C42h E 2 C42 2 C21 2 C14 2 C21^2 2 C42^5 2 C7 2 C6 2 C21^4 2 C14^3 2 C21^5 2 C42^11 2 C7^2 2 C42^13 2 C3 2 C14^5 2 C21^8 2 C42^17 2 C7^3 2 C42^19 2 C21^10 C2 i 2 S42 2 S21 2 S14 2 S21^2 2 S42^5 2 S7 2 S6 2 S21^4 2 S14^3 2 S21^5 2 S42^11 2 S7^2 2 S42^13 2 S3 2 S14^5 2 S21^8 2 S42^17 2 S7^3 2 S42^19 2 S21^10 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 ..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 ... ... ..... ....... ......... ........... ............. E1g * 2.00000 1.97766 1.91115 1.80194 1.65248 1.46610 1.24698 1.00000 0.73068 0.44504 0.14946 -0.14946 -0.44504 -0.73068 -1.00000 -1.24698 -1.46610 -1.65248 -1.80194 -1.91115 -1.97766 -2.00000 2.00000 -1.97766 -1.91115 -1.80194 -1.65248 -1.46610 -1.24698 -1.00000 -0.73068 -0.44504 -0.14946 0.14946 0.44504 0.73068 1.00000 1.24698 1.46610 1.65248 1.80194 1.91115 1.97766 -2.00000 TT. ... ..TT. ....... ......TT. ........... ..........TT. E2g * 2.00000 1.91115 1.65248 1.24698 0.73068 0.14946 -0.44504 -1.00000 -1.46610 -1.80194 -1.97766 -1.97766 -1.80194 -1.46610 -1.00000 -0.44504 0.14946 0.73068 1.24698 1.65248 1.91115 2.00000 2.00000 1.91115 1.65248 1.24698 0.73068 0.14946 -0.44504 -1.00000 -1.46610 -1.80194 -1.97766 -1.97766 -1.80194 -1.46610 -1.00000 -0.44504 0.14946 0.73068 1.24698 1.65248 1.91115 2.00000 ... ... TT... ....... ....TT... ........... ........TT... E3g * 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 ... ... ..... ....... ..TT..... ........... ......TT..... E4g * 2.00000 1.65248 0.73068 -0.44504 -1.46610 -1.97766 -1.80194 -1.00000 0.14946 1.24698 1.91115 1.91115 1.24698 0.14946 -1.00000 -1.80194 -1.97766 -1.46610 -0.44504 0.73068 1.65248 2.00000 2.00000 1.65248 0.73068 -0.44504 -1.46610 -1.97766 -1.80194 -1.00000 0.14946 1.24698 1.91115 1.91115 1.24698 0.14946 -1.00000 -1.80194 -1.97766 -1.46610 -0.44504 0.73068 1.65248 2.00000 ... ... ..... ....... TT....... ........... ....TT....... E5g * 2.00000 1.46610 0.14946 -1.24698 -1.97766 -1.65248 -0.44504 1.00000 1.91115 1.80194 0.73068 -0.73068 -1.80194 -1.91115 -1.00000 0.44504 1.65248 1.97766 1.24698 -0.14946 -1.46610 -2.00000 2.00000 -1.46610 -0.14946 1.24698 1.97766 1.65248 0.44504 -1.00000 -1.91115 -1.80194 -0.73068 0.73068 1.80194 1.91115 1.00000 -0.44504 -1.65248 -1.97766 -1.24698 0.14946 1.46610 -2.00000 ... ... ..... ....... ......... ........... ..TT......... E6g * 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 ... ... ..... ....... ......... ........... TT........... E7g * 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 ... ... ..... ....... ......... ........... ............. E8g * 2.00000 0.73068 -1.46610 -1.80194 0.14946 1.91115 1.24698 -1.00000 -1.97766 -0.44504 1.65248 1.65248 -0.44504 -1.97766 -1.00000 1.24698 1.91115 0.14946 -1.80194 -1.46610 0.73068 2.00000 2.00000 0.73068 -1.46610 -1.80194 0.14946 1.91115 1.24698 -1.00000 -1.97766 -0.44504 1.65248 1.65248 -0.44504 -1.97766 -1.00000 1.24698 1.91115 0.14946 -1.80194 -1.46610 0.73068 2.00000 ... ... ..... ....... ......... ........... ............. E9g * 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 ... ... ..... ....... ......... ........... ............. E10g * 2.00000 0.14946 -1.97766 -0.44504 1.91115 0.73068 -1.80194 -1.00000 1.65248 1.24698 -1.46610 -1.46610 1.24698 1.65248 -1.00000 -1.80194 0.73068 1.91115 -0.44504 -1.97766 0.14946 2.00000 2.00000 0.14946 -1.97766 -0.44504 1.91115 0.73068 -1.80194 -1.00000 1.65248 1.24698 -1.46610 -1.46610 1.24698 1.65248 -1.00000 -1.80194 0.73068 1.91115 -0.44504 -1.97766 0.14946 2.00000 ... ... ..... ....... ......... ........... ............. E11g * 2.00000 -0.14946 -1.97766 0.44504 1.91115 -0.73068 -1.80194 1.00000 1.65248 -1.24698 -1.46610 1.46610 1.24698 -1.65248 -1.00000 1.80194 0.73068 -1.91115 -0.44504 1.97766 0.14946 -2.00000 2.00000 0.14946 1.97766 -0.44504 -1.91115 0.73068 1.80194 -1.00000 -1.65248 1.24698 1.46610 -1.46610 -1.24698 1.65248 1.00000 -1.80194 -0.73068 1.91115 0.44504 -1.97766 -0.14946 -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 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 -0.73068 -1.46610 1.80194 0.14946 -1.91115 1.24698 1.00000 -1.97766 0.44504 1.65248 -1.65248 -0.44504 1.97766 -1.00000 -1.24698 1.91115 -0.14946 -1.80194 1.46610 0.73068 -2.00000 2.00000 0.73068 1.46610 -1.80194 -0.14946 1.91115 -1.24698 -1.00000 1.97766 -0.44504 -1.65248 1.65248 0.44504 -1.97766 1.00000 1.24698 -1.91115 0.14946 1.80194 -1.46610 -0.73068 -2.00000 ... ... ..... ....... ......... ........... ............. E14g * 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 ... ... ..... ....... ......... ........... ............. E15g * 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 ... ... ..... ....... ......... ........... ............. E16g * 2.00000 -1.46610 0.14946 1.24698 -1.97766 1.65248 -0.44504 -1.00000 1.91115 -1.80194 0.73068 0.73068 -1.80194 1.91115 -1.00000 -0.44504 1.65248 -1.97766 1.24698 0.14946 -1.46610 2.00000 2.00000 -1.46610 0.14946 1.24698 -1.97766 1.65248 -0.44504 -1.00000 1.91115 -1.80194 0.73068 0.73068 -1.80194 1.91115 -1.00000 -0.44504 1.65248 -1.97766 1.24698 0.14946 -1.46610 2.00000 ... ... ..... ....... ......... ........... ............. E17g * 2.00000 -1.65248 0.73068 0.44504 -1.46610 1.97766 -1.80194 1.00000 0.14946 -1.24698 1.91115 -1.91115 1.24698 -0.14946 -1.00000 1.80194 -1.97766 1.46610 -0.44504 -0.73068 1.65248 -2.00000 2.00000 1.65248 -0.73068 -0.44504 1.46610 -1.97766 1.80194 -1.00000 -0.14946 1.24698 -1.91115 1.91115 -1.24698 0.14946 1.00000 -1.80194 1.97766 -1.46610 0.44504 0.73068 -1.65248 -2.00000 ... ... ..... ....... ......... ........... ............. E18g * 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 ... ... ..... ....... ......... ........... ............. E19g * 2.00000 -1.91115 1.65248 -1.24698 0.73068 -0.14946 -0.44504 1.00000 -1.46610 1.80194 -1.97766 1.97766 -1.80194 1.46610 -1.00000 0.44504 0.14946 -0.73068 1.24698 -1.65248 1.91115 -2.00000 2.00000 1.91115 -1.65248 1.24698 -0.73068 0.14946 0.44504 -1.00000 1.46610 -1.80194 1.97766 -1.97766 1.80194 -1.46610 1.00000 -0.44504 -0.14946 0.73068 -1.24698 1.65248 -1.91115 -2.00000 ... ... ..... ....... ......... ........... ............. E20g * 2.00000 -1.97766 1.91115 -1.80194 1.65248 -1.46610 1.24698 -1.00000 0.73068 -0.44504 0.14946 0.14946 -0.44504 0.73068 -1.00000 1.24698 -1.46610 1.65248 -1.80194 1.91115 -1.97766 2.00000 2.00000 -1.97766 1.91115 -1.80194 1.65248 -1.46610 1.24698 -1.00000 0.73068 -0.44504 0.14946 0.14946 -0.44504 0.73068 -1.00000 1.24698 -1.46610 1.65248 -1.80194 1.91115 -1.97766 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 ... ..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 ... ... ..... ....... ......... ........... ............. E1u * 2.00000 1.97766 1.91115 1.80194 1.65248 1.46610 1.24698 1.00000 0.73068 0.44504 0.14946 -0.14946 -0.44504 -0.73068 -1.00000 -1.24698 -1.46610 -1.65248 -1.80194 -1.91115 -1.97766 -2.00000 -2.00000 1.97766 1.91115 1.80194 1.65248 1.46610 1.24698 1.00000 0.73068 0.44504 0.14946 -0.14946 -0.44504 -0.73068 -1.00000 -1.24698 -1.46610 -1.65248 -1.80194 -1.91115 -1.97766 2.00000 ... TT. ..... ....TT. ......... ........TT. ............. E2u * 2.00000 1.91115 1.65248 1.24698 0.73068 0.14946 -0.44504 -1.00000 -1.46610 -1.80194 -1.97766 -1.97766 -1.80194 -1.46610 -1.00000 -0.44504 0.14946 0.73068 1.24698 1.65248 1.91115 2.00000 -2.00000 -1.91115 -1.65248 -1.24698 -0.73068 -0.14946 0.44504 1.00000 1.46610 1.80194 1.97766 1.97766 1.80194 1.46610 1.00000 0.44504 -0.14946 -0.73068 -1.24698 -1.65248 -1.91115 -2.00000 ... ... ..... ..TT... ......... ......TT... ............. E3u * 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 ... ... ..... TT..... ......... ....TT..... ............. E4u * 2.00000 1.65248 0.73068 -0.44504 -1.46610 -1.97766 -1.80194 -1.00000 0.14946 1.24698 1.91115 1.91115 1.24698 0.14946 -1.00000 -1.80194 -1.97766 -1.46610 -0.44504 0.73068 1.65248 2.00000 -2.00000 -1.65248 -0.73068 0.44504 1.46610 1.97766 1.80194 1.00000 -0.14946 -1.24698 -1.91115 -1.91115 -1.24698 -0.14946 1.00000 1.80194 1.97766 1.46610 0.44504 -0.73068 -1.65248 -2.00000 ... ... ..... ....... ......... ..TT....... ............. E5u * 2.00000 1.46610 0.14946 -1.24698 -1.97766 -1.65248 -0.44504 1.00000 1.91115 1.80194 0.73068 -0.73068 -1.80194 -1.91115 -1.00000 0.44504 1.65248 1.97766 1.24698 -0.14946 -1.46610 -2.00000 -2.00000 1.46610 0.14946 -1.24698 -1.97766 -1.65248 -0.44504 1.00000 1.91115 1.80194 0.73068 -0.73068 -1.80194 -1.91115 -1.00000 0.44504 1.65248 1.97766 1.24698 -0.14946 -1.46610 2.00000 ... ... ..... ....... ......... TT......... ............. E6u * 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 ... ... ..... ....... ......... ........... ............. E7u * 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 ... ... ..... ....... ......... ........... ............. E8u * 2.00000 0.73068 -1.46610 -1.80194 0.14946 1.91115 1.24698 -1.00000 -1.97766 -0.44504 1.65248 1.65248 -0.44504 -1.97766 -1.00000 1.24698 1.91115 0.14946 -1.80194 -1.46610 0.73068 2.00000 -2.00000 -0.73068 1.46610 1.80194 -0.14946 -1.91115 -1.24698 1.00000 1.97766 0.44504 -1.65248 -1.65248 0.44504 1.97766 1.00000 -1.24698 -1.91115 -0.14946 1.80194 1.46610 -0.73068 -2.00000 ... ... ..... ....... ......... ........... ............. E9u * 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 ... ... ..... ....... ......... ........... ............. E10u * 2.00000 0.14946 -1.97766 -0.44504 1.91115 0.73068 -1.80194 -1.00000 1.65248 1.24698 -1.46610 -1.46610 1.24698 1.65248 -1.00000 -1.80194 0.73068 1.91115 -0.44504 -1.97766 0.14946 2.00000 -2.00000 -0.14946 1.97766 0.44504 -1.91115 -0.73068 1.80194 1.00000 -1.65248 -1.24698 1.46610 1.46610 -1.24698 -1.65248 1.00000 1.80194 -0.73068 -1.91115 0.44504 1.97766 -0.14946 -2.00000 ... ... ..... ....... ......... ........... ............. E11u * 2.00000 -0.14946 -1.97766 0.44504 1.91115 -0.73068 -1.80194 1.00000 1.65248 -1.24698 -1.46610 1.46610 1.24698 -1.65248 -1.00000 1.80194 0.73068 -1.91115 -0.44504 1.97766 0.14946 -2.00000 -2.00000 -0.14946 -1.97766 0.44504 1.91115 -0.73068 -1.80194 1.00000 1.65248 -1.24698 -1.46610 1.46610 1.24698 -1.65248 -1.00000 1.80194 0.73068 -1.91115 -0.44504 1.97766 0.14946 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 -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 -0.73068 -1.46610 1.80194 0.14946 -1.91115 1.24698 1.00000 -1.97766 0.44504 1.65248 -1.65248 -0.44504 1.97766 -1.00000 -1.24698 1.91115 -0.14946 -1.80194 1.46610 0.73068 -2.00000 -2.00000 -0.73068 -1.46610 1.80194 0.14946 -1.91115 1.24698 1.00000 -1.97766 0.44504 1.65248 -1.65248 -0.44504 1.97766 -1.00000 -1.24698 1.91115 -0.14946 -1.80194 1.46610 0.73068 2.00000 ... ... ..... ....... ......... ........... ............. E14u * 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 ... ... ..... ....... ......... ........... ............. E15u * 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 ... ... ..... ....... ......... ........... ............. E16u * 2.00000 -1.46610 0.14946 1.24698 -1.97766 1.65248 -0.44504 -1.00000 1.91115 -1.80194 0.73068 0.73068 -1.80194 1.91115 -1.00000 -0.44504 1.65248 -1.97766 1.24698 0.14946 -1.46610 2.00000 -2.00000 1.46610 -0.14946 -1.24698 1.97766 -1.65248 0.44504 1.00000 -1.91115 1.80194 -0.73068 -0.73068 1.80194 -1.91115 1.00000 0.44504 -1.65248 1.97766 -1.24698 -0.14946 1.46610 -2.00000 ... ... ..... ....... ......... ........... ............. E17u * 2.00000 -1.65248 0.73068 0.44504 -1.46610 1.97766 -1.80194 1.00000 0.14946 -1.24698 1.91115 -1.91115 1.24698 -0.14946 -1.00000 1.80194 -1.97766 1.46610 -0.44504 -0.73068 1.65248 -2.00000 -2.00000 -1.65248 0.73068 0.44504 -1.46610 1.97766 -1.80194 1.00000 0.14946 -1.24698 1.91115 -1.91115 1.24698 -0.14946 -1.00000 1.80194 -1.97766 1.46610 -0.44504 -0.73068 1.65248 2.00000 ... ... ..... ....... ......... ........... ............. E18u * 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 ... ... ..... ....... ......... ........... ............. E19u * 2.00000 -1.91115 1.65248 -1.24698 0.73068 -0.14946 -0.44504 1.00000 -1.46610 1.80194 -1.97766 1.97766 -1.80194 1.46610 -1.00000 0.44504 0.14946 -0.73068 1.24698 -1.65248 1.91115 -2.00000 -2.00000 -1.91115 1.65248 -1.24698 0.73068 -0.14946 -0.44504 1.00000 -1.46610 1.80194 -1.97766 1.97766 -1.80194 1.46610 -1.00000 0.44504 0.14946 -0.73068 1.24698 -1.65248 1.91115 2.00000 ... ... ..... ....... ......... ........... ............. E20u * 2.00000 -1.97766 1.91115 -1.80194 1.65248 -1.46610 1.24698 -1.00000 0.73068 -0.44504 0.14946 0.14946 -0.44504 0.73068 -1.00000 1.24698 -1.46610 1.65248 -1.80194 1.91115 -1.97766 2.00000 -2.00000 1.97766 -1.91115 1.80194 -1.65248 1.46610 -1.24698 1.00000 -0.73068 0.44504 -0.14946 -0.14946 0.44504 -0.73068 1.00000 -1.24698 1.46610 -1.65248 1.80194 -1.91115 1.97766 -2.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.977661652450 = 2*cos(2*π/42) = 2*cos(π/21) 1.911145611572 = 2*cos(4*π/42) = 2*cos(2*π/21) 1.801937735805 = 2*cos(6*π/42) = 2*cos(π/7) 1.652477548632 = 2*cos(8*π/42) = 2*cos(4*π/21) 1.466103743660 = 2*cos(10*π/42) = 2*cos(5*π/21) 1.246979603717 = 2*cos(12*π/42) = 2*cos(2*π/7) 0.730682048733 = 2*cos(16*π/42) = 2*cos(8*π/21) 0.445041867913 = 2*cos(18*π/42) = 2*cos(3*π/7) 0.149460187173 = 2*cos(20*π/42) = 2*cos(10*π/21) Symmetry of Rotations and Cartesian products Ag R+d+g+i+k+m Rz, z2, z4, z6 E1g R+d+g+i+k+m {Rx, Ry}, {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 C42h point group is 84, and the order of the principal axis (C42) is 42. The group has 44 irreducible representations. β The C42h point group is generated by two symmetry elements, which are canonically chosen as C42 and i. Other possible choices are C42 and σh, or less commonly S42 with either C2 or σh. γ The lowest nonvanishing multipole moment in C42h is 4 (quadrupole moment). δ This is an Abelian point group (the commutative law holds between all symmetry operations). The C42h 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 80 cases have been combined into 40 two-dimensional representations that are no longer irreducible but have real-valued characters. Accordingly, 40 pairs of left and right rotations have been combined into one two-membered pseudo-class each. ζ The 40 reducible “E” representations almost behave like true irreducible representations. Their norm, however, is twice the group order. Therefore, they have been marked with an asterisk in the table. This is essential when trying to decompose a reducible representation into “irreducible” ones using the familiar projection formula. η Some of the characters in the table are irrational because the order of the principal axis is neither 1,2,3,4 nor 6. These irrational values can be expressed as cosine values, or as solutions of algebraic equations with a leading coefficient of 1. All characters are algebraic integers of a degree much less than half the order of the principal axis. θ The point group corresponds to a polygon inconstructible by the classical means of ruler and compass. Yet it becomes constructible if angle trisection is allowed, e.g., with neusis construction or origami. This is because the order of the principal axis is given by a product of any number of different Pierpont primes (...,5,7,13,17,19,37,73,97,109,163,...) times arbitrary powers of two and three. All characters of this group can be expressed using complex numbers, elementary arithmetic operations, square roots and third roots.
| C40h | ||
| C41h | ||
| C42 C42v | C42h | D42 D42h D42d S42 |
| C43h | ||
| C44h |
This Character Table for the C42h point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.