Character table for the C75 point group

C75     E        2 C75    2 C75^2  2 C25    2 C75^4  2 C15    2 C25^2  2 C75^7  2 C75^8  2 C25^3  2 C15^2  2 C75^11 2 C25^4  2 C75^13 2 C75^14 2 C5     2 C75^16 2 C75^17 2 C25^6  2 C75^19 2 C15^4  2 C25^7  2 C75^22 2 C75^23 2 C25^8  2 C3     2 C75^26 2 C25^9  2 C75^28 2 C75^29 2 C5^2   2 C75^31 2 C75^32 2 C25^11 2 C75^34 2 C15^7  2 C25^12 2 C75^37    <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> 
A       1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  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 ............T
E1  *   2.00000  1.99299  1.97199  1.93717  1.88875  1.82709  1.75261  1.66584  1.56739  1.45794  1.33826  1.20920  1.07165  0.92659  0.77503  0.61803  0.45670  0.29217  0.12558 -0.04188 -0.20906 -0.37476 -0.53784 -0.69714 -0.85156 -1.00000 -1.14143 -1.27485 -1.39933 -1.51399 -1.61803 -1.71073 -1.79142 -1.85955 -1.91464 -1.95630 -1.98423 -1.99825     TT. TT. ..TT. ....TT. ......TT. ........TT. ..........TT.
E2  *   2.00000  1.97199  1.88875  1.75261  1.56739  1.33826  1.07165  0.77503  0.45670  0.12558 -0.20906 -0.53784 -0.85156 -1.14143 -1.39933 -1.61803 -1.79142 -1.91464 -1.98423 -1.99825 -1.95630 -1.85955 -1.71073 -1.51399 -1.27485 -1.00000 -0.69714 -0.37476 -0.04188  0.29217  0.61803  0.92659  1.20920  1.45794  1.66584  1.82709  1.93717  1.99299     ... ... TT... ..TT... ....TT... ......TT... ........TT...
E3  *   2.00000  1.93717  1.75261  1.45794  1.07165  0.61803  0.12558 -0.37476 -0.85156 -1.27485 -1.61803 -1.85955 -1.98423 -1.98423 -1.85955 -1.61803 -1.27485 -0.85156 -0.37476  0.12558  0.61803  1.07165  1.45794  1.75261  1.93717  2.00000  1.93717  1.75261  1.45794  1.07165  0.61803  0.12558 -0.37476 -0.85156 -1.27485 -1.61803 -1.85955 -1.98423     ... ... ..... TT..... ..TT..... ....TT..... ......TT.....
E4  *   2.00000  1.88875  1.56739  1.07165  0.45670 -0.20906 -0.85156 -1.39933 -1.79142 -1.98423 -1.95630 -1.71073 -1.27485 -0.69714 -0.04188  0.61803  1.20920  1.66584  1.93717  1.99299  1.82709  1.45794  0.92659  0.29217 -0.37476 -1.00000 -1.51399 -1.85955 -1.99825 -1.91464 -1.61803 -1.14143 -0.53784  0.12558  0.77503  1.33826  1.75261  1.97199     ... ... ..... ....... TT....... ..TT....... ....TT.......
E5  *   2.00000  1.82709  1.33826  0.61803 -0.20906 -1.00000 -1.61803 -1.95630 -1.95630 -1.61803 -1.00000 -0.20906  0.61803  1.33826  1.82709  2.00000  1.82709  1.33826  0.61803 -0.20906 -1.00000 -1.61803 -1.95630 -1.95630 -1.61803 -1.00000 -0.20906  0.61803  1.33826  1.82709  2.00000  1.82709  1.33826  0.61803 -0.20906 -1.00000 -1.61803 -1.95630     ... ... ..... ....... ......... TT......... ..TT.........
E6  *   2.00000  1.75261  1.07165  0.12558 -0.85156 -1.61803 -1.98423 -1.85955 -1.27485 -0.37476  0.61803  1.45794  1.93717  1.93717  1.45794  0.61803 -0.37476 -1.27485 -1.85955 -1.98423 -1.61803 -0.85156  0.12558  1.07165  1.75261  2.00000  1.75261  1.07165  0.12558 -0.85156 -1.61803 -1.98423 -1.85955 -1.27485 -0.37476  0.61803  1.45794  1.93717     ... ... ..... ....... ......... ........... TT...........
E7  *   2.00000  1.66584  0.77503 -0.37476 -1.39933 -1.95630 -1.85955 -1.14143 -0.04188  1.07165  1.82709  1.97199  1.45794  0.45670 -0.69714 -1.61803 -1.99825 -1.71073 -0.85156  0.29217  1.33826  1.93717  1.88875  1.20920  0.12558 -1.00000 -1.79142 -1.98423 -1.51399 -0.53784  0.61803  1.56739  1.99299  1.75261  0.92659 -0.20906 -1.27485 -1.91464     ... ... ..... ....... ......... ........... .............
E8  *   2.00000  1.56739  0.45670 -0.85156 -1.79142 -1.95630 -1.27485 -0.04188  1.20920  1.93717  1.82709  0.92659 -0.37476 -1.51399 -1.99825 -1.61803 -0.53784  0.77503  1.75261  1.97199  1.33826  0.12558 -1.14143 -1.91464 -1.85955 -1.00000  0.29217  1.45794  1.99299  1.66584  0.61803 -0.69714 -1.71073 -1.98423 -1.39933 -0.20906  1.07165  1.88875     ... ... ..... ....... ......... ........... .............
E9  *   2.00000  1.45794  0.12558 -1.27485 -1.98423 -1.61803 -0.37476  1.07165  1.93717  1.75261  0.61803 -0.85156 -1.85955 -1.85955 -0.85156  0.61803  1.75261  1.93717  1.07165 -0.37476 -1.61803 -1.98423 -1.27485  0.12558  1.45794  2.00000  1.45794  0.12558 -1.27485 -1.98423 -1.61803 -0.37476  1.07165  1.93717  1.75261  0.61803 -0.85156 -1.85955     ... ... ..... ....... ......... ........... .............
E10 *   2.00000  1.33826 -0.20906 -1.61803 -1.95630 -1.00000  0.61803  1.82709  1.82709  0.61803 -1.00000 -1.95630 -1.61803 -0.20906  1.33826  2.00000  1.33826 -0.20906 -1.61803 -1.95630 -1.00000  0.61803  1.82709  1.82709  0.61803 -1.00000 -1.95630 -1.61803 -0.20906  1.33826  2.00000  1.33826 -0.20906 -1.61803 -1.95630 -1.00000  0.61803  1.82709     ... ... ..... ....... ......... ........... .............
E11 *   2.00000  1.20920 -0.53784 -1.85955 -1.71073 -0.20906  1.45794  1.97199  0.92659 -0.85156 -1.95630 -1.51399  0.12558  1.66584  1.88875  0.61803 -1.14143 -1.99825 -1.27485  0.45670  1.82709  1.75261  0.29217 -1.39933 -1.98423 -1.00000  0.77503  1.93717  1.56739 -0.04188 -1.61803 -1.91464 -0.69714  1.07165  1.99299  1.33826 -0.37476 -1.79142     ... ... ..... ....... ......... ........... .............
E12 *   2.00000  1.07165 -0.85156 -1.98423 -1.27485  0.61803  1.93717  1.45794 -0.37476 -1.85955 -1.61803  0.12558  1.75261  1.75261  0.12558 -1.61803 -1.85955 -0.37476  1.45794  1.93717  0.61803 -1.27485 -1.98423 -0.85156  1.07165  2.00000  1.07165 -0.85156 -1.98423 -1.27485  0.61803  1.93717  1.45794 -0.37476 -1.85955 -1.61803  0.12558  1.75261     ... ... ..... ....... ......... ........... .............
E13 *   2.00000  0.92659 -1.14143 -1.98423 -0.69714  1.33826  1.93717  0.45670 -1.51399 -1.85955 -0.20906  1.66584  1.75261 -0.04188 -1.79142 -1.61803  0.29217  1.88875  1.45794 -0.53784 -1.95630 -1.27485  0.77503  1.99299  1.07165 -1.00000 -1.99825 -0.85156  1.20920  1.97199  0.61803 -1.39933 -1.91464 -0.37476  1.56739  1.82709  0.12558 -1.71073     ... ... ..... ....... ......... ........... .............
E14 *   2.00000  0.77503 -1.39933 -1.85955 -0.04188  1.82709  1.45794 -0.69714 -1.99825 -0.85156  1.33826  1.88875  0.12558 -1.79142 -1.51399  0.61803  1.99299  0.92659 -1.27485 -1.91464 -0.20906  1.75261  1.56739 -0.53784 -1.98423 -1.00000  1.20920  1.93717  0.29217 -1.71073 -1.61803  0.45670  1.97199  1.07165 -1.14143 -1.95630 -0.37476  1.66584     ... ... ..... ....... ......... ........... .............
E15 *   2.00000  0.61803 -1.61803 -1.61803  0.61803  2.00000  0.61803 -1.61803 -1.61803  0.61803  2.00000  0.61803 -1.61803 -1.61803  0.61803  2.00000  0.61803 -1.61803 -1.61803  0.61803  2.00000  0.61803 -1.61803 -1.61803  0.61803  2.00000  0.61803 -1.61803 -1.61803  0.61803  2.00000  0.61803 -1.61803 -1.61803  0.61803  2.00000  0.61803 -1.61803     ... ... ..... ....... ......... ........... .............
E16 *   2.00000  0.45670 -1.79142 -1.27485  1.20920  1.82709 -0.37476 -1.99825 -0.53784  1.75261  1.33826 -1.14143 -1.85955  0.29217  1.99299  0.61803 -1.71073 -1.39933  1.07165  1.88875 -0.20906 -1.98423 -0.69714  1.66584  1.45794 -1.00000 -1.91464  0.12558  1.97199  0.77503 -1.61803 -1.51399  0.92659  1.93717 -0.04188 -1.95630 -0.85156  1.56739     ... ... ..... ....... ......... ........... .............
E17 *   2.00000  0.29217 -1.91464 -0.85156  1.66584  1.33826 -1.27485 -1.71073  0.77503  1.93717 -0.20906 -1.99825 -0.37476  1.88875  0.92659 -1.61803 -1.39933  1.20920  1.75261 -0.69714 -1.95630  0.12558  1.99299  0.45670 -1.85955 -1.00000  1.56739  1.45794 -1.14143 -1.79142  0.61803  1.97199 -0.04188 -1.98423 -0.53784  1.82709  1.07165 -1.51399     ... ... ..... ....... ......... ........... .............
E18 *   2.00000  0.12558 -1.98423 -0.37476  1.93717  0.61803 -1.85955 -0.85156  1.75261  1.07165 -1.61803 -1.27485  1.45794  1.45794 -1.27485 -1.61803  1.07165  1.75261 -0.85156 -1.85955  0.61803  1.93717 -0.37476 -1.98423  0.12558  2.00000  0.12558 -1.98423 -0.37476  1.93717  0.61803 -1.85955 -0.85156  1.75261  1.07165 -1.61803 -1.27485  1.45794     ... ... ..... ....... ......... ........... .............
E19 *   2.00000 -0.04188 -1.99825  0.12558  1.99299 -0.20906 -1.98423  0.29217  1.97199 -0.37476 -1.95630  0.45670  1.93717 -0.53784 -1.91464  0.61803  1.88875 -0.69714 -1.85955  0.77503  1.82709 -0.85156 -1.79142  0.92659  1.75261 -1.00000 -1.71073  1.07165  1.66584 -1.14143 -1.61803  1.20920  1.56739 -1.27485 -1.51399  1.33826  1.45794 -1.39933     ... ... ..... ....... ......... ........... .............
E20 *   2.00000 -0.20906 -1.95630  0.61803  1.82709 -1.00000 -1.61803  1.33826  1.33826 -1.61803 -1.00000  1.82709  0.61803 -1.95630 -0.20906  2.00000 -0.20906 -1.95630  0.61803  1.82709 -1.00000 -1.61803  1.33826  1.33826 -1.61803 -1.00000  1.82709  0.61803 -1.95630 -0.20906  2.00000 -0.20906 -1.95630  0.61803  1.82709 -1.00000 -1.61803  1.33826     ... ... ..... ....... ......... ........... .............
E21 *   2.00000 -0.37476 -1.85955  1.07165  1.45794 -1.61803 -0.85156  1.93717  0.12558 -1.98423  0.61803  1.75261 -1.27485 -1.27485  1.75261  0.61803 -1.98423  0.12558  1.93717 -0.85156 -1.61803  1.45794  1.07165 -1.85955 -0.37476  2.00000 -0.37476 -1.85955  1.07165  1.45794 -1.61803 -0.85156  1.93717  0.12558 -1.98423  0.61803  1.75261 -1.27485     ... ... ..... ....... ......... ........... .............
E22 *   2.00000 -0.53784 -1.71073  1.45794  0.92659 -1.95630  0.12558  1.88875 -1.14143 -1.27485  1.82709  0.29217 -1.98423  0.77503  1.56739 -1.61803 -0.69714  1.99299 -0.37476 -1.79142  1.33826  1.07165 -1.91464 -0.04188  1.93717 -1.00000 -1.39933  1.75261  0.45670 -1.99825  0.61803  1.66584 -1.51399 -0.85156  1.97199 -0.20906 -1.85955  1.20920     ... ... ..... ....... ......... ........... .............
E23 *   2.00000 -0.69714 -1.51399  1.75261  0.29217 -1.95630  1.07165  1.20920 -1.91464  0.12558  1.82709 -1.39933 -0.85156  1.99299 -0.53784 -1.61803  1.66584  0.45670 -1.98423  0.92659  1.33826 -1.85955 -0.04188  1.88875 -1.27485 -1.00000  1.97199 -0.37476 -1.71073  1.56739  0.61803 -1.99825  0.77503  1.45794 -1.79142 -0.20906  1.93717 -1.14143     ... ... ..... ....... ......... ........... .............
E24 *   2.00000 -0.85156 -1.27485  1.93717 -0.37476 -1.61803  1.75261  0.12558 -1.85955  1.45794  0.61803 -1.98423  1.07165  1.07165 -1.98423  0.61803  1.45794 -1.85955  0.12558  1.75261 -1.61803 -0.37476  1.93717 -1.27485 -0.85156  2.00000 -0.85156 -1.27485  1.93717 -0.37476 -1.61803  1.75261  0.12558 -1.85955  1.45794  0.61803 -1.98423  1.07165     ... ... ..... ....... ......... ........... .............
E25 *   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     ... ... ..... ....... ......... ........... .............
E26 *   2.00000 -1.14143 -0.69714  1.93717 -1.51399 -0.20906  1.75261 -1.79142  0.29217  1.45794 -1.95630  0.77503  1.07165 -1.99825  1.20920  0.61803 -1.91464  1.56739  0.12558 -1.71073  1.82709 -0.37476 -1.39933  1.97199 -0.85156 -1.00000  1.99299 -1.27485 -0.53784  1.88875 -1.61803 -0.04188  1.66584 -1.85955  0.45670  1.33826 -1.98423  0.92659     ... ... ..... ....... ......... ........... .............
E27 *   2.00000 -1.27485 -0.37476  1.75261 -1.85955  0.61803  1.07165 -1.98423  1.45794  0.12558 -1.61803  1.93717 -0.85156 -0.85156  1.93717 -1.61803  0.12558  1.45794 -1.98423  1.07165  0.61803 -1.85955  1.75261 -0.37476 -1.27485  2.00000 -1.27485 -0.37476  1.75261 -1.85955  0.61803  1.07165 -1.98423  1.45794  0.12558 -1.61803  1.93717 -0.85156     ... ... ..... ....... ......... ........... .............
E28 *   2.00000 -1.39933 -0.04188  1.45794 -1.99825  1.33826  0.12558 -1.51399  1.99299 -1.27485 -0.20906  1.56739 -1.98423  1.20920  0.29217 -1.61803  1.97199 -1.14143 -0.37476  1.66584 -1.95630  1.07165  0.45670 -1.71073  1.93717 -1.00000 -0.53784  1.75261 -1.91464  0.92659  0.61803 -1.79142  1.88875 -0.85156 -0.69714  1.82709 -1.85955  0.77503     ... ... ..... ....... ......... ........... .............
E29 *   2.00000 -1.51399  0.29217  1.07165 -1.91464  1.82709 -0.85156 -0.53784  1.66584 -1.98423  1.33826 -0.04188 -1.27485  1.97199 -1.71073  0.61803  0.77503 -1.79142  1.93717 -1.14143 -0.20906  1.45794 -1.99825  1.56739 -0.37476 -1.00000  1.88875 -1.85955  0.92659  0.45670 -1.61803  1.99299 -1.39933  0.12558  1.20920 -1.95630  1.75261 -0.69714     ... ... ..... ....... ......... ........... .............
E30 *   2.00000 -1.61803  0.61803  0.61803 -1.61803  2.00000 -1.61803  0.61803  0.61803 -1.61803  2.00000 -1.61803  0.61803  0.61803 -1.61803  2.00000 -1.61803  0.61803  0.61803 -1.61803  2.00000 -1.61803  0.61803  0.61803 -1.61803  2.00000 -1.61803  0.61803  0.61803 -1.61803  2.00000 -1.61803  0.61803  0.61803 -1.61803  2.00000 -1.61803  0.61803     ... ... ..... ....... ......... ........... .............
E31 *   2.00000 -1.71073  0.92659  0.12558 -1.14143  1.82709 -1.98423  1.56739 -0.69714 -0.37476  1.33826 -1.91464  1.93717 -1.39933  0.45670  0.61803 -1.51399  1.97199 -1.85955  1.20920 -0.20906 -0.85156  1.66584 -1.99825  1.75261 -1.00000 -0.04188  1.07165 -1.79142  1.99299 -1.61803  0.77503  0.29217 -1.27485  1.88875 -1.95630  1.45794 -0.53784     ... ... ..... ....... ......... ........... .............
E32 *   2.00000 -1.79142  1.20920 -0.37476 -0.53784  1.33826 -1.85955  1.99299 -1.71073  1.07165 -0.20906 -0.69714  1.45794 -1.91464  1.97199 -1.61803  0.92659 -0.04188 -0.85156  1.56739 -1.95630  1.93717 -1.51399  0.77503  0.12558 -1.00000  1.66584 -1.98423  1.88875 -1.39933  0.61803  0.29217 -1.14143  1.75261 -1.99825  1.82709 -1.27485  0.45670     ... ... ..... ....... ......... ........... .............
E33 *   2.00000 -1.85955  1.45794 -0.85156  0.12558  0.61803 -1.27485  1.75261 -1.98423  1.93717 -1.61803  1.07165 -0.37476 -0.37476  1.07165 -1.61803  1.93717 -1.98423  1.75261 -1.27485  0.61803  0.12558 -0.85156  1.45794 -1.85955  2.00000 -1.85955  1.45794 -0.85156  0.12558  0.61803 -1.27485  1.75261 -1.98423  1.93717 -1.61803  1.07165 -0.37476     ... ... ..... ....... ......... ........... .............
E34 *   2.00000 -1.91464  1.66584 -1.27485  0.77503 -0.20906 -0.37476  0.92659 -1.39933  1.75261 -1.95630  1.99299 -1.85955  1.56739 -1.14143  0.61803 -0.04188 -0.53784  1.07165 -1.51399  1.82709 -1.98423  1.97199 -1.79142  1.45794 -1.00000  0.45670  0.12558 -0.69714  1.20920 -1.61803  1.88875 -1.99825  1.93717 -1.71073  1.33826 -0.85156  0.29217     ... ... ..... ....... ......... ........... .............
E35 *   2.00000 -1.95630  1.82709 -1.61803  1.33826 -1.00000  0.61803 -0.20906 -0.20906  0.61803 -1.00000  1.33826 -1.61803  1.82709 -1.95630  2.00000 -1.95630  1.82709 -1.61803  1.33826 -1.00000  0.61803 -0.20906 -0.20906  0.61803 -1.00000  1.33826 -1.61803  1.82709 -1.95630  2.00000 -1.95630  1.82709 -1.61803  1.33826 -1.00000  0.61803 -0.20906     ... ... ..... ....... ......... ........... .............
E36 *   2.00000 -1.98423  1.93717 -1.85955  1.75261 -1.61803  1.45794 -1.27485  1.07165 -0.85156  0.61803 -0.37476  0.12558  0.12558 -0.37476  0.61803 -0.85156  1.07165 -1.27485  1.45794 -1.61803  1.75261 -1.85955  1.93717 -1.98423  2.00000 -1.98423  1.93717 -1.85955  1.75261 -1.61803  1.45794 -1.27485  1.07165 -0.85156  0.61803 -0.37476  0.12558     ... ... ..... ....... ......... ........... .............
E37 *   2.00000 -1.99825  1.99299 -1.98423  1.97199 -1.95630  1.93717 -1.91464  1.88875 -1.85955  1.82709 -1.79142  1.75261 -1.71073  1.66584 -1.61803  1.56739 -1.51399  1.45794 -1.39933  1.33826 -1.27485  1.20920 -1.14143  1.07165 -1.00000  0.92659 -0.85156  0.77503 -0.69714  0.61803 -0.53784  0.45670 -0.37476  0.29217 -0.20906  0.12558 -0.04188     ... ... ..... ....... ......... ........... .............

 Irrational character values:  1.992985718499 = 2*cos(2*π/75)
                               1.971992074141 = 2*cos(4*π/75)
                               1.937166322257 = 2*cos(6*π/75) = 2*cos(2*π/25)
                               1.888752740475 = 2*cos(8*π/75)
                               1.827090915285 = 2*cos(10*π/75) = 2*cos(2*π/15) = (1+√5+√30−6*√5)/4
                               1.752613360088 = 2*cos(12*π/75) = 2*cos(4*π/25)
                               1.665842481420 = 2*cos(14*π/75)
                               1.567386914652 = 2*cos(16*π/75)
                               1.457937254843 = 2*cos(18*π/75) = 2*cos(6*π/25)
                               1.338261212718 = 2*cos(20*π/75) = 2*cos(4*π/15) = (1−√5+√30+6*√5)/4
                               1.209198229725 = 2*cos(22*π/75)
                               1.071653589958 = 2*cos(24*π/75) = 2*cos(8*π/25)
                               0.926592070240 = 2*cos(26*π/75)
                               0.775031172904 = 2*cos(28*π/75)
                               0.618033988750 = 2*cos(30*π/75) = 2*cos(2*π/5) = (√5−1)/2
                               0.456701740221 = 2*cos(32*π/75)
                               0.292166057125 = 2*cos(34*π/75)
                               0.125581039059 = 2*cos(36*π/75) = 2*cos(12*π/25)
                              -0.041884839767 = 2*cos(38*π/75)
                              -0.209056926535 = 2*cos(40*π/75) = 2*cos(8*π/15) = −(√30−6*√5−√5−1)/4
                              -0.374762629171 = 2*cos(42*π/75) = 2*cos(14*π/25)
                              -0.537839641231 = 2*cos(44*π/75)
                              -0.697144094644 = 2*cos(46*π/75)
                              -0.851558583130 = 2*cos(48*π/75) = 2*cos(16*π/25)
                              -1.141427135369 = 2*cos(52*π/75)
                              -1.274847979497 = 2*cos(54*π/75) = 2*cos(18*π/25)
                              -1.399326681027 = 2*cos(56*π/75)
                              -1.513990111304 = 2*cos(58*π/75)
                              -1.618033988750 = 2*cos(60*π/75) = 2*cos(4*π/5) = −(√5+1)/2
                              -1.710728520321 = 2*cos(62*π/75)
                              -1.791423520479 = 2*cos(64*π/75)
                              -1.859552971777 = 2*cos(66*π/75) = 2*cos(22*π/25)
                              -1.914638995064 = 2*cos(68*π/75)
                              -1.956295201468 = 2*cos(70*π/75) = 2*cos(14*π/15) = −(√5−1+√30+6*√5)/4
                              -1.984229402629 = 2*cos(72*π/75) = 2*cos(24*π/25)
                              -1.998245660198 = 2*cos(74*π/75)



 Symmetry of Rotations and Cartesian products

A    R+p+d+f+g+h+i+j+k+l+m  Rz, z, z2, z3, z4, z5, z6 
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      {x2y2, xy}, {z(x2y2), xyz}, {z2(x2y2), xyz2}, {z3(x2y2), xyz3}, {z4(x2y2), xyz4} 
E3   f+g+h+i+j+k+l+m        {x(x2−3y2), y(3x2y2)}, {xz(x2−3y2), yz(3x2y2)}, {xz2(x2−3y2), yz2(3x2y2)}, {xz3(x2−3y2), yz3(3x2y2)} 
E4   g+h+i+j+k+l+m          {(x2y2)2−4x2y2, xy(x2y2)}, {z((x2y2)2−4x2y2), xyz(x2y2)}, {z2((x2y2)2−4x2y2), xyz2(x2y2)} 
E5   h+i+j+k+l+m            {x(x2−(5+2√5)y2)(x2−(5−2√5)y2), y((5+2√5)x2y2)((5−2√5)x2y2)}, {xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), yz((5+2√5)x2y2)((5−2√5)x2y2)} 
E6   i+j+k+l+m              {x2(x2−3y2)2y2(3x2y2)2, xy(x2−3y2)(3x2y2)} 
E7   j+k+l+m 
E8   k+l+m 
E9   l+m 
E10  m 

 Notes:

    α  The order of the C75 point group is 75, and the order of the principal axis (C75) is 75. The group has 38 irreducible representations.

    β  The C75 point group is generated by one single symmetry element, C75. Therefore, it is a cyclic group.

    γ  The lowest nonvanishing multipole moment in C75 is 2 (dipole moment).

    δ  This is an Abelian point group (the commutative law holds between all symmetry operations).
       The C75 group is Abelian because it contains only one symmetry element, all the powers of which necessarily commute (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 74 cases have been combined into 37 two-dimensional representations that are no longer irreducible but have real-valued characters.
       Accordingly, 37 pairs of left and right rotations have been combined into one two-membered pseudo-class each.

    ζ  The 37 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  The point group is chiral, as it does not contain any mirroring operation.

    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.
       For this group, however, some of the irrational characters cannot be expressed by a closed algebraic form using real numbers only.

This Character Table for the C75 point group was created by Gernot Katzer.

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