504 2.00000 -0.44504 -1.80194 1.24698 1.24698 -1.80194 -0.44504 2.00000 -0.44504 -1.80194 1.24698 1.24698 -1.80194 -0.44504 2.00000 -0.44504 -1.80194 1.24698 1.24698 -1.80194 -0.44504 2.00000 ... ... ..... ....... ......... ........... ............. E17g * 2.00000 -0.66056 -1.56366 1.69345 0.44504 -1.98742 0.86777 1.41421 -1.80194 -0.22393 1.94986 -1.06406 -1.24698 1.88777 0.00000 -1.88777 1.24698 1.06406 -1.94986 0.22393 1.80194 -1.41421 -0.86777 1.98742 -0.44504 -1.69345 1.56366 0.66056 -2.00000 2.00000 0.66056 1.56366 -1.69345 -0.44504 1.98742 -0.86777 -1.41421 1.80194 0.22393 -1.94986 1.06406 1.24698 -1.88777 0.00000 1.88777 -1.24698 -1.06406 1.94986 -0.22393 -1.80194 1.41421 0.86777 -1.98742 0.44504 1.69345 -1.56366 -0.66056 -2.00000 ... ... ..... ....... ......... ........... ............. E18g * 2.00000 -0.86777 -1.24698 1.94986 -0.44504 -1.56366 1.80194 0.00000 -1.80194 1.56366 0.44504 -1.94986 1.24698 0.86777 -2.00000 0.86777 1.24698 -1.94986 0.44504 1.56366 -1.80194 0.00000 1.80194 -1.56366 -0.44504 1.94986 -1.24698 -0.86777 2.00000 2.00000 -0.86777 -1.24698 1.94986 -0.44504 -1.56366 1.80194 0.00000 -1.80194 1.56366 0.44504 -1.94986 1.24698 0.86777 -2.00000 0.86777 1.24698 -1.94986 0.44504 1.56366 -1.80194 0.00000 1.80194 -1.56366 -0.44504 1.94986 -1.24698 -0.86777 2.00000 ... ... ..... ....... ......... ........... ............. E19g * 2.00000 -1.06406 -0.86777 1.98742 -1.24698 -0.66056 1.94986 -1.41421 -0.44504 1.88777 -1.56366 -0.22393 1.80194 -1.69345 0.00000 1.69345 -1.80194 0.22393 1.56366 -1.88777 0.44504 1.41421 -1.94986 0.66056 1.24698 -1.98742 0.86777 1.06406 -2.00000 2.00000 1.06406 0.86777 -1.98742 1.24698 0.66056 -1.94986 1.41421 0.44504 -1.88777 1.56366 0.22393 -1.80194 1.69345 0.00000 -1.69345 1.80194 -0.22393 -1.56366 1.88777 -0.44504 -1.41421 1.94986 -0.66056 -1.24698 1.98742 -0.86777 -1.06406 -2.00000 ... ... ..... ....... ......... ........... ............. E20g * 2.00000 -1.24698 -0.44504 1.80194 -1.80194 0.44504 1.24698 -2.00000 1.24698 0.44504 -1.80194 1.80194 -0.44504 -1.24698 2.00000 -1.24698 -0.44504 1.80194 -1.80194 0.44504 1.24698 -2.00000 1.24698 0.44504 -1.80194 1.80194 -0.44504 -1.24698 2.00000 2.00000 -1.24698 -0.44504 1.80194 -1.80194 0.44504 1.24698 -2.00000 1.24698 0.44504 -1.80194 1.80194 -0.44504 -1.24698 2.00000 -1.24698 -0.44504 1.80194 -1.80194 0.44504 1.24698 -2.00000 1.24698 0.44504 -1.80194 1.80194 -0.44504 -1.24698 2.00000 ... ... ..... ....... ......... ........... ............. E21g * 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 -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 1.41421 0.00000 -1.41421 -2.00000 ... ... ..... ....... ......... ........... ............. E22g * 2.00000 -1.56366 0.44504 0.86777 -1.80194 1.94986 -1.24698 0.00000 1.24698 -1.94986 1.80194 -0.86777 -0.44504 1.56366 -2.00000 1.56366 -0.44504 -0.86777 1.80194 -1.94986 1.24698 0.00000 -1.24698 1.94986 -1.80194 0.86777 0.44504 -1.56366 2.00000 2.00000 -1.56366 0.44504 0.86777 -1.80194 1.94986 -1.24698 0.00000 1.24698 -1.94986 1.80194 -0.86777 -0.44504 1.56366 -2.00000 1.56366 -0.44504 -0.86777 1.80194 -1.94986 1.24698 0.00000 -1.24698 1.94986 -1.80194 0.86777 0.44504 -1.56366 2.00000 ... ... ..... ....... ......... ........... ............. E23g * 2.00000 -1.69345 0.86777 0.22393 -1.24698 1.88777 -1.94986 1.41421 -0.44504 -0.66056 1.56366 -1.98742 1.80194 -1.06406 0.00000 1.06406 -1.80194 1.98742 -1.56366 0.66056 0.44504 -1.41421 1.94986 -1.88777 1.24698 -0.22393 -0.86777 1.69345 -2.00000 2.00000 1.69345 -0.86777 -0.22393 1.24698 -1.88777 1.94986 -1.41421 0.44504 0.66056 -1.56366 1.98742 -1.80194 1.06406 0.00000 -1.06406 1.80194 -1.98742 1.56366 -0.66056 -0.44504 1.41421 -1.94986 1.88777 -1.24698 0.22393 0.86777 -1.69345 -2.00000 ... ... ..... ....... ......... ........... ............. E24g * 2.00000 -1.80194 1.24698 -0.44504 -0.44504 1.24698 -1.80194 2.00000 -1.80194 1.24698 -0.44504 -0.44504 1.24698 -1.80194 2.00000 -1.80194 1.24698 -0.44504 -0.44504 1.24698 -1.80194 2.00000 -1.80194 1.24698 -0.44504 -0.44504 1.24698 -1.80194 2.00000 2.00000 -1.80194 1.24698 -0.44504 -0.44504 1.24698 -1.80194 2.00000 -1.80194 1.24698 -0.44504 -0.44504 1.24698 -1.80194 2.00000 -1.80194 1.24698 -0.44504 -0.44504 1.24698 -1.80194 2.00000 -1.80194 1.24698 -0.44504 -0.44504 1.24698 -1.80194 2.00000 ... ... ..... ....... ......... ........... ............. E25g * 2.00000 -1.88777 1.56366 -1.06406 0.44504 0.22393 -0.86777 1.41421 -1.80194 1.98742 -1.94986 1.69345 -1.24698 0.66056 0.00000 -0.66056 1.24698 -1.69345 1.94986 -1.98742 1.80194 -1.41421 0.86777 -0.22393 -0.44504 1.06406 -1.56366 1.88777 -2.00000 2.00000 1.88777 -1.56366 1.06406 -0.44504 -0.22393 0.86777 -1.41421 1.80194 -1.98742 1.94986 -1.69345 1.24698 -0.66056 0.00000 0.66056 -1.24698 1.69345 -1.94986 1.98742 -1.80194 1.41421 -0.86777 0.22393 0.44504 -1.06406 1.56366 -1.88777 -2.00000 ... ... ..... ....... ......... ........... ............. E26g * 2.00000 -1.94986 1.80194 -1.56366 1.24698 -0.86777 0.44504 0.00000 -0.44504 0.86777 -1.24698 1.56366 -1.80194 1.94986 -2.00000 1.94986 -1.80194 1.56366 -1.24698 0.86777 -0.44504 0.00000 0.44504 -0.86777 1.24698 -1.56366 1.80194 -1.94986 2.00000 2.00000 -1.94986 1.80194 -1.56366 1.24698 -0.86777 0.44504 0.00000 -0.44504 0.86777 -1.24698 1.56366 -1.80194 1.94986 -2.00000 1.94986 -1.80194 1.56366 -1.24698 0.86777 -0.44504 0.00000 0.44504 -0.86777 1.24698 -1.56366 1.80194 -1.94986 2.00000 ... ... ..... ....... ......... ........... ............. E27g * 2.00000 -1.98742 1.94986 -1.88777 1.80194 -1.69345 1.56366 -1.41421 1.24698 -1.06406 0.86777 -0.66056 0.44504 -0.22393 0.00000 0.22393 -0.44504 0.66056 -0.86777 1.06406 -1.24698 1.41421 -1.56366 1.69345 -1.80194 1.88777 -1.94986 1.98742 -2.00000 2.00000 1.98742 -1.94986 1.88777 -1.80194 1.69345 -1.56366 1.41421 -1.24698 1.06406 -0.86777 0.66056 -0.44504 0.22393 0.00000 -0.22393 0.44504 -0.66056 0.86777 -1.06406 1.24698 -1.41421 1.56366 -1.69345 1.80194 -1.88777 1.94986 -1.98742 -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 -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 1.00000 -1.00000 ... ... ..... ....... ......... ........... ............. E1u * 2.00000 1.98742 1.94986 1.88777 1.80194 1.69345 1.56366 1.41421 1.24698 1.06406 0.86777 0.66056 0.44504 0.22393 0.00000 -0.22393 -0.44504 -0.66056 -0.86777 -1.06406 -1.24698 -1.41421 -1.56366 -1.69345 -1.80194 -1.88777 -1.94986 -1.98742 -2.00000 -2.00000 1.98742 1.94986 1.88777 1.80194 1.69345 1.56366 1.41421 1.24698 1.06406 0.86777 0.66056 0.44504 0.22393 0.00000 -0.22393 -0.44504 -0.66056 -0.86777 -1.06406 -1.24698 -1.41421 -1.56366 -1.69345 -1.80194 -1.88777 -1.94986 -1.98742 2.00000 ... TT. ..... ....TT. ......... ........TT. ............. E2u * 2.00000 1.94986 1.80194 1.56366 1.24698 0.86777 0.44504 0.00000 -0.44504 -0.86777 -1.24698 -1.56366 -1.80194 -1.94986 -2.00000 -1.94986 -1.80194 -1.56366 -1.24698 -0.86777 -0.44504 0.00000 0.44504 0.86777 1.24698 1.56366 1.80194 1.94986 2.00000 -2.00000 -1.94986 -1.80194 -1.56366 -1.24698 -0.86777 -0.44504 0.00000 0.44504 0.86777 1.24698 1.56366 1.80194 1.94986 2.00000 1.94986 1.80194 1.56366 1.24698 0.86777 0.44504 0.00000 -0.44504 -0.86777 -1.24698 -1.56366 -1.80194 -1.94986 -2.00000 ... ... ..... ..TT... ......... ......TT... ............. E3u * 2.00000 1.88777 1.56366 1.06406 0.44504 -0.22393 -0.86777 -1.41421 -1.80194 -1.98742 -1.94986 -1.69345 -1.24698 -0.66056 0.00000 0.66056 1.24698 1.69345 1.94986 1.98742 1.80194 1.41421 0.86777 0.22393 -0.44504 -1.06406 -1.56366 -1.88777 -2.00000 -2.00000 1.88777 1.56366 1.06406 0.44504 -0.22393 -0.86777 -1.41421 -1.80194 -1.98742 -1.94986 -1.69345 -1.24698 -0.66056 0.00000 0.66056 1.24698 1.69345 1.94986 1.98742 1.80194 1.41421 0.86777 0.22393 -0.44504 -1.06406 -1.56366 -1.88777 2.00000 ... ... ..... TT..... ......... ....TT..... ............. E4u * 2.00000 1.80194 1.24698 0.44504 -0.44504 -1.24698 -1.80194 -2.00000 -1.80194 -1.24698 -0.44504 0.44504 1.24698 1.80194 2.00000 1.80194 1.24698 0.44504 -0.44504 -1.24698 -1.80194 -2.00000 -1.80194 -1.24698 -0.44504 0.44504 1.24698 1.80194 2.00000 -2.00000 -1.80194 -1.24698 -0.44504 0.44504 1.24698 1.80194 2.00000 1.80194 1.24698 0.44504 -0.44504 -1.24698 -1.80194 -2.00000 -1.80194 -1.24698 -0.44504 0.44504 1.24698 1.80194 2.00000 1.80194 1.24698 0.44504 -0.44504 -1.24698 -1.80194 -2.00000 ... ... ..... ....... ......... ..TT....... ............. E5u * 2.00000 1.69345 0.86777 -0.22393 -1.24698 -1.88777 -1.94986 -1.41421 -0.44504 0.66056 1.56366 1.98742 1.80194 1.06406 0.00000 -1.06406 -1.80194 -1.98742 -1.56366 -0.66056 0.44504 1.41421 1.94986 1.88777 1.24698 0.22393 -0.86777 -1.69345 -2.00000 -2.00000 1.69345 0.86777 -0.22393 -1.24698 -1.88777 -1.94986 -1.41421 -0.44504 0.66056 1.56366 1.98742 1.80194 1.06406 0.00000 -1.06406 -1.80194 -1.98742 -1.56366 -0.66056 0.44504 1.41421 1.94986 1.88777 1.24698 0.22393 -0.86777 -1.69345 2.00000 ... ... ..... ....... ......... TT......... ............. E6u * 2.00000 1.56366 0.44504 -0.86777 -1.80194 -1.94986 -1.24698 0.00000 1.24698 1.94986 1.80194 0.86777 -0.44504 -1.56366 -2.00000 -1.56366 -0.44504 0.86777 1.80194 1.94986 1.24698 0.00000 -1.24698 -1.94986 -1.80194 -0.86777 0.44504 1.56366 2.00000 -2.00000 -1.56366 -0.44504 0.86777 1.80194 1.94986 1.24698 0.00000 -1.24698 -1.94986 -1.80194 -0.86777 0.44504 1.56366 2.00000 1.56366 0.44504 -0.86777 -1.80194 -1.94986 -1.24698 0.00000 1.24698 1.94986 1.80194 0.86777 -0.44504 -1.56366 -2.00000 ... ... ..... ....... ......... ........... ............. E7u * 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 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 1.41421 0.00000 -1.41421 2.00000 ... ... ..... ....... ......... ........... ............. E8u * 2.00000 1.24698 -0.44504 -1.80194 -1.80194 -0.44504 1.24698 2.00000 1.24698 -0.44504 -1.80194 -1.80194 -0.44504 1.24698 2.00000 1.24698 -0.44504 -1.80194 -1.80194 -0.44504 1.24698 2.00000 1.24698 -0.44504 -1.80194 -1.80194 -0.44504 1.24698 2.00000 -2.00000 -1.24698 0.44504 1.80194 1.80194 0.44504 -1.24698 -2.00000 -1.24698 0.44504 1.80194 1.80194 0.44504 -1.24698 -2.00000 -1.24698 0.44504 1.80194 1.80194 0.44504 -1.24698 -2.00000 -1.24698 0.44504 1.80194 1.80194 0.44504 -1.24698 -2.00000 ... ... ..... ....... ......... ........... ............. E9u * 2.00000 1.06406 -0.86777 -1.98742 -1.24698 0.66056 1.94986 1.41421 -0.44504 -1.88777 -1.56366 0.22393 1.80194 1.69345 0.00000 -1.69345 -1.80194 -0.22393 1.56366 1.88777 0.44504 -1.41421 -1.94986 -0.66056 1.24698 1.98742 0.86777 -1.06406 -2.00000 -2.00000 1.06406 -0.86777 -1.98742 -1.24698 0.66056 1.94986 1.41421 -0.44504 -1.88777 -1.56366 0.22393 1.80194 1.69345 0.00000 -1.69345 -1.80194 -0.22393 1.56366 1.88777 0.44504 -1.41421 -1.94986 -0.66056 1.24698 1.98742 0.86777 -1.06406 2.00000 ... ... ..... ....... ......... ........... ............. E10u * 2.00000 0.86777 -1.24698 -1.94986 -0.44504 1.56366 1.80194 0.00000 -1.80194 -1.56366 0.44504 1.94986 1.24698 -0.86777 -2.00000 -0.86777 1.24698 1.94986 0.44504 -1.56366 -1.80194 0.00000 1.80194 1.56366 -0.44504 -1.94986 -1.24698 0.86777 2.00000 -2.00000 -0.86777 1.24698 1.94986 0.44504 -1.56366 -1.80194 0.00000 1.80194 1.56366 -0.44504 -1.94986 -1.24698 0.86777 2.00000 0.86777 -1.24698 -1.94986 -0.44504 1.56366 1.80194 0.00000 -1.80194 -1.56366 0.44504 1.94986 1.24698 -0.86777 -2.00000 ... ... ..... ....... ......... ........... ............. E11u * 2.00000 0.66056 -1.56366 -1.69345 0.44504 1.98742 0.86777 -1.41421 -1.80194 0.22393 1.94986 1.06406 -1.24698 -1.88777 0.00000 1.88777 1.24698 -1.06406 -1.94986 -0.22393 1.80194 1.41421 -0.86777 -1.98742 -0.44504 1.69345 1.56366 -0.66056 -2.00000 -2.00000 0.66056 -1.56366 -1.69345 0.44504 1.98742 0.86777 -1.41421 -1.80194 0.22393 1.94986 1.06406 -1.24698 -1.88777 0.00000 1.88777 1.24698 -1.06406 -1.94986 -0.22393 1.80194 1.41421 -0.86777 -1.98742 -0.44504 1.69345 1.56366 -0.66056 2.00000 ... ... ..... ....... ......... ........... ............. E12u * 2.00000 0.44504 -1.80194 -1.24698 1.24698 1.80194 -0.44504 -2.00000 -0.44504 1.80194 1.24698 -1.24698 -1.80194 0.44504 2.00000 0.44504 -1.80194 -1.24698 1.24698 1.80194 -0.44504 -2.00000 -0.44504 1.80194 1.24698 -1.24698 -1.80194 0.44504 2.00000 -2.00000 -0.44504 1.80194 1.24698 -1.24698 -1.80194 0.44504 2.00000 0.44504 -1.80194 -1.24698 1.24698 1.80194 -0.44504 -2.00000 -0.44504 1.80194 1.24698 -1.24698 -1.80194 0.44504 2.00000 0.44504 -1.80194 -1.24698 1.24698 1.80194 -0.44504 -2.00000 ... ... ..... ....... ......... ........... ............. E13u * 2.00000 0.22393 -1.94986 -0.66056 1.80194 1.06406 -1.56366 -1.41421 1.24698 1.69345 -0.86777 -1.88777 0.44504 1.98742 0.00000 -1.98742 -0.44504 1.88777 0.86777 -1.69345 -1.24698 1.41421 1.56366 -1.06406 -1.80194 0.66056 1.94986 -0.22393 -2.00000 -2.00000 0.22393 -1.94986 -0.66056 1.80194 1.06406 -1.56366 -1.41421 1.24698 1.69345 -0.86777 -1.88777 0.44504 1.98742 0.00000 -1.98742 -0.44504 1.88777 0.86777 -1.69345 -1.24698 1.41421 1.56366 -1.06406 -1.80194 0.66056 1.94986 -0.22393 2.00000 ... ... ..... ....... ......... ........... ............. E14u * 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 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 0.00000 2.00000 0.00000 -2.00000 ... ... ..... ....... ......... ........... ............. E15u * 2.00000 -0.22393 -1.94986 0.66056 1.80194 -1.06406 -1.56366 1.41421 1.24698 -1.69345 -0.86777 1.88777 0.44504 -1.98742 0.00000 1.98742 -0.44504 -1.88777 0.86777 1.69345 -1.24698 -1.41421 1.56366 1.06406 -1.80194 -0.66056 1.94986 0.22393 -2.00000 -2.00000 -0.22393 -1.94986 0.66056 1.80194 -1.06406 -1.56366 1.41421 1.24698 -1.69345 -0.86777 1.88777 0.44504 -1.98742 0.00000 1.98742 -0.44504 -1.88777 0.86777 1.69345 -1.24698 -1.41421 1.56366 1.06406 -1.80194 -0.66056 1.94986 0.22393 2.00000 ... ... ..... ....... ......... ........... ............. E16u * 2.00000 -0.44504 -1.80194 1.24698 1.24698 -1.80194 -0.44504 2.00000 -0.44504 -1.80194 1.24698 1.24698 -1.80194 -0.44504 2.00000 -0.44504 -1.80194 1.24698 1.24698 -1.80194 -0.44504 2.00000 -0.44504 -1.80194 1.24698 1.24698 -1.80194 -0.44504 2.00000 -2.00000 0.44504 1.80194 -1.24698 -1.24698 1.80194 0.44504 -2.00000 0.44504 1.80194 -1.24698 -1.24698 1.80194 0.44504 -2.00000 0.44504 1.80194 -1.24698 -1.24698 1.80194 0.44504 -2.00000 0.44504 1.80194 -1.24698 -1.24698 1.80194 0.44504 -2.00000 ... ... ..... ....... ......... ........... ............. E17u * 2.00000 -0.66056 -1.56366 1.69345 0.44504 -1.98742 0.86777 1.41421 -1.80194 -0.22393 1.94986 -1.06406 -1.24698 1.88777 0.00000 -1.88777 1.24698 1.06406 -1.94986 0.22393 1.80194 -1.41421 -0.86777 1.98742 -0.44504 -1.69345 1.56366 0.66056 -2.00000 -2.00000 -0.66056 -1.56366 1.69345 0.44504 -1.98742 0.86777 1.41421 -1.80194 -0.22393 1.94986 -1.06406 -1.24698 1.88777 0.00000 -1.88777 1.24698 1.06406 -1.94986 0.22393 1.80194 -1.41421 -0.86777 1.98742 -0.44504 -1.69345 1.56366 0.66056 2.00000 ... ... ..... ....... ......... ........... ............. E18u * 2.00000 -0.86777 -1.24698 1.94986 -0.44504 -1.56366 1.80194 0.00000 -1.80194 1.56366 0.44504 -1.94986 1.24698 0.86777 -2.00000 0.86777 1.24698 -1.94986 0.44504 1.56366 -1.80194 0.00000 1.80194 -1.56366 -0.44504 1.94986 -1.24698 -0.86777 2.00000 -2.00000 0.86777 1.24698 -1.94986 0.44504 1.56366 -1.80194 0.00000 1.80194 -1.56366 -0.44504 1.94986 -1.24698 -0.86777 2.00000 -0.86777 -1.24698 1.94986 -0.44504 -1.56366 1.80194 0.00000 -1.80194 1.56366 0.44504 -1.94986 1.24698 0.86777 -2.00000 ... ... ..... ....... ......... ........... ............. E19u * 2.00000 -1.06406 -0.86777 1.98742 -1.24698 -0.66056 1.94986 -1.41421 -0.44504 1.88777 -1.56366 -0.22393 1.80194 -1.69345 0.00000 1.69345 -1.80194 0.22393 1.56366 -1.88777 0.44504 1.41421 -1.94986 0.66056 1.24698 -1.98742 0.86777 1.06406 -2.00000 -2.00000 -1.06406 -0.86777 1.98742 -1.24698 -0.66056 1.94986 -1.41421 -0.44504 1.88777 -1.56366 -0.22393 1.80194 -1.69345 0.00000 1.69345 -1.80194 0.22393 1.56366 -1.88777 0.44504 1.41421 -1.94986 0.66056 1.24698 -1.98742 0.86777 1.06406 2.00000 ... ... ..... ....... ......... ........... ............. E20u * 2.00000 -1.24698 -0.44504 1.80194 -1.80194 0.44504 1.24698 -2.00000 1.24698 0.44504 -1.80194 1.80194 -0.44504 -1.24698 2.00000 -1.24698 -0.44504 1.80194 -1.80194 0.44504 1.24698 -2.00000 1.24698 0.44504 -1.80194 1.80194 -0.44504 -1.24698 2.00000 -2.00000 1.24698 0.44504 -1.80194 1.80194 -0.44504 -1.24698 2.00000 -1.24698 -0.44504 1.80194 -1.80194 0.44504 1.24698 -2.00000 1.24698 0.44504 -1.80194 1.80194 -0.44504 -1.24698 2.00000 -1.24698 -0.44504 1.80194 -1.80194 0.44504 1.24698 -2.00000 ... ... ..... ....... ......... ........... ............. E21u * 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 -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 -1.41421 0.00000 1.41421 2.00000 ... ... ..... ....... ......... ........... ............. E22u * 2.00000 -1.56366 0.44504 0.86777 -1.80194 1.94986 -1.24698 0.00000 1.24698 -1.94986 1.80194 -0.86777 -0.44504 1.56366 -2.00000 1.56366 -0.44504 -0.86777 1.80194 -1.94986 1.24698 0.00000 -1.24698 1.94986 -1.80194 0.86777 0.44504 -1.56366 2.00000 -2.00000 1.56366 -0.44504 -0.86777 1.80194 -1.94986 1.24698 0.00000 -1.24698 1.94986 -1.80194 0.86777 0.44504 -1.56366 2.00000 -1.56366 0.44504 0.86777 -1.80194 1.94986 -1.24698 0.00000 1.24698 -1.94986 1.80194 -0.86777 -0.44504 1.56366 -2.00000 ... ... ..... ....... ......... ........... ............. E23u * 2.00000 -1.69345 0.86777 0.22393 -1.24698 1.88777 -1.94986 1.41421 -0.44504 -0.66056 1.56366 -1.98742 1.80194 -1.06406 0.00000 1.06406 -1.80194 1.98742 -1.56366 0.66056 0.44504 -1.41421 1.94986 -1.88777 1.24698 -0.22393 -0.86777 1.69345 -2.00000 -2.00000 -1.69345 0.86777 0.22393 -1.24698 1.88777 -1.94986 1.41421 -0.44504 -0.66056 1.56366 -1.98742 1.80194 -1.06406 0.00000 1.06406 -1.80194 1.98742 -1.56366 0.66056 0.44504 -1.41421 1.94986 -1.88777 1.24698 -0.22393 -0.86777 1.69345 2.00000 ... ... ..... ....... ......... ........... ............. E24u * 2.00000 -1.80194 1.24698 -0.44504 -0.44504 1.24698 -1.80194 2.00000 -1.80194 1.24698 -0.44504 -0.44504 1.24698 -1.80194 2.00000 -1.80194 1.24698 -0.44504 -0.44504 1.24698 -1.80194 2.00000 -1.80194 1.24698 -0.44504 -0.44504 1.24698 -1.80194 2.00000 -2.00000 1.80194 -1.24698 0.44504 0.44504 -1.24698 1.80194 -2.00000 1.80194 -1.24698 0.44504 0.44504 -1.24698 1.80194 -2.00000 1.80194 -1.24698 0.44504 0.44504 -1.24698 1.80194 -2.00000 1.80194 -1.24698 0.44504 0.44504 -1.24698 1.80194 -2.00000 ... ... ..... ....... ......... ........... ............. E25u * 2.00000 -1.88777 1.56366 -1.06406 0.44504 0.22393 -0.86777 1.41421 -1.80194 1.98742 -1.94986 1.69345 -1.24698 0.66056 0.00000 -0.66056 1.24698 -1.69345 1.94986 -1.98742 1.80194 -1.41421 0.86777 -0.22393 -0.44504 1.06406 -1.56366 1.88777 -2.00000 -2.00000 -1.88777 1.56366 -1.06406 0.44504 0.22393 -0.86777 1.41421 -1.80194 1.98742 -1.94986 1.69345 -1.24698 0.66056 0.00000 -0.66056 1.24698 -1.69345 1.94986 -1.98742 1.80194 -1.41421 0.86777 -0.22393 -0.44504 1.06406 -1.56366 1.88777 2.00000 ... ... ..... ....... ......... ........... ............. E26u * 2.00000 -1.94986 1.80194 -1.56366 1.24698 -0.86777 0.44504 0.00000 -0.44504 0.86777 -1.24698 1.56366 -1.80194 1.94986 -2.00000 1.94986 -1.80194 1.56366 -1.24698 0.86777 -0.44504 0.00000 0.44504 -0.86777 1.24698 -1.56366 1.80194 -1.94986 2.00000 -2.00000 1.94986 -1.80194 1.56366 -1.24698 0.86777 -0.44504 0.00000 0.44504 -0.86777 1.24698 -1.56366 1.80194 -1.94986 2.00000 -1.94986 1.80194 -1.56366 1.24698 -0.86777 0.44504 0.00000 -0.44504 0.86777 -1.24698 1.56366 -1.80194 1.94986 -2.00000 ... ... ..... ....... ......... ........... ............. E27u * 2.00000 -1.98742 1.94986 -1.88777 1.80194 -1.69345 1.56366 -1.41421 1.24698 -1.06406 0.86777 -0.66056 0.44504 -0.22393 0.00000 0.22393 -0.44504 0.66056 -0.86777 1.06406 -1.24698 1.41421 -1.56366 1.69345 -1.80194 1.88777 -1.94986 1.98742 -2.00000 -2.00000 -1.98742 1.94986 -1.88777 1.80194 -1.69345 1.56366 -1.41421 1.24698 -1.06406 0.86777 -0.66056 0.44504 -0.22393 0.00000 0.22393 -0.44504 0.66056 -0.86777 1.06406 -1.24698 1.41421 -1.56366 1.69345 -1.80194 1.88777 -1.94986 1.98742 2.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.987424419786 = 2*cos(2*π/56) = 2*cos(π/28) 1.949855824364 = 2*cos(4*π/56) = 2*cos(π/14) 1.887766660617 = 2*cos(6*π/56) = 2*cos(3*π/28) 1.801937735805 = 2*cos(8*π/56) = 2*cos(π/7) 1.693448398457 = 2*cos(10*π/56) = 2*cos(5*π/28) 1.563662964936 = 2*cos(12*π/56) = 2*cos(3*π/14) 1.414213562373 = 2*cos(14*π/56) = 2*cos(π/4) = √2 1.246979603717 = 2*cos(16*π/56) = 2*cos(2*π/7) 1.064064153031 = 2*cos(18*π/56) = 2*cos(9*π/28) 0.867767478235 = 2*cos(20*π/56) = 2*cos(5*π/14) 0.660558123910 = 2*cos(22*π/56) = 2*cos(11*π/28) 0.445041867913 = 2*cos(24*π/56) = 2*cos(3*π/7) 0.223928952207 = 2*cos(26*π/56) = 2*cos(13*π/28) 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_56h point group is 112, and the order of the principal axis (C₅₆) is 56. The group has 58 irreducible representations. β The C_56h point group is generated by two symmetry elements, which are canonically chosen as C₅₆ and i. Other possible choices are C₅₆ and σ_h, or less commonly S₅₆ with either i or σ_h. γ The lowest nonvanishing multipole moment in C_56h is 4 (quadrupole moment). δ This is an Abelian point group (the commutative law holds between all symmetry operations). The C_56h 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 108 cases have been combined into 54 two-dimensional representations that are no longer irreducible but have real-valued characters. Accordingly, 54 pairs of left and right rotations have been combined into one two-membered pseudo-class each. ζ The 54 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. ι The regular heptagon is the lowest regular polygon that cannot be constructed by compass and ruler alone, which has mystified mathematicians since antiquity. The reason for the inconstructibility of the heptagon is that cos(2π/7) has an algebraic degree of 3 and thus can not be represented by square roots and integer numbers. An algebraic representation becomes possible if cubic roots and complex numbers are used: 2*cos(2π/7) = (³√28+i*84*√3 + ³√28−i*84*√3 − 2)/6. This complex expression for a real value is hardly useful. Therefore, regular polygons of order 14,21,28,35,42 etc. are also inconstructible, and their cosines have no representation in real radicals.

This Character Table for the C_56h point group was created by Gernot Katzer.

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

	C_54h
	C_55h
C₅₆ C_56v	C_56h	D₅₆ D_56h D_56d S₅₆
	C_57h
	C_58h

Character table for the C56h point group

Character table for the C_56h point group