C67v E 2 C67 2 C67^2 2 C67^3 2 C67^4 2 C67^5 2 C67^6 2 C67^7 2 C67^8 2 C67^9 2 C67^10 2 C67^11 2 C67^12 2 C67^13 2 C67^14 2 C67^15 2 C67^16 2 C67^17 2 C67^18 2 C67^19 2 C67^20 2 C67^21 2 C67^22 2 C67^23 2 C67^24 2 C67^25 2 C67^26 2 C67^27 2 C67^28 2 C67^29 2 C67^30 2 C67^31 2 C67^32 2 C67^33 67 sv <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 ... ..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 ..T ... ..... ....... ......... ........... ............. E1 2.00000 1.99121 1.96493 1.92137 1.86093 1.78414 1.69166 1.58432 1.46306 1.32894 1.18314 1.02695 0.86173 0.68893 0.51009 0.32675 0.14055 -0.04689 -0.23391 -0.41888 -0.60017 -0.77618 -0.94538 -1.10626 -1.25743 -1.39754 -1.52537 -1.63980 -1.73981 -1.82454 -1.89323 -1.94529 -1.98025 -1.99780 0.00000 TT. TT. ..TT. ....TT. ......TT. ........TT. ..........TT. E2 2.00000 1.96493 1.86093 1.69166 1.46306 1.18314 0.86173 0.51009 0.14055 -0.23391 -0.60017 -0.94538 -1.25743 -1.52537 -1.73981 -1.89323 -1.98025 -1.99780 -1.94529 -1.82454 -1.63980 -1.39754 -1.10626 -0.77618 -0.41888 -0.04689 0.32675 0.68893 1.02695 1.32894 1.58432 1.78414 1.92137 1.99121 0.00000 ... ... TT... ..TT... ....TT... ......TT... ........TT... E3 2.00000 1.92137 1.69166 1.32894 0.86173 0.32675 -0.23391 -0.77618 -1.25743 -1.63980 -1.89323 -1.99780 -1.94529 -1.73981 -1.39754 -0.94538 -0.41888 0.14055 0.68893 1.18314 1.58432 1.86093 1.99121 1.96493 1.78414 1.46306 1.02695 0.51009 -0.04689 -0.60017 -1.10626 -1.52537 -1.82454 -1.98025 0.00000 ... ... ..... TT..... ..TT..... ....TT..... ......TT..... E4 2.00000 1.86093 1.46306 0.86173 0.14055 -0.60017 -1.25743 -1.73981 -1.98025 -1.94529 -1.63980 -1.10626 -0.41888 0.32675 1.02695 1.58432 1.92137 1.99121 1.78414 1.32894 0.68893 -0.04689 -0.77618 -1.39754 -1.82454 -1.99780 -1.89323 -1.52537 -0.94538 -0.23391 0.51009 1.18314 1.69166 1.96493 0.00000 ... ... ..... ....... TT....... ..TT....... ....TT....... E5 2.00000 1.78414 1.18314 0.32675 -0.60017 -1.39754 -1.89323 -1.98025 -1.63980 -0.94538 -0.04689 0.86173 1.58432 1.96493 1.92137 1.46306 0.68893 -0.23391 -1.10626 -1.73981 -1.99780 -1.82454 -1.25743 -0.41888 0.51009 1.32894 1.86093 1.99121 1.69166 1.02695 0.14055 -0.77618 -1.52537 -1.94529 0.00000 ... ... ..... ....... ......... TT......... ..TT......... E6 2.00000 1.69166 0.86173 -0.23391 -1.25743 -1.89323 -1.94529 -1.39754 -0.41888 0.68893 1.58432 1.99121 1.78414 1.02695 -0.04689 -1.10626 -1.82454 -1.98025 -1.52537 -0.60017 0.51009 1.46306 1.96493 1.86093 1.18314 0.14055 -0.94538 -1.73981 -1.99780 -1.63980 -0.77618 0.32675 1.32894 1.92137 0.00000 ... ... ..... ....... ......... ........... TT........... E7 2.00000 1.58432 0.51009 -0.77618 -1.73981 -1.98025 -1.39754 -0.23391 1.02695 1.86093 1.92137 1.18314 -0.04689 -1.25743 -1.94529 -1.82454 -0.94538 0.32675 1.46306 1.99121 1.69166 0.68893 -0.60017 -1.63980 -1.99780 -1.52537 -0.41888 0.86173 1.78414 1.96493 1.32894 0.14055 -1.10626 -1.89323 0.00000 ... ... ..... ....... ......... ........... ............. E8 2.00000 1.46306 0.14055 -1.25743 -1.98025 -1.63980 -0.41888 1.02695 1.92137 1.78414 0.68893 -0.77618 -1.82454 -1.89323 -0.94538 0.51009 1.69166 1.96493 1.18314 -0.23391 -1.52537 -1.99780 -1.39754 -0.04689 1.32894 1.99121 1.58432 0.32675 -1.10626 -1.94529 -1.73981 -0.60017 0.86173 1.86093 0.00000 ... ... ..... ....... ......... ........... ............. E9 2.00000 1.32894 -0.23391 -1.63980 -1.94529 -0.94538 0.68893 1.86093 1.78414 0.51009 -1.10626 -1.98025 -1.52537 -0.04689 1.46306 1.99121 1.18314 -0.41888 -1.73981 -1.89323 -0.77618 0.86173 1.92137 1.69166 0.32675 -1.25743 -1.99780 -1.39754 0.14055 1.58432 1.96493 1.02695 -0.60017 -1.82454 0.00000 ... ... ..... ....... ......... ........... ............. E10 2.00000 1.18314 -0.60017 -1.89323 -1.63980 -0.04689 1.58432 1.92137 0.68893 -1.10626 -1.99780 -1.25743 0.51009 1.86093 1.69166 0.14055 -1.52537 -1.94529 -0.77618 1.02695 1.99121 1.32894 -0.41888 -1.82454 -1.73981 -0.23391 1.46306 1.96493 0.86173 -0.94538 -1.98025 -1.39754 0.32675 1.78414 0.00000 ... ... ..... ....... ......... ........... ............. E11 2.00000 1.02695 -0.94538 -1.99780 -1.10626 0.86173 1.99121 1.18314 -0.77618 -1.98025 -1.25743 0.68893 1.96493 1.32894 -0.60017 -1.94529 -1.39754 0.51009 1.92137 1.46306 -0.41888 -1.89323 -1.52537 0.32675 1.86093 1.58432 -0.23391 -1.82454 -1.63980 0.14055 1.78414 1.69166 -0.04689 -1.73981 0.00000 ... ... ..... ....... ......... ........... ............. E12 2.00000 0.86173 -1.25743 -1.94529 -0.41888 1.58432 1.78414 -0.04689 -1.82454 -1.52537 0.51009 1.96493 1.18314 -0.94538 -1.99780 -0.77618 1.32894 1.92137 0.32675 -1.63980 -1.73981 0.14055 1.86093 1.46306 -0.60017 -1.98025 -1.10626 1.02695 1.99121 0.68893 -1.39754 -1.89323 -0.23391 1.69166 0.00000 ... ... ..... ....... ......... ........... ............. E13 2.00000 0.68893 -1.52537 -1.73981 0.32675 1.96493 1.02695 -1.25743 -1.89323 -0.04689 1.86093 1.32894 -0.94538 -1.98025 -0.41888 1.69166 1.58432 -0.60017 -1.99780 -0.77618 1.46306 1.78414 -0.23391 -1.94529 -1.10626 1.18314 1.92137 0.14055 -1.82454 -1.39754 0.86173 1.99121 0.51009 -1.63980 0.00000 ... ... ..... ....... ......... ........... ............. E14 2.00000 0.51009 -1.73981 -1.39754 1.02695 1.92137 -0.04689 -1.94529 -0.94538 1.46306 1.69166 -0.60017 -1.99780 -0.41888 1.78414 1.32894 -1.10626 -1.89323 0.14055 1.96493 0.86173 -1.52537 -1.63980 0.68893 1.99121 0.32675 -1.82454 -1.25743 1.18314 1.86093 -0.23391 -1.98025 -0.77618 1.58432 0.00000 ... ... ..... ....... ......... ........... ............. E15 2.00000 0.32675 -1.89323 -0.94538 1.58432 1.46306 -1.10626 -1.82454 0.51009 1.99121 0.14055 -1.94529 -0.77618 1.69166 1.32894 -1.25743 -1.73981 0.68893 1.96493 -0.04689 -1.98025 -0.60017 1.78414 1.18314 -1.39754 -1.63980 0.86173 1.92137 -0.23391 -1.99780 -0.41888 1.86093 1.02695 -1.52537 0.00000 ... ... ..... ....... ......... ........... ............. E16 2.00000 0.14055 -1.98025 -0.41888 1.92137 0.68893 -1.82454 -0.94538 1.69166 1.18314 -1.52537 -1.39754 1.32894 1.58432 -1.10626 -1.73981 0.86173 1.86093 -0.60017 -1.94529 0.32675 1.99121 -0.04689 -1.99780 -0.23391 1.96493 0.51009 -1.89323 -0.77618 1.78414 1.02695 -1.63980 -1.25743 1.46306 0.00000 ... ... ..... ....... ......... ........... ............. E17 2.00000 -0.04689 -1.99780 0.14055 1.99121 -0.23391 -1.98025 0.32675 1.96493 -0.41888 -1.94529 0.51009 1.92137 -0.60017 -1.89323 0.68893 1.86093 -0.77618 -1.82454 0.86173 1.78414 -0.94538 -1.73981 1.02695 1.69166 -1.10626 -1.63980 1.18314 1.58432 -1.25743 -1.52537 1.32894 1.46306 -1.39754 0.00000 ... ... ..... ....... ......... ........... ............. E18 2.00000 -0.23391 -1.94529 0.68893 1.78414 -1.10626 -1.52537 1.46306 1.18314 -1.73981 -0.77618 1.92137 0.32675 -1.99780 0.14055 1.96493 -0.60017 -1.82454 1.02695 1.58432 -1.39754 -1.25743 1.69166 0.86173 -1.89323 -0.41888 1.99121 -0.04689 -1.98025 0.51009 1.86093 -0.94538 -1.63980 1.32894 0.00000 ... ... ..... ....... ......... ........... ............. E19 2.00000 -0.41888 -1.82454 1.18314 1.32894 -1.73981 -0.60017 1.99121 -0.23391 -1.89323 1.02695 1.46306 -1.63980 -0.77618 1.96493 -0.04689 -1.94529 0.86173 1.58432 -1.52537 -0.94538 1.92137 0.14055 -1.98025 0.68893 1.69166 -1.39754 -1.10626 1.86093 0.32675 -1.99780 0.51009 1.78414 -1.25743 0.00000 ... ... ..... ....... ......... ........... ............. E20 2.00000 -0.60017 -1.63980 1.58432 0.68893 -1.99780 0.51009 1.69166 -1.52537 -0.77618 1.99121 -0.41888 -1.73981 1.46306 0.86173 -1.98025 0.32675 1.78414 -1.39754 -0.94538 1.96493 -0.23391 -1.82454 1.32894 1.02695 -1.94529 0.14055 1.86093 -1.25743 -1.10626 1.92137 -0.04689 -1.89323 1.18314 0.00000 ... ... ..... ....... ......... ........... ............. E21 2.00000 -0.77618 -1.39754 1.86093 -0.04689 -1.82454 1.46306 0.68893 -1.99780 0.86173 1.32894 -1.89323 0.14055 1.78414 -1.52537 -0.60017 1.99121 -0.94538 -1.25743 1.92137 -0.23391 -1.73981 1.58432 0.51009 -1.98025 1.02695 1.18314 -1.94529 0.32675 1.69166 -1.63980 -0.41888 1.96493 -1.10626 0.00000 ... ... ..... ....... ......... ........... ............. E22 2.00000 -0.94538 -1.10626 1.99121 -0.77618 -1.25743 1.96493 -0.60017 -1.39754 1.92137 -0.41888 -1.52537 1.86093 -0.23391 -1.63980 1.78414 -0.04689 -1.73981 1.69166 0.14055 -1.82454 1.58432 0.32675 -1.89323 1.46306 0.51009 -1.94529 1.32894 0.68893 -1.98025 1.18314 0.86173 -1.99780 1.02695 0.00000 ... ... ..... ....... ......... ........... ............. E23 2.00000 -1.10626 -0.77618 1.96493 -1.39754 -0.41888 1.86093 -1.63980 -0.04689 1.69166 -1.82454 0.32675 1.46306 -1.94529 0.68893 1.18314 -1.99780 1.02695 0.86173 -1.98025 1.32894 0.51009 -1.89323 1.58432 0.14055 -1.73981 1.78414 -0.23391 -1.52537 1.92137 -0.60017 -1.25743 1.99121 -0.94538 0.00000 ... ... ..... ....... ......... ........... ............. E24 2.00000 -1.25743 -0.41888 1.78414 -1.82454 0.51009 1.18314 -1.99780 1.32894 0.32675 -1.73981 1.86093 -0.60017 -1.10626 1.99121 -1.39754 -0.23391 1.69166 -1.89323 0.68893 1.02695 -1.98025 1.46306 0.14055 -1.63980 1.92137 -0.77618 -0.94538 1.96493 -1.52537 -0.04689 1.58432 -1.94529 0.86173 0.00000 ... ... ..... ....... ......... ........... ............. E25 2.00000 -1.39754 -0.04689 1.46306 -1.99780 1.32894 0.14055 -1.52537 1.99121 -1.25743 -0.23391 1.58432 -1.98025 1.18314 0.32675 -1.63980 1.96493 -1.10626 -0.41888 1.69166 -1.94529 1.02695 0.51009 -1.73981 1.92137 -0.94538 -0.60017 1.78414 -1.89323 0.86173 0.68893 -1.82454 1.86093 -0.77618 0.00000 ... ... ..... ....... ......... ........... ............. E26 2.00000 -1.52537 0.32675 1.02695 -1.89323 1.86093 -0.94538 -0.41888 1.58432 -1.99780 1.46306 -0.23391 -1.10626 1.92137 -1.82454 0.86173 0.51009 -1.63980 1.99121 -1.39754 0.14055 1.18314 -1.94529 1.78414 -0.77618 -0.60017 1.69166 -1.98025 1.32894 -0.04689 -1.25743 1.96493 -1.73981 0.68893 0.00000 ... ... ..... ....... ......... ........... ............. E27 2.00000 -1.63980 0.68893 0.51009 -1.52537 1.99121 -1.73981 0.86173 0.32675 -1.39754 1.96493 -1.82454 1.02695 0.14055 -1.25743 1.92137 -1.89323 1.18314 -0.04689 -1.10626 1.86093 -1.94529 1.32894 -0.23391 -0.94538 1.78414 -1.98025 1.46306 -0.41888 -0.77618 1.69166 -1.99780 1.58432 -0.60017 0.00000 ... ... ..... ....... ......... ........... ............. E28 2.00000 -1.73981 1.02695 -0.04689 -0.94538 1.69166 -1.99780 1.78414 -1.10626 0.14055 0.86173 -1.63980 1.99121 -1.82454 1.18314 -0.23391 -0.77618 1.58432 -1.98025 1.86093 -1.25743 0.32675 0.68893 -1.52537 1.96493 -1.89323 1.32894 -0.41888 -0.60017 1.46306 -1.94529 1.92137 -1.39754 0.51009 0.00000 ... ... ..... ....... ......... ........... ............. E29 2.00000 -1.82454 1.32894 -0.60017 -0.23391 1.02695 -1.63980 1.96493 -1.94529 1.58432 -0.94538 0.14055 0.68893 -1.39754 1.86093 -1.99780 1.78414 -1.25743 0.51009 0.32675 -1.10626 1.69166 -1.98025 1.92137 -1.52537 0.86173 -0.04689 -0.77618 1.46306 -1.89323 1.99121 -1.73981 1.18314 -0.41888 0.00000 ... ... ..... ....... ......... ........... ............. E30 2.00000 -1.89323 1.58432 -1.10626 0.51009 0.14055 -0.77618 1.32894 -1.73981 1.96493 -1.98025 1.78414 -1.39754 0.86173 -0.23391 -0.41888 1.02695 -1.52537 1.86093 -1.99780 1.92137 -1.63980 1.18314 -0.60017 -0.04689 0.68893 -1.25743 1.69166 -1.94529 1.99121 -1.82454 1.46306 -0.94538 0.32675 0.00000 ... ... ..... ....... ......... ........... ............. E31 2.00000 -1.94529 1.78414 -1.52537 1.18314 -0.77618 0.32675 0.14055 -0.60017 1.02695 -1.39754 1.69166 -1.89323 1.99121 -1.98025 1.86093 -1.63980 1.32894 -0.94538 0.51009 -0.04689 -0.41888 0.86173 -1.25743 1.58432 -1.82454 1.96493 -1.99780 1.92137 -1.73981 1.46306 -1.10626 0.68893 -0.23391 0.00000 ... ... ..... ....... ......... ........... ............. E32 2.00000 -1.98025 1.92137 -1.82454 1.69166 -1.52537 1.32894 -1.10626 0.86173 -0.60017 0.32675 -0.04689 -0.23391 0.51009 -0.77618 1.02695 -1.25743 1.46306 -1.63980 1.78414 -1.89323 1.96493 -1.99780 1.99121 -1.94529 1.86093 -1.73981 1.58432 -1.39754 1.18314 -0.94538 0.68893 -0.41888 0.14055 0.00000 ... ... ..... ....... ......... ........... ............. E33 2.00000 -1.99780 1.99121 -1.98025 1.96493 -1.94529 1.92137 -1.89323 1.86093 -1.82454 1.78414 -1.73981 1.69166 -1.63980 1.58432 -1.52537 1.46306 -1.39754 1.32894 -1.25743 1.18314 -1.10626 1.02695 -0.94538 0.86173 -0.77618 0.68893 -0.60017 0.51009 -0.41888 0.32675 -0.23391 0.14055 -0.04689 0.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.991211964044 = 2*cos(2*π/67) 1.964925085751 = 2*cos(4*π/67) 1.921370375154 = 2*cos(6*π/67) 1.860930592614 = 2*cos(8*π/67) 1.784136885115 = 2*cos(10*π/67) 1.691664118518 = 2*cos(12*π/67) 1.584324946822 = 2*cos(14*π/67) 1.463062670527 = 2*cos(16*π/67) 1.328942946878 = 2*cos(18*π/67) 1.183144424827 = 2*cos(20*π/67) 1.026948387030 = 2*cos(22*π/67) 0.861727489882 = 2*cos(24*π/67) 0.688933700569 = 2*cos(26*π/67) 0.510085537123 = 2*cos(28*π/67) 0.326754723637 = 2*cos(30*π/67) 0.140552377891 = 2*cos(32*π/67) -0.046885147207 = 2*cos(34*π/67) -0.233910643944 = 2*cos(36*π/67) -0.418880525532 = 2*cos(38*π/67) -0.600169270001 = 2*cos(40*π/67) -0.776183705344 = 2*cos(42*π/67) -0.945377010377 = 2*cos(44*π/67) -1.106262308250 = 2*cos(46*π/67) -1.257425733182 = 2*cos(48*π/67) -1.397538855558 = 2*cos(50*π/67) -1.525370356221 = 2*cos(52*π/67) -1.639796847347 = 2*cos(54*π/67) -1.739812744818 = 2*cos(56*π/67) -1.824539105330 = 2*cos(58*π/67) -1.893231350581 = 2*cos(60*π/67) -1.945285810650 = 2*cos(62*π/67) -1.980245029069 = 2*cos(64*π/67) -1.997801782971 = 2*cos(66*π/67) 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 Rz E1 R+p+d+f+g+h+i+j+k+l+m {Rx, Ry}, {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 C67v point group is 134, and the order of the principal axis (C67) is 67. The group has 35 irreducible representations. β The C67v point group is isomorphic to D67. γ The C67v point group is generated by two symmetry elements, C67 and any σv. Also, the group may be generated from any two σv planes. δ The lowest nonvanishing multipole moment in C67v is 2 (dipole moment). ε This point group is non-Abelian (some symmetry operations are not commutative). Therefore, the character table contains multi-membered classes and degenerate irreducible representations. ζ 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 just less than half the order of the principal axis. For this group, however, none of the irrational characters can be expressed by a closed algebraic form using real numbers only.
| C65v | ||
| C66v | ||
| C67 | C67v | C67h D67 D67h D67d |
| C68v | ||
| C69v |
This Character Table for the C67v point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.