Character Table for Point Group D78d

Character table for the D_78d point group

D78d E 2 S156 2 C78 2 S52 2 C39 2 S156^5 2 C26 2 S156^7 2 C39^2 2 S52^3 2 C78^5 2 S156^11 2 C13 2 S12 2 C78^7 2 S52^5 2 C39^4 2 S156^17 2 C26^3 2 S156^19 2 C39^5 2 S52^7 2 C78^11 2 S156^23 2 C13^2 2 S156^25 2 C6 2 S52^9 2 C39^7 2 S156^29 2 C26^5 2 S156^31 2 C39^8 2 S52^11 2 C78^17 2 S156^35 2 C13^3 2 S156^37 2 C78^19 2 S4 2 C39^10 2 S156^41 2 C26^7 2 S156^43 2 C39^11 2 S52^15 2 C78^23 2 S156^47 2 C13^4 2 S156^49 2 C78^25 2 S52^17 2 C3 2 S156^53 2 C26^9 2 S156^55 2 C39^14 2 S52^19 2 C78^29 2 S156^59 2 C13^5 2 S156^61 2 C78^31 2 S52^21 2 C39^16 2 S12^5 2 C26^11 2 S156^67 2 C39^17 2 S52^23 2 C78^35 2 S156^71 2 C13^6 2 S156^73 2 C78^37 2 S52^25 2 C39^19 2 S156^77 C2 78 C2' 78 sd <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> A1 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 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 A2 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 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 ... ..... ....... ......... ........... ............. B1 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 1.00000 -1.00000 ... ... ..... ....... ......... ........... ............. B2 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -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 ............. E1 2.00000 1.99838 1.99351 1.98542 1.97410 1.95958 1.94188 1.92104 1.89707 1.87003 1.83996 1.80690 1.77091 1.73205 1.69038 1.64597 1.59889 1.54921 1.49702 1.44240 1.38545 1.32625 1.26489 1.20148 1.13613 1.06893 1.00000 0.92945 0.85739 0.78393 0.70921 0.63334 0.55643 0.47863 0.40005 0.32082 0.24107 0.16093 0.08053 0.00000 -0.08053 -0.16093 -0.24107 -0.32082 -0.40005 -0.47863 -0.55643 -0.63334 -0.70921 -0.78393 -0.85739 -0.92945 -1.00000 -1.06893 -1.13613 -1.20148 -1.26489 -1.32625 -1.38545 -1.44240 -1.49702 -1.54921 -1.59889 -1.64597 -1.69038 -1.73205 -1.77091 -1.80690 -1.83996 -1.87003 -1.89707 -1.92104 -1.94188 -1.95958 -1.97410 -1.98542 -1.99351 -1.99838 -2.00000 0.00000 0.00000 ... TT. ..... ....TT. ......... ........TT. ............. E2 2.00000 1.99351 1.97410 1.94188 1.89707 1.83996 1.77091 1.69038 1.59889 1.49702 1.38545 1.26489 1.13613 1.00000 0.85739 0.70921 0.55643 0.40005 0.24107 0.08053 -0.08053 -0.24107 -0.40005 -0.55643 -0.70921 -0.85739 -1.00000 -1.13613 -1.26489 -1.38545 -1.49702 -1.59889 -1.69038 -1.77091 -1.83996 -1.89707 -1.94188 -1.97410 -1.99351 -2.00000 -1.99351 -1.97410 -1.94188 -1.89707 -1.83996 -1.77091 -1.69038 -1.59889 -1.49702 -1.38545 -1.26489 -1.13613 -1.00000 -0.85739 -0.70921 -0.55643 -0.40005 -0.24107 -0.08053 0.08053 0.24107 0.40005 0.55643 0.70921 0.85739 1.00000 1.13613 1.26489 1.38545 1.49702 1.59889 1.69038 1.77091 1.83996 1.89707 1.94188 1.97410 1.99351 2.00000 0.00000 0.00000 ... ... TT... ....... ....TT... ........... ........TT... E3 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 -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 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..... ............. E4 2.00000 1.97410 1.89707 1.77091 1.59889 1.38545 1.13613 0.85739 0.55643 0.24107 -0.08053 -0.40005 -0.70921 -1.00000 -1.26489 -1.49702 -1.69038 -1.83996 -1.94188 -1.99351 -1.99351 -1.94188 -1.83996 -1.69038 -1.49702 -1.26489 -1.00000 -0.70921 -0.40005 -0.08053 0.24107 0.55643 0.85739 1.13613 1.38545 1.59889 1.77091 1.89707 1.97410 2.00000 1.97410 1.89707 1.77091 1.59889 1.38545 1.13613 0.85739 0.55643 0.24107 -0.08053 -0.40005 -0.70921 -1.00000 -1.26489 -1.49702 -1.69038 -1.83996 -1.94188 -1.99351 -1.99351 -1.94188 -1.83996 -1.69038 -1.49702 -1.26489 -1.00000 -0.70921 -0.40005 -0.08053 0.24107 0.55643 0.85739 1.13613 1.38545 1.59889 1.77091 1.89707 1.97410 2.00000 0.00000 0.00000 ... ... ..... ....... TT....... ........... ....TT....... E5 2.00000 1.95958 1.83996 1.64597 1.38545 1.06893 0.70921 0.32082 -0.08053 -0.47863 -0.85739 -1.20148 -1.49702 -1.73205 -1.89707 -1.98542 -1.99351 -1.92104 -1.77091 -1.54921 -1.26489 -0.92945 -0.55643 -0.16093 0.24107 0.63334 1.00000 1.32625 1.59889 1.80690 1.94188 1.99838 1.97410 1.87003 1.69038 1.44240 1.13613 0.78393 0.40005 0.00000 -0.40005 -0.78393 -1.13613 -1.44240 -1.69038 -1.87003 -1.97410 -1.99838 -1.94188 -1.80690 -1.59889 -1.32625 -1.00000 -0.63334 -0.24107 0.16093 0.55643 0.92945 1.26489 1.54921 1.77091 1.92104 1.99351 1.98542 1.89707 1.73205 1.49702 1.20148 0.85739 0.47863 0.08053 -0.32082 -0.70921 -1.06893 -1.38545 -1.64597 -1.83996 -1.95958 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... TT......... ............. E6 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 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 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........... E7 2.00000 1.92104 1.69038 1.32625 0.85739 0.32082 -0.24107 -0.78393 -1.26489 -1.64597 -1.89707 -1.99838 -1.94188 -1.73205 -1.38545 -0.92945 -0.40005 0.16093 0.70921 1.20148 1.59889 1.87003 1.99351 1.95958 1.77091 1.44240 1.00000 0.47863 -0.08053 -0.63334 -1.13613 -1.54921 -1.83996 -1.98542 -1.97410 -1.80690 -1.49702 -1.06893 -0.55643 0.00000 0.55643 1.06893 1.49702 1.80690 1.97410 1.98542 1.83996 1.54921 1.13613 0.63334 0.08053 -0.47863 -1.00000 -1.44240 -1.77091 -1.95958 -1.99351 -1.87003 -1.59889 -1.20148 -0.70921 -0.16093 0.40005 0.92945 1.38545 1.73205 1.94188 1.99838 1.89707 1.64597 1.26489 0.78393 0.24107 -0.32082 -0.85739 -1.32625 -1.69038 -1.92104 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E8 2.00000 1.89707 1.59889 1.13613 0.55643 -0.08053 -0.70921 -1.26489 -1.69038 -1.94188 -1.99351 -1.83996 -1.49702 -1.00000 -0.40005 0.24107 0.85739 1.38545 1.77091 1.97410 1.97410 1.77091 1.38545 0.85739 0.24107 -0.40005 -1.00000 -1.49702 -1.83996 -1.99351 -1.94188 -1.69038 -1.26489 -0.70921 -0.08053 0.55643 1.13613 1.59889 1.89707 2.00000 1.89707 1.59889 1.13613 0.55643 -0.08053 -0.70921 -1.26489 -1.69038 -1.94188 -1.99351 -1.83996 -1.49702 -1.00000 -0.40005 0.24107 0.85739 1.38545 1.77091 1.97410 1.97410 1.77091 1.38545 0.85739 0.24107 -0.40005 -1.00000 -1.49702 -1.83996 -1.99351 -1.94188 -1.69038 -1.26489 -0.70921 -0.08053 0.55643 1.13613 1.59889 1.89707 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E9 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 -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 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 ... ... ..... ....... ......... ........... ............. E10 2.00000 1.83996 1.38545 0.70921 -0.08053 -0.85739 -1.49702 -1.89707 -1.99351 -1.77091 -1.26489 -0.55643 0.24107 1.00000 1.59889 1.94188 1.97410 1.69038 1.13613 0.40005 -0.40005 -1.13613 -1.69038 -1.97410 -1.94188 -1.59889 -1.00000 -0.24107 0.55643 1.26489 1.77091 1.99351 1.89707 1.49702 0.85739 0.08053 -0.70921 -1.38545 -1.83996 -2.00000 -1.83996 -1.38545 -0.70921 0.08053 0.85739 1.49702 1.89707 1.99351 1.77091 1.26489 0.55643 -0.24107 -1.00000 -1.59889 -1.94188 -1.97410 -1.69038 -1.13613 -0.40005 0.40005 1.13613 1.69038 1.97410 1.94188 1.59889 1.00000 0.24107 -0.55643 -1.26489 -1.77091 -1.99351 -1.89707 -1.49702 -0.85739 -0.08053 0.70921 1.38545 1.83996 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E11 2.00000 1.80690 1.26489 0.47863 -0.40005 -1.20148 -1.77091 -1.99838 -1.83996 -1.32625 -0.55643 0.32082 1.13613 1.73205 1.99351 1.87003 1.38545 0.63334 -0.24107 -1.06893 -1.69038 -1.98542 -1.89707 -1.44240 -0.70921 0.16093 1.00000 1.64597 1.97410 1.92104 1.49702 0.78393 -0.08053 -0.92945 -1.59889 -1.95958 -1.94188 -1.54921 -0.85739 0.00000 0.85739 1.54921 1.94188 1.95958 1.59889 0.92945 0.08053 -0.78393 -1.49702 -1.92104 -1.97410 -1.64597 -1.00000 -0.16093 0.70921 1.44240 1.89707 1.98542 1.69038 1.06893 0.24107 -0.63334 -1.38545 -1.87003 -1.99351 -1.73205 -1.13613 -0.32082 0.55643 1.32625 1.83996 1.99838 1.77091 1.20148 0.40005 -0.47863 -1.26489 -1.80690 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E12 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 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 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 ... ... ..... ....... ......... ........... ............. E13 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 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 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 1.73205 1.00000 0.00000 -1.00000 -1.73205 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E14 2.00000 1.69038 0.85739 -0.24107 -1.26489 -1.89707 -1.94188 -1.38545 -0.40005 0.70921 1.59889 1.99351 1.77091 1.00000 -0.08053 -1.13613 -1.83996 -1.97410 -1.49702 -0.55643 0.55643 1.49702 1.97410 1.83996 1.13613 0.08053 -1.00000 -1.77091 -1.99351 -1.59889 -0.70921 0.40005 1.38545 1.94188 1.89707 1.26489 0.24107 -0.85739 -1.69038 -2.00000 -1.69038 -0.85739 0.24107 1.26489 1.89707 1.94188 1.38545 0.40005 -0.70921 -1.59889 -1.99351 -1.77091 -1.00000 0.08053 1.13613 1.83996 1.97410 1.49702 0.55643 -0.55643 -1.49702 -1.97410 -1.83996 -1.13613 -0.08053 1.00000 1.77091 1.99351 1.59889 0.70921 -0.40005 -1.38545 -1.94188 -1.89707 -1.26489 -0.24107 0.85739 1.69038 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E15 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 -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 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 ... ... ..... ....... ......... ........... ............. 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E75 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 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 -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..... E76 2.00000 -1.99351 1.97410 -1.94188 1.89707 -1.83996 1.77091 -1.69038 1.59889 -1.49702 1.38545 -1.26489 1.13613 -1.00000 0.85739 -0.70921 0.55643 -0.40005 0.24107 -0.08053 -0.08053 0.24107 -0.40005 0.55643 -0.70921 0.85739 -1.00000 1.13613 -1.26489 1.38545 -1.49702 1.59889 -1.69038 1.77091 -1.83996 1.89707 -1.94188 1.97410 -1.99351 2.00000 -1.99351 1.97410 -1.94188 1.89707 -1.83996 1.77091 -1.69038 1.59889 -1.49702 1.38545 -1.26489 1.13613 -1.00000 0.85739 -0.70921 0.55643 -0.40005 0.24107 -0.08053 -0.08053 0.24107 -0.40005 0.55643 -0.70921 0.85739 -1.00000 1.13613 -1.26489 1.38545 -1.49702 1.59889 -1.69038 1.77091 -1.83996 1.89707 -1.94188 1.97410 -1.99351 2.00000 0.00000 0.00000 ... ... ..... ..TT... ......... ......TT... ............. E77 2.00000 -1.99838 1.99351 -1.98542 1.97410 -1.95958 1.94188 -1.92104 1.89707 -1.87003 1.83996 -1.80690 1.77091 -1.73205 1.69038 -1.64597 1.59889 -1.54921 1.49702 -1.44240 1.38545 -1.32625 1.26489 -1.20148 1.13613 -1.06893 1.00000 -0.92945 0.85739 -0.78393 0.70921 -0.63334 0.55643 -0.47863 0.40005 -0.32082 0.24107 -0.16093 0.08053 0.00000 -0.08053 0.16093 -0.24107 0.32082 -0.40005 0.47863 -0.55643 0.63334 -0.70921 0.78393 -0.85739 0.92945 -1.00000 1.06893 -1.13613 1.20148 -1.26489 1.32625 -1.38545 1.44240 -1.49702 1.54921 -1.59889 1.64597 -1.69038 1.73205 -1.77091 1.80690 -1.83996 1.87003 -1.89707 1.92104 -1.94188 1.95958 -1.97410 1.98542 -1.99351 1.99838 -2.00000 0.00000 0.00000 TT. ... ..TT. ....... ......TT. ........... ..........TT. Irrational character values: 1.998377996343 = 2*cos(2*π/156) = 2*cos(π/78) 1.993514616268 = 2*cos(4*π/156) = 2*cos(π/39) 1.985417748196 = 2*cos(6*π/156) = 2*cos(π/26) 1.974100525276 = 2*cos(8*π/156) = 2*cos(2*π/39) 1.959581304085 = 2*cos(10*π/156) = 2*cos(5*π/78) 1.941883634852 = 2*cos(12*π/156) = 2*cos(π/13) 1.921036223263 = 2*cos(14*π/156) = 2*cos(7*π/78) 1.897072883894 = 2*cos(16*π/156) = 2*cos(4*π/39) 1.870032485371 = 2*cos(18*π/156) = 2*cos(3*π/26) 1.839958887318 = 2*cos(20*π/156) = 2*cos(5*π/39) 1.806900869221 = 2*cos(22*π/156) = 2*cos(11*π/78) 1.770912051306 = 2*cos(24*π/156) = 2*cos(2*π/13) 1.732050807569 = 2*cos(26*π/156) = 2*cos(π/6) = √3 1.690380171088 = 2*cos(28*π/156) = 2*cos(7*π/39) 1.645967731787 = 2*cos(30*π/156) = 2*cos(5*π/26) 1.598885526807 = 2*cos(32*π/156) = 2*cos(8*π/39) 1.549209923655 = 2*cos(34*π/156) = 2*cos(17*π/78) 1.497021496342 = 2*cos(36*π/156) = 2*cos(3*π/13) 1.442404894688 = 2*cos(38*π/156) = 2*cos(19*π/78) 1.385448707019 = 2*cos(40*π/156) = 2*cos(10*π/39) 1.326245316482 = 2*cos(42*π/156) = 2*cos(7*π/26) 1.264890751191 = 2*cos(44*π/156) = 2*cos(11*π/39) 1.201484528476 = 2*cos(46*π/156) = 2*cos(23*π/78) 1.136129493462 = 2*cos(48*π/156) = 2*cos(4*π/13) 1.068931652256 = 2*cos(50*π/156) = 2*cos(25*π/78) 0.929446344088 = 2*cos(54*π/156) = 2*cos(9*π/26) 0.857385122806 = 2*cos(56*π/156) = 2*cos(14*π/39) 0.783933219720 = 2*cos(58*π/156) = 2*cos(29*π/78) 0.709209774085 = 2*cos(60*π/156) = 2*cos(5*π/13) 0.633335987603 = 2*cos(62*π/156) = 2*cos(31*π/78) 0.556434927833 = 2*cos(64*π/156) = 2*cos(16*π/39) 0.478631328575 = 2*cos(66*π/156) = 2*cos(11*π/26) 0.400051387552 = 2*cos(68*π/156) = 2*cos(17*π/39) 0.320822561716 = 2*cos(70*π/156) = 2*cos(35*π/78) 0.241073360511 = 2*cos(72*π/156) = 2*cos(6*π/13) 0.160933137433 = 2*cos(74*π/156) = 2*cos(37*π/78) 0.080531880219 = 2*cos(76*π/156) = 2*cos(19*π/39) Symmetry of Rotations and Cartesian products A1 d+g+i+k+m z², z⁴, z⁶ A2 R R_z B2 p+f+h+j+l z, z³, z⁵ E1 p+f+h+j+l {x, y}, {xz², yz²}, {xz⁴, yz⁴} E2 d+g+i+k+m {x²−y², xy}, {z²(x²−y²), xyz²}, {z⁴(x²−y²), xyz⁴} E3 f+h+j+l {x(x²−3y²), y(3x²−y²)}, {xz²(x²−3y²), yz²(3x²−y²)} E4 g+i+k+m {(x²−y²)²−4x²y², xy(x²−y²)}, {z²((x²−y²)²−4x²y²), xyz²(x²−y²)} E5 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)} E6 i+k+m {x²(x²−3y²)²−y²(3x²−y²)², xy(x²−3y²)(3x²−y²)} E7 j+l E8 k+m E9 l E10 m E69 m E70 l E71 k+m E72 j+l E73 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)} E74 h+j+l {z((x²−y²)²−4x²y²), xyz(x²−y²)} E75 g+i+k+m {xz(x²−3y²), yz(3x²−y²)}, {xz³(x²−3y²), yz³(3x²−y²)} E76 f+h+j+l {z(x²−y²), xyz}, {z³(x²−y²), xyz³} E77 R+d+g+i+k+m {R_x, R_y}, {xz, yz}, {xz³, yz³}, {xz⁵, yz⁵} Notes: α The order of the D_78d point group is 312, and the order of the principal axis (S₁₅₆) is 156. The group has 81 irreducible representations. β The D_78d point group is isomorphic to C_156v and D₁₅₆. γ The D_78d point group is generated by two symmetry elements, S₁₅₆ and either a perpendicular C₂^′ or a vertical σ_d. Also, the group may be generated from a C₂^′ plus a σ_d (some pairs will yield smaller groups, though; choosing a minimum angle is safe). δ The group contains one set of twofold symmetry axes (C₂^′) perpendicular to the principal (z) axis. Both x and y axes are members of that set. ε The single σ_d set of symmetry planes contains neither the xz nor the yz planes; but it contains the median plane (x+y)z. ζ The lowest nonvanishing multipole moment in D_78d is 4 (quadrupole moment). η This point group is non-Abelian (some symmetry operations are not commutative). Therefore, the character table contains multi-membered classes and degenerate irreducible representations. θ 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. ι 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. While some characters of this group can be expressed using real integers and square roots alone, others need complex numbers and third roots.

This Character Table for the D_78d point group was created by Gernot Katzer.

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

	D_76d
	D_77d
C₇₈ C_78v C_78h D₇₈ D_78h	D_78d	S₇₈
	D_79d
	D_80d

Character table for the D78d point group

Character table for the D_78d point group