Character Table for Point Group D52h

Character table for the D_52h point group

D52h E 2 C52 2 C26 2 C52^3 2 C13 2 C52^5 2 C26^3 2 C52^7 2 C13^2 2 C52^9 2 C26^5 2 C52^11 2 C13^3 2 C4 2 C26^7 2 C52^15 2 C13^4 2 C52^17 2 C26^9 2 C52^19 2 C13^5 2 C52^21 2 C26^11 2 C52^23 2 C13^6 2 C52^25 C2 26 C2' 26 C2" i 2 S52 2 S26 2 S52^3 2 S13 2 S52^5 2 S26^3 2 S52^7 2 S13^2 2 S52^9 2 S26^5 2 S52^11 2 S13^3 2 S4 2 S26^7 2 S52^15 2 S13^4 2 S52^17 2 S26^9 2 S52^19 2 S13^5 2 S52^21 2 S26^11 2 S52^23 2 S13^6 2 S52^25 sh 26 sv 26 sd <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> A1g 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 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 A2g 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 -1.00000 -1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 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 ... ..... ....... ......... ........... ............. B1g 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 1.00000 -1.00000 ... ... ..... ....... ......... ........... ............. B2g 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -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 1.98542 1.94188 1.87003 1.77091 1.64597 1.49702 1.32625 1.13613 0.92945 0.70921 0.47863 0.24107 0.00000 -0.24107 -0.47863 -0.70921 -0.92945 -1.13613 -1.32625 -1.49702 -1.64597 -1.77091 -1.87003 -1.94188 -1.98542 -2.00000 0.00000 0.00000 2.00000 -1.98542 -1.94188 -1.87003 -1.77091 -1.64597 -1.49702 -1.32625 -1.13613 -0.92945 -0.70921 -0.47863 -0.24107 0.00000 0.24107 0.47863 0.70921 0.92945 1.13613 1.32625 1.49702 1.64597 1.77091 1.87003 1.94188 1.98542 -2.00000 0.00000 0.00000 TT. ... ..TT. ....... ......TT. ........... ..........TT. E2g 2.00000 1.94188 1.77091 1.49702 1.13613 0.70921 0.24107 -0.24107 -0.70921 -1.13613 -1.49702 -1.77091 -1.94188 -2.00000 -1.94188 -1.77091 -1.49702 -1.13613 -0.70921 -0.24107 0.24107 0.70921 1.13613 1.49702 1.77091 1.94188 2.00000 0.00000 0.00000 2.00000 1.94188 1.77091 1.49702 1.13613 0.70921 0.24107 -0.24107 -0.70921 -1.13613 -1.49702 -1.77091 -1.94188 -2.00000 -1.94188 -1.77091 -1.49702 -1.13613 -0.70921 -0.24107 0.24107 0.70921 1.13613 1.49702 1.77091 1.94188 2.00000 0.00000 0.00000 ... ... TT... ....... ....TT... ........... ........TT... E3g 2.00000 1.87003 1.49702 0.92945 0.24107 -0.47863 -1.13613 -1.64597 -1.94188 -1.98542 -1.77091 -1.32625 -0.70921 0.00000 0.70921 1.32625 1.77091 1.98542 1.94188 1.64597 1.13613 0.47863 -0.24107 -0.92945 -1.49702 -1.87003 -2.00000 0.00000 0.00000 2.00000 -1.87003 -1.49702 -0.92945 -0.24107 0.47863 1.13613 1.64597 1.94188 1.98542 1.77091 1.32625 0.70921 0.00000 -0.70921 -1.32625 -1.77091 -1.98542 -1.94188 -1.64597 -1.13613 -0.47863 0.24107 0.92945 1.49702 1.87003 -2.00000 0.00000 0.00000 ... ... ..... ....... ..TT..... ........... ......TT..... E4g 2.00000 1.77091 1.13613 0.24107 -0.70921 -1.49702 -1.94188 -1.94188 -1.49702 -0.70921 0.24107 1.13613 1.77091 2.00000 1.77091 1.13613 0.24107 -0.70921 -1.49702 -1.94188 -1.94188 -1.49702 -0.70921 0.24107 1.13613 1.77091 2.00000 0.00000 0.00000 2.00000 1.77091 1.13613 0.24107 -0.70921 -1.49702 -1.94188 -1.94188 -1.49702 -0.70921 0.24107 1.13613 1.77091 2.00000 1.77091 1.13613 0.24107 -0.70921 -1.49702 -1.94188 -1.94188 -1.49702 -0.70921 0.24107 1.13613 1.77091 2.00000 0.00000 0.00000 ... ... ..... ....... TT....... ........... ....TT....... E5g 2.00000 1.64597 0.70921 -0.47863 -1.49702 -1.98542 -1.77091 -0.92945 0.24107 1.32625 1.94188 1.87003 1.13613 0.00000 -1.13613 -1.87003 -1.94188 -1.32625 -0.24107 0.92945 1.77091 1.98542 1.49702 0.47863 -0.70921 -1.64597 -2.00000 0.00000 0.00000 2.00000 -1.64597 -0.70921 0.47863 1.49702 1.98542 1.77091 0.92945 -0.24107 -1.32625 -1.94188 -1.87003 -1.13613 0.00000 1.13613 1.87003 1.94188 1.32625 0.24107 -0.92945 -1.77091 -1.98542 -1.49702 -0.47863 0.70921 1.64597 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ..TT......... E6g 2.00000 1.49702 0.24107 -1.13613 -1.94188 -1.77091 -0.70921 0.70921 1.77091 1.94188 1.13613 -0.24107 -1.49702 -2.00000 -1.49702 -0.24107 1.13613 1.94188 1.77091 0.70921 -0.70921 -1.77091 -1.94188 -1.13613 0.24107 1.49702 2.00000 0.00000 0.00000 2.00000 1.49702 0.24107 -1.13613 -1.94188 -1.77091 -0.70921 0.70921 1.77091 1.94188 1.13613 -0.24107 -1.49702 -2.00000 -1.49702 -0.24107 1.13613 1.94188 1.77091 0.70921 -0.70921 -1.77091 -1.94188 -1.13613 0.24107 1.49702 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... TT........... E7g 2.00000 1.32625 -0.24107 -1.64597 -1.94188 -0.92945 0.70921 1.87003 1.77091 0.47863 -1.13613 -1.98542 -1.49702 0.00000 1.49702 1.98542 1.13613 -0.47863 -1.77091 -1.87003 -0.70921 0.92945 1.94188 1.64597 0.24107 -1.32625 -2.00000 0.00000 0.00000 2.00000 -1.32625 0.24107 1.64597 1.94188 0.92945 -0.70921 -1.87003 -1.77091 -0.47863 1.13613 1.98542 1.49702 0.00000 -1.49702 -1.98542 -1.13613 0.47863 1.77091 1.87003 0.70921 -0.92945 -1.94188 -1.64597 -0.24107 1.32625 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E8g 2.00000 1.13613 -0.70921 -1.94188 -1.49702 0.24107 1.77091 1.77091 0.24107 -1.49702 -1.94188 -0.70921 1.13613 2.00000 1.13613 -0.70921 -1.94188 -1.49702 0.24107 1.77091 1.77091 0.24107 -1.49702 -1.94188 -0.70921 1.13613 2.00000 0.00000 0.00000 2.00000 1.13613 -0.70921 -1.94188 -1.49702 0.24107 1.77091 1.77091 0.24107 -1.49702 -1.94188 -0.70921 1.13613 2.00000 1.13613 -0.70921 -1.94188 -1.49702 0.24107 1.77091 1.77091 0.24107 -1.49702 -1.94188 -0.70921 1.13613 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E9g 2.00000 0.92945 -1.13613 -1.98542 -0.70921 1.32625 1.94188 0.47863 -1.49702 -1.87003 -0.24107 1.64597 1.77091 0.00000 -1.77091 -1.64597 0.24107 1.87003 1.49702 -0.47863 -1.94188 -1.32625 0.70921 1.98542 1.13613 -0.92945 -2.00000 0.00000 0.00000 2.00000 -0.92945 1.13613 1.98542 0.70921 -1.32625 -1.94188 -0.47863 1.49702 1.87003 0.24107 -1.64597 -1.77091 0.00000 1.77091 1.64597 -0.24107 -1.87003 -1.49702 0.47863 1.94188 1.32625 -0.70921 -1.98542 -1.13613 0.92945 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E10g 2.00000 0.70921 -1.49702 -1.77091 0.24107 1.94188 1.13613 -1.13613 -1.94188 -0.24107 1.77091 1.49702 -0.70921 -2.00000 -0.70921 1.49702 1.77091 -0.24107 -1.94188 -1.13613 1.13613 1.94188 0.24107 -1.77091 -1.49702 0.70921 2.00000 0.00000 0.00000 2.00000 0.70921 -1.49702 -1.77091 0.24107 1.94188 1.13613 -1.13613 -1.94188 -0.24107 1.77091 1.49702 -0.70921 -2.00000 -0.70921 1.49702 1.77091 -0.24107 -1.94188 -1.13613 1.13613 1.94188 0.24107 -1.77091 -1.49702 0.70921 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E11g 2.00000 0.47863 -1.77091 -1.32625 1.13613 1.87003 -0.24107 -1.98542 -0.70921 1.64597 1.49702 -0.92945 -1.94188 0.00000 1.94188 0.92945 -1.49702 -1.64597 0.70921 1.98542 0.24107 -1.87003 -1.13613 1.32625 1.77091 -0.47863 -2.00000 0.00000 0.00000 2.00000 -0.47863 1.77091 1.32625 -1.13613 -1.87003 0.24107 1.98542 0.70921 -1.64597 -1.49702 0.92945 1.94188 0.00000 -1.94188 -0.92945 1.49702 1.64597 -0.70921 -1.98542 -0.24107 1.87003 1.13613 -1.32625 -1.77091 0.47863 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E12g 2.00000 0.24107 -1.94188 -0.70921 1.77091 1.13613 -1.49702 -1.49702 1.13613 1.77091 -0.70921 -1.94188 0.24107 2.00000 0.24107 -1.94188 -0.70921 1.77091 1.13613 -1.49702 -1.49702 1.13613 1.77091 -0.70921 -1.94188 0.24107 2.00000 0.00000 0.00000 2.00000 0.24107 -1.94188 -0.70921 1.77091 1.13613 -1.49702 -1.49702 1.13613 1.77091 -0.70921 -1.94188 0.24107 2.00000 0.24107 -1.94188 -0.70921 1.77091 1.13613 -1.49702 -1.49702 1.13613 1.77091 -0.70921 -1.94188 0.24107 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E13g 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 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E14g 2.00000 -0.24107 -1.94188 0.70921 1.77091 -1.13613 -1.49702 1.49702 1.13613 -1.77091 -0.70921 1.94188 0.24107 -2.00000 0.24107 1.94188 -0.70921 -1.77091 1.13613 1.49702 -1.49702 -1.13613 1.77091 0.70921 -1.94188 -0.24107 2.00000 0.00000 0.00000 2.00000 -0.24107 -1.94188 0.70921 1.77091 -1.13613 -1.49702 1.49702 1.13613 -1.77091 -0.70921 1.94188 0.24107 -2.00000 0.24107 1.94188 -0.70921 -1.77091 1.13613 1.49702 -1.49702 -1.13613 1.77091 0.70921 -1.94188 -0.24107 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E15g 2.00000 -0.47863 -1.77091 1.32625 1.13613 -1.87003 -0.24107 1.98542 -0.70921 -1.64597 1.49702 0.92945 -1.94188 0.00000 1.94188 -0.92945 -1.49702 1.64597 0.70921 -1.98542 0.24107 1.87003 -1.13613 -1.32625 1.77091 0.47863 -2.00000 0.00000 0.00000 2.00000 0.47863 1.77091 -1.32625 -1.13613 1.87003 0.24107 -1.98542 0.70921 1.64597 -1.49702 -0.92945 1.94188 0.00000 -1.94188 0.92945 1.49702 -1.64597 -0.70921 1.98542 -0.24107 -1.87003 1.13613 1.32625 -1.77091 -0.47863 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E16g 2.00000 -0.70921 -1.49702 1.77091 0.24107 -1.94188 1.13613 1.13613 -1.94188 0.24107 1.77091 -1.49702 -0.70921 2.00000 -0.70921 -1.49702 1.77091 0.24107 -1.94188 1.13613 1.13613 -1.94188 0.24107 1.77091 -1.49702 -0.70921 2.00000 0.00000 0.00000 2.00000 -0.70921 -1.49702 1.77091 0.24107 -1.94188 1.13613 1.13613 -1.94188 0.24107 1.77091 -1.49702 -0.70921 2.00000 -0.70921 -1.49702 1.77091 0.24107 -1.94188 1.13613 1.13613 -1.94188 0.24107 1.77091 -1.49702 -0.70921 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E17g 2.00000 -0.92945 -1.13613 1.98542 -0.70921 -1.32625 1.94188 -0.47863 -1.49702 1.87003 -0.24107 -1.64597 1.77091 0.00000 -1.77091 1.64597 0.24107 -1.87003 1.49702 0.47863 -1.94188 1.32625 0.70921 -1.98542 1.13613 0.92945 -2.00000 0.00000 0.00000 2.00000 0.92945 1.13613 -1.98542 0.70921 1.32625 -1.94188 0.47863 1.49702 -1.87003 0.24107 1.64597 -1.77091 0.00000 1.77091 -1.64597 -0.24107 1.87003 -1.49702 -0.47863 1.94188 -1.32625 -0.70921 1.98542 -1.13613 -0.92945 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E18g 2.00000 -1.13613 -0.70921 1.94188 -1.49702 -0.24107 1.77091 -1.77091 0.24107 1.49702 -1.94188 0.70921 1.13613 -2.00000 1.13613 0.70921 -1.94188 1.49702 0.24107 -1.77091 1.77091 -0.24107 -1.49702 1.94188 -0.70921 -1.13613 2.00000 0.00000 0.00000 2.00000 -1.13613 -0.70921 1.94188 -1.49702 -0.24107 1.77091 -1.77091 0.24107 1.49702 -1.94188 0.70921 1.13613 -2.00000 1.13613 0.70921 -1.94188 1.49702 0.24107 -1.77091 1.77091 -0.24107 -1.49702 1.94188 -0.70921 -1.13613 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E19g 2.00000 -1.32625 -0.24107 1.64597 -1.94188 0.92945 0.70921 -1.87003 1.77091 -0.47863 -1.13613 1.98542 -1.49702 0.00000 1.49702 -1.98542 1.13613 0.47863 -1.77091 1.87003 -0.70921 -0.92945 1.94188 -1.64597 0.24107 1.32625 -2.00000 0.00000 0.00000 2.00000 1.32625 0.24107 -1.64597 1.94188 -0.92945 -0.70921 1.87003 -1.77091 0.47863 1.13613 -1.98542 1.49702 0.00000 -1.49702 1.98542 -1.13613 -0.47863 1.77091 -1.87003 0.70921 0.92945 -1.94188 1.64597 -0.24107 -1.32625 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E20g 2.00000 -1.49702 0.24107 1.13613 -1.94188 1.77091 -0.70921 -0.70921 1.77091 -1.94188 1.13613 0.24107 -1.49702 2.00000 -1.49702 0.24107 1.13613 -1.94188 1.77091 -0.70921 -0.70921 1.77091 -1.94188 1.13613 0.24107 -1.49702 2.00000 0.00000 0.00000 2.00000 -1.49702 0.24107 1.13613 -1.94188 1.77091 -0.70921 -0.70921 1.77091 -1.94188 1.13613 0.24107 -1.49702 2.00000 -1.49702 0.24107 1.13613 -1.94188 1.77091 -0.70921 -0.70921 1.77091 -1.94188 1.13613 0.24107 -1.49702 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E21g 2.00000 -1.64597 0.70921 0.47863 -1.49702 1.98542 -1.77091 0.92945 0.24107 -1.32625 1.94188 -1.87003 1.13613 0.00000 -1.13613 1.87003 -1.94188 1.32625 -0.24107 -0.92945 1.77091 -1.98542 1.49702 -0.47863 -0.70921 1.64597 -2.00000 0.00000 0.00000 2.00000 1.64597 -0.70921 -0.47863 1.49702 -1.98542 1.77091 -0.92945 -0.24107 1.32625 -1.94188 1.87003 -1.13613 0.00000 1.13613 -1.87003 1.94188 -1.32625 0.24107 0.92945 -1.77091 1.98542 -1.49702 0.47863 0.70921 -1.64597 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E22g 2.00000 -1.77091 1.13613 -0.24107 -0.70921 1.49702 -1.94188 1.94188 -1.49702 0.70921 0.24107 -1.13613 1.77091 -2.00000 1.77091 -1.13613 0.24107 0.70921 -1.49702 1.94188 -1.94188 1.49702 -0.70921 -0.24107 1.13613 -1.77091 2.00000 0.00000 0.00000 2.00000 -1.77091 1.13613 -0.24107 -0.70921 1.49702 -1.94188 1.94188 -1.49702 0.70921 0.24107 -1.13613 1.77091 -2.00000 1.77091 -1.13613 0.24107 0.70921 -1.49702 1.94188 -1.94188 1.49702 -0.70921 -0.24107 1.13613 -1.77091 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E23g 2.00000 -1.87003 1.49702 -0.92945 0.24107 0.47863 -1.13613 1.64597 -1.94188 1.98542 -1.77091 1.32625 -0.70921 0.00000 0.70921 -1.32625 1.77091 -1.98542 1.94188 -1.64597 1.13613 -0.47863 -0.24107 0.92945 -1.49702 1.87003 -2.00000 0.00000 0.00000 2.00000 1.87003 -1.49702 0.92945 -0.24107 -0.47863 1.13613 -1.64597 1.94188 -1.98542 1.77091 -1.32625 0.70921 0.00000 -0.70921 1.32625 -1.77091 1.98542 -1.94188 1.64597 -1.13613 0.47863 0.24107 -0.92945 1.49702 -1.87003 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E24g 2.00000 -1.94188 1.77091 -1.49702 1.13613 -0.70921 0.24107 0.24107 -0.70921 1.13613 -1.49702 1.77091 -1.94188 2.00000 -1.94188 1.77091 -1.49702 1.13613 -0.70921 0.24107 0.24107 -0.70921 1.13613 -1.49702 1.77091 -1.94188 2.00000 0.00000 0.00000 2.00000 -1.94188 1.77091 -1.49702 1.13613 -0.70921 0.24107 0.24107 -0.70921 1.13613 -1.49702 1.77091 -1.94188 2.00000 -1.94188 1.77091 -1.49702 1.13613 -0.70921 0.24107 0.24107 -0.70921 1.13613 -1.49702 1.77091 -1.94188 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E25g 2.00000 -1.98542 1.94188 -1.87003 1.77091 -1.64597 1.49702 -1.32625 1.13613 -0.92945 0.70921 -0.47863 0.24107 0.00000 -0.24107 0.47863 -0.70921 0.92945 -1.13613 1.32625 -1.49702 1.64597 -1.77091 1.87003 -1.94188 1.98542 -2.00000 0.00000 0.00000 2.00000 1.98542 -1.94188 1.87003 -1.77091 1.64597 -1.49702 1.32625 -1.13613 0.92945 -0.70921 0.47863 -0.24107 0.00000 0.24107 -0.47863 0.70921 -0.92945 1.13613 -1.32625 1.49702 -1.64597 1.77091 -1.87003 1.94188 -1.98542 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. A1u 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 ... ... ..... ....... ......... ........... ............. 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E24u 2.00000 -1.94188 1.77091 -1.49702 1.13613 -0.70921 0.24107 0.24107 -0.70921 1.13613 -1.49702 1.77091 -1.94188 2.00000 -1.94188 1.77091 -1.49702 1.13613 -0.70921 0.24107 0.24107 -0.70921 1.13613 -1.49702 1.77091 -1.94188 2.00000 0.00000 0.00000 -2.00000 1.94188 -1.77091 1.49702 -1.13613 0.70921 -0.24107 -0.24107 0.70921 -1.13613 1.49702 -1.77091 1.94188 -2.00000 1.94188 -1.77091 1.49702 -1.13613 0.70921 -0.24107 -0.24107 0.70921 -1.13613 1.49702 -1.77091 1.94188 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E25u 2.00000 -1.98542 1.94188 -1.87003 1.77091 -1.64597 1.49702 -1.32625 1.13613 -0.92945 0.70921 -0.47863 0.24107 0.00000 -0.24107 0.47863 -0.70921 0.92945 -1.13613 1.32625 -1.49702 1.64597 -1.77091 1.87003 -1.94188 1.98542 -2.00000 0.00000 0.00000 -2.00000 -1.98542 1.94188 -1.87003 1.77091 -1.64597 1.49702 -1.32625 1.13613 -0.92945 0.70921 -0.47863 0.24107 0.00000 -0.24107 0.47863 -0.70921 0.92945 -1.13613 1.32625 -1.49702 1.64597 -1.77091 1.87003 -1.94188 1.98542 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.985417748196 = 2*cos(2*π/52) = 2*cos(π/26) 1.941883634852 = 2*cos(4*π/52) = 2*cos(π/13) 1.870032485371 = 2*cos(6*π/52) = 2*cos(3*π/26) 1.770912051306 = 2*cos(8*π/52) = 2*cos(2*π/13) 1.645967731787 = 2*cos(10*π/52) = 2*cos(5*π/26) 1.497021496342 = 2*cos(12*π/52) = 2*cos(3*π/13) 1.326245316482 = 2*cos(14*π/52) = 2*cos(7*π/26) 1.136129493462 = 2*cos(16*π/52) = 2*cos(4*π/13) 0.929446344088 = 2*cos(18*π/52) = 2*cos(9*π/26) 0.709209774085 = 2*cos(20*π/52) = 2*cos(5*π/13) 0.478631328575 = 2*cos(22*π/52) = 2*cos(11*π/26) 0.241073360511 = 2*cos(24*π/52) = 2*cos(6*π/13) Symmetry of Rotations and Cartesian products A1g d+g+i+k+m z², z⁴, z⁶ A2g R R_z 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 A2u 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 D_52h point group is 208, and the order of the principal axis (C₅₂) is 52. The group has 58 irreducible representations. β The D_52h point group is generated by three symmetry elements that are canonically chosen C₅₂, C₂^′ and i. Other choices include σ_h instead of i, or any of C₂^″, σ_v or σ_d instead of C₂^′. Also, some ternary combinations of C₂^′, C₂^″, σ_v and σ_d act as generators. Lastly, the S₅₂ can be chosen, together with i or σ_h and any one of C₂^′, C₂^″, σ_v or σ_d. γ There are two different sets of twofold symmetry axes perpendicular to the principal axis (z axis in standard orientation). By convention, the set denoted as C₂^′ contains both the x and y axes. δ There are two different sets of symmetry planes containing the principal axis (z axis in standard orientation). By convention, the set denoted as σ_v contains both the xz and the yz planes. ε The lowest nonvanishing multipole moment in D_52h is 4 (quadrupole moment). ζ This point group is non-Abelian (some symmetry operations are not commutative). Therefore, the character table contains multi-membered classes and degenerate irreducible representations. η Some of the characters in the table are irrational because the order of the principal axis is neither 1,2,3,4 nor 6. These irrational values can be expressed as cosine values, or as solutions of algebraic equations with a leading coefficient of 1. All characters are algebraic integers of a degree much less than half the order of the principal axis. θ The point group corresponds to a polygon inconstructible by the classical means of ruler and compass. Yet it becomes constructible if angle trisection is allowed, e.g., with neusis construction or origami. This is because the order of the principal axis is given by a product of any number of different Pierpont primes (...,5,7,13,17,19,37,73,97,109,163,...) times arbitrary powers of two and three. All characters of this group can be expressed using complex numbers, elementary arithmetic operations, square roots and third roots.

This Character Table for the D_52h point group was created by Gernot Katzer.

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

	D_50h
	D_51h
C₅₂ C_52v C_52h D₅₂	D_52h	D_52d S₅₂
	D_53h
	D_54h

Character table for the D52h point group

Character table for the D_52h point group