Character Table for Point Group C78v

Character table for the C_78v point group

C78v E 2 C78 2 C39 2 C26 2 C39^2 2 C78^5 2 C13 2 C78^7 2 C39^4 2 C26^3 2 C39^5 2 C78^11 2 C13^2 2 C6 2 C39^7 2 C26^5 2 C39^8 2 C78^17 2 C13^3 2 C78^19 2 C39^10 2 C26^7 2 C39^11 2 C78^23 2 C13^4 2 C78^25 2 C3 2 C26^9 2 C39^14 2 C78^29 2 C13^5 2 C78^31 2 C39^16 2 C26^11 2 C39^17 2 C78^35 2 C13^6 2 C78^37 2 C39^19 C2 39 sv 39 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 ... ..T ....T ......T ........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 ..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 ... ... ..... ....... ......... ........... ............. 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 ... ... ..... ....... ......... ........... ............. E1 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. ....TT. ......TT. ........TT. ..........TT. E2 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... ....TT... ......TT... ........TT... E3 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..... ..TT..... ....TT..... ......TT..... E4 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 ... ... ..... ....... TT....... ..TT....... ....TT....... E5 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 ... ... ..... ....... ......... TT......... ..TT......... E6 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 ... ... ..... ....... ......... ........... TT........... E7 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 ... ... ..... ....... ......... ........... ............. E8 2.00000 1.59889 0.55643 -0.70921 -1.69038 -1.99351 -1.49702 -0.40005 0.85739 1.77091 1.97410 1.38545 0.24107 -1.00000 -1.83996 -1.94188 -1.26489 -0.08053 1.13613 1.89707 1.89707 1.13613 -0.08053 -1.26489 -1.94188 -1.83996 -1.00000 0.24107 1.38545 1.97410 1.77091 0.85739 -0.40005 -1.49702 -1.99351 -1.69038 -0.70921 0.55643 1.59889 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E9 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 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 ... ... ..... ....... ......... ........... ............. E10 2.00000 1.38545 -0.08053 -1.49702 -1.99351 -1.26489 0.24107 1.59889 1.97410 1.13613 -0.40005 -1.69038 -1.94188 -1.00000 0.55643 1.77091 1.89707 0.85739 -0.70921 -1.83996 -1.83996 -0.70921 0.85739 1.89707 1.77091 0.55643 -1.00000 -1.94188 -1.69038 -0.40005 1.13613 1.97410 1.59889 0.24107 -1.26489 -1.99351 -1.49702 -0.08053 1.38545 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E11 2.00000 1.26489 -0.40005 -1.77091 -1.83996 -0.55643 1.13613 1.99351 1.38545 -0.24107 -1.69038 -1.89707 -0.70921 1.00000 1.97410 1.49702 -0.08053 -1.59889 -1.94188 -0.85739 0.85739 1.94188 1.59889 0.08053 -1.49702 -1.97410 -1.00000 0.70921 1.89707 1.69038 0.24107 -1.38545 -1.99351 -1.13613 0.55643 1.83996 1.77091 0.40005 -1.26489 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E12 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 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 ... ... ..... ....... ......... ........... ............. E13 2.00000 1.00000 -1.00000 -2.00000 -1.00000 1.00000 2.00000 1.00000 -1.00000 -2.00000 -1.00000 1.00000 2.00000 1.00000 -1.00000 -2.00000 -1.00000 1.00000 2.00000 1.00000 -1.00000 -2.00000 -1.00000 1.00000 2.00000 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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E14 2.00000 0.85739 -1.26489 -1.94188 -0.40005 1.59889 1.77091 -0.08053 -1.83996 -1.49702 0.55643 1.97410 1.13613 -1.00000 -1.99351 -0.70921 1.38545 1.89707 0.24107 -1.69038 -1.69038 0.24107 1.89707 1.38545 -0.70921 -1.99351 -1.00000 1.13613 1.97410 0.55643 -1.49702 -1.83996 -0.08053 1.77091 1.59889 -0.40005 -1.94188 -1.26489 0.85739 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E15 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 0.70921 -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 ... ... ..... ....... ......... ........... ............. E16 2.00000 0.55643 -1.69038 -1.49702 0.85739 1.97410 0.24107 -1.83996 -1.26489 1.13613 1.89707 -0.08053 -1.94188 -1.00000 1.38545 1.77091 -0.40005 -1.99351 -0.70921 1.59889 1.59889 -0.70921 -1.99351 -0.40005 1.77091 1.38545 -1.00000 -1.94188 -0.08053 1.89707 1.13613 -1.26489 -1.83996 0.24107 1.97410 0.85739 -1.49702 -1.69038 0.55643 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E17 2.00000 0.40005 -1.83996 -1.13613 1.38545 1.69038 -0.70921 -1.97410 -0.08053 1.94188 0.85739 -1.59889 -1.49702 1.00000 1.89707 -0.24107 -1.99351 -0.55643 1.77091 1.26489 -1.26489 -1.77091 0.55643 1.99351 0.24107 -1.89707 -1.00000 1.49702 1.59889 -0.85739 -1.94188 0.08053 1.97410 0.70921 -1.69038 -1.38545 1.13613 1.83996 -0.40005 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E18 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 0.24107 -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 ... ... ..... ....... ......... ........... ............. E19 2.00000 0.08053 -1.99351 -0.24107 1.97410 0.40005 -1.94188 -0.55643 1.89707 0.70921 -1.83996 -0.85739 1.77091 1.00000 -1.69038 -1.13613 1.59889 1.26489 -1.49702 -1.38545 1.38545 1.49702 -1.26489 -1.59889 1.13613 1.69038 -1.00000 -1.77091 0.85739 1.83996 -0.70921 -1.89707 0.55643 1.94188 -0.40005 -1.97410 0.24107 1.99351 -0.08053 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E20 2.00000 -0.08053 -1.99351 0.24107 1.97410 -0.40005 -1.94188 0.55643 1.89707 -0.70921 -1.83996 0.85739 1.77091 -1.00000 -1.69038 1.13613 1.59889 -1.26489 -1.49702 1.38545 1.38545 -1.49702 -1.26489 1.59889 1.13613 -1.69038 -1.00000 1.77091 0.85739 -1.83996 -0.70921 1.89707 0.55643 -1.94188 -0.40005 1.97410 0.24107 -1.99351 -0.08053 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E21 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 -0.24107 -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 ... ... ..... ....... ......... ........... ............. E22 2.00000 -0.40005 -1.83996 1.13613 1.38545 -1.69038 -0.70921 1.97410 -0.08053 -1.94188 0.85739 1.59889 -1.49702 -1.00000 1.89707 0.24107 -1.99351 0.55643 1.77091 -1.26489 -1.26489 1.77091 0.55643 -1.99351 0.24107 1.89707 -1.00000 -1.49702 1.59889 0.85739 -1.94188 -0.08053 1.97410 -0.70921 -1.69038 1.38545 1.13613 -1.83996 -0.40005 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E23 2.00000 -0.55643 -1.69038 1.49702 0.85739 -1.97410 0.24107 1.83996 -1.26489 -1.13613 1.89707 0.08053 -1.94188 1.00000 1.38545 -1.77091 -0.40005 1.99351 -0.70921 -1.59889 1.59889 0.70921 -1.99351 0.40005 1.77091 -1.38545 -1.00000 1.94188 -0.08053 -1.89707 1.13613 1.26489 -1.83996 -0.24107 1.97410 -0.85739 -1.49702 1.69038 0.55643 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E24 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 -0.70921 -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 ... ... ..... ....... ......... ........... ............. E25 2.00000 -0.85739 -1.26489 1.94188 -0.40005 -1.59889 1.77091 0.08053 -1.83996 1.49702 0.55643 -1.97410 1.13613 1.00000 -1.99351 0.70921 1.38545 -1.89707 0.24107 1.69038 -1.69038 -0.24107 1.89707 -1.38545 -0.70921 1.99351 -1.00000 -1.13613 1.97410 -0.55643 -1.49702 1.83996 -0.08053 -1.77091 1.59889 0.40005 -1.94188 1.26489 0.85739 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E26 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -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 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E27 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 -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 ... ... ..... ....... ......... ........... ............. E28 2.00000 -1.26489 -0.40005 1.77091 -1.83996 0.55643 1.13613 -1.99351 1.38545 0.24107 -1.69038 1.89707 -0.70921 -1.00000 1.97410 -1.49702 -0.08053 1.59889 -1.94188 0.85739 0.85739 -1.94188 1.59889 -0.08053 -1.49702 1.97410 -1.00000 -0.70921 1.89707 -1.69038 0.24107 1.38545 -1.99351 1.13613 0.55643 -1.83996 1.77091 -0.40005 -1.26489 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E29 2.00000 -1.38545 -0.08053 1.49702 -1.99351 1.26489 0.24107 -1.59889 1.97410 -1.13613 -0.40005 1.69038 -1.94188 1.00000 0.55643 -1.77091 1.89707 -0.85739 -0.70921 1.83996 -1.83996 0.70921 0.85739 -1.89707 1.77091 -0.55643 -1.00000 1.94188 -1.69038 0.40005 1.13613 -1.97410 1.59889 -0.24107 -1.26489 1.99351 -1.49702 0.08053 1.38545 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E30 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 -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 ... ... ..... ....... ......... ........... ............. E31 2.00000 -1.59889 0.55643 0.70921 -1.69038 1.99351 -1.49702 0.40005 0.85739 -1.77091 1.97410 -1.38545 0.24107 1.00000 -1.83996 1.94188 -1.26489 0.08053 1.13613 -1.89707 1.89707 -1.13613 -0.08053 1.26489 -1.94188 1.83996 -1.00000 -0.24107 1.38545 -1.97410 1.77091 -0.85739 -0.40005 1.49702 -1.99351 1.69038 -0.70921 -0.55643 1.59889 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E32 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 ... ... ..... ....... ......... ........... ............. E33 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 ... ... ..... ....... ......... ........... ............. E34 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 ... ... ..... ....... ......... ........... ............. E35 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 ... ... ..... ....... ......... ........... ............. E36 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 ... ... ..... ....... ......... ........... ............. E37 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 ... ... ..... ....... ......... ........... ............. E38 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 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.993514616268 = 2*cos(2*π/78) = 2*cos(π/39) 1.974100525276 = 2*cos(4*π/78) = 2*cos(2*π/39) 1.941883634852 = 2*cos(6*π/78) = 2*cos(π/13) 1.897072883894 = 2*cos(8*π/78) = 2*cos(4*π/39) 1.839958887318 = 2*cos(10*π/78) = 2*cos(5*π/39) 1.770912051306 = 2*cos(12*π/78) = 2*cos(2*π/13) 1.690380171088 = 2*cos(14*π/78) = 2*cos(7*π/39) 1.598885526807 = 2*cos(16*π/78) = 2*cos(8*π/39) 1.497021496342 = 2*cos(18*π/78) = 2*cos(3*π/13) 1.385448707019 = 2*cos(20*π/78) = 2*cos(10*π/39) 1.264890751191 = 2*cos(22*π/78) = 2*cos(11*π/39) 1.136129493462 = 2*cos(24*π/78) = 2*cos(4*π/13) 0.857385122806 = 2*cos(28*π/78) = 2*cos(14*π/39) 0.709209774085 = 2*cos(30*π/78) = 2*cos(5*π/13) 0.556434927833 = 2*cos(32*π/78) = 2*cos(16*π/39) 0.400051387552 = 2*cos(34*π/78) = 2*cos(17*π/39) 0.241073360511 = 2*cos(36*π/78) = 2*cos(6*π/13) 0.080531880219 = 2*cos(38*π/78) = 2*cos(19*π/39) Symmetry of Rotations and Cartesian products A1 p+d+f+g+h+i+j+k+l+m z, z², z³, z⁴, z⁵, z⁶ A2 R R_z E1 R+p+d+f+g+h+i+j+k+l+m {R_x, R_y}, {x, y}, {xz, yz}, {xz², yz²}, {xz³, yz³}, {xz⁴, yz⁴}, {xz⁵, yz⁵} E2 d+f+g+h+i+j+k+l+m {x²−y², xy}, {z(x²−y²), xyz}, {z²(x²−y²), xyz²}, {z³(x²−y²), xyz³}, {z⁴(x²−y²), xyz⁴} E3 f+g+h+i+j+k+l+m {x(x²−3y²), y(3x²−y²)}, {xz(x²−3y²), yz(3x²−y²)}, {xz²(x²−3y²), yz²(3x²−y²)}, {xz³(x²−3y²), yz³(3x²−y²)} E4 g+h+i+j+k+l+m {(x²−y²)²−4x²y², xy(x²−y²)}, {z((x²−y²)²−4x²y²), xyz(x²−y²)}, {z²((x²−y²)²−4x²y²), xyz²(x²−y²)} E5 h+i+j+k+l+m {x(x²−(5+2√5)y²)(x²−(5−2√5)y²), y((5+2√5)x²−y²)((5−2√5)x²−y²)}, {xz(x²−(5+2√5)y²)(x²−(5−2√5)y²), yz((5+2√5)x²−y²)((5−2√5)x²−y²)} E6 i+j+k+l+m {x²(x²−3y²)²−y²(3x²−y²)², xy(x²−3y²)(3x²−y²)} E7 j+k+l+m E8 k+l+m E9 l+m E10 m Notes: α The order of the C_78v point group is 156, and the order of the principal axis (C₇₈) is 78. The group has 42 irreducible representations. β The C_78v point group is isomorphic to D_39d, D_39h and D₇₈. γ The C_78v point group is generated by two symmetry elements, C₇₈ and any σ_v (or, non-canonically, any σ_d). Also, the group may be generated from a σ_v plus a σ_d (some pairs will yield smaller groups, though; choosing a minimum angle is safe). 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 has the xz plane as a member, while the yz plane is a member of the σ_d set. ε The lowest nonvanishing multipole moment in C_78v is 2 (dipole 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 C_78v point group was created by Gernot Katzer.

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

	C_76v
	C_77v
C₇₈	C_78v	C_78h D₇₈ D_78h D_78d S₇₈
	C_79v
	C_80v

Character table for the C78v point group

Character table for the C_78v point group