Character Table for Point Group C74v

Character table for the C_74v point group

C74v E 2 C74 2 C37 2 C74^3 2 C37^2 2 C74^5 2 C37^3 2 C74^7 2 C37^4 2 C74^9 2 C37^5 2 C74^11 2 C37^6 2 C74^13 2 C37^7 2 C74^15 2 C37^8 2 C74^17 2 C37^9 2 C74^19 2 C37^10 2 C74^21 2 C37^11 2 C74^23 2 C37^12 2 C74^25 2 C37^13 2 C74^27 2 C37^14 2 C74^29 2 C37^15 2 C74^31 2 C37^16 2 C74^33 2 C37^17 2 C74^35 2 C37^18 C2 37 sv 37 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 ... ..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 ..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 ... ... ..... ....... ......... ........... ............. 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 ... ... ..... ....... ......... ........... ............. E1 2.00000 1.99279 1.97123 1.93547 1.88575 1.82246 1.74603 1.65702 1.55607 1.44391 1.32135 1.18927 1.04861 0.90041 0.74571 0.58565 0.42136 0.25404 0.08488 -0.08488 -0.25404 -0.42136 -0.58565 -0.74571 -0.90041 -1.04861 -1.18927 -1.32135 -1.44391 -1.55607 -1.65702 -1.74603 -1.82246 -1.88575 -1.93547 -1.97123 -1.99279 -2.00000 0.00000 0.00000 TT. TT. ..TT. ....TT. ......TT. ........TT. ..........TT. E2 2.00000 1.97123 1.88575 1.74603 1.55607 1.32135 1.04861 0.74571 0.42136 0.08488 -0.25404 -0.58565 -0.90041 -1.18927 -1.44391 -1.65702 -1.82246 -1.93547 -1.99279 -1.99279 -1.93547 -1.82246 -1.65702 -1.44391 -1.18927 -0.90041 -0.58565 -0.25404 0.08488 0.42136 0.74571 1.04861 1.32135 1.55607 1.74603 1.88575 1.97123 2.00000 0.00000 0.00000 ... ... TT... ..TT... ....TT... ......TT... ........TT... E3 2.00000 1.93547 1.74603 1.44391 1.04861 0.58565 0.08488 -0.42136 -0.90041 -1.32135 -1.65702 -1.88575 -1.99279 -1.97123 -1.82246 -1.55607 -1.18927 -0.74571 -0.25404 0.25404 0.74571 1.18927 1.55607 1.82246 1.97123 1.99279 1.88575 1.65702 1.32135 0.90041 0.42136 -0.08488 -0.58565 -1.04861 -1.44391 -1.74603 -1.93547 -2.00000 0.00000 0.00000 ... ... ..... TT..... ..TT..... ....TT..... ......TT..... E4 2.00000 1.88575 1.55607 1.04861 0.42136 -0.25404 -0.90041 -1.44391 -1.82246 -1.99279 -1.93547 -1.65702 -1.18927 -0.58565 0.08488 0.74571 1.32135 1.74603 1.97123 1.97123 1.74603 1.32135 0.74571 0.08488 -0.58565 -1.18927 -1.65702 -1.93547 -1.99279 -1.82246 -1.44391 -0.90041 -0.25404 0.42136 1.04861 1.55607 1.88575 2.00000 0.00000 0.00000 ... ... ..... ....... TT....... ..TT....... ....TT....... E5 2.00000 1.82246 1.32135 0.58565 -0.25404 -1.04861 -1.65702 -1.97123 -1.93547 -1.55607 -0.90041 -0.08488 0.74571 1.44391 1.88575 1.99279 1.74603 1.18927 0.42136 -0.42136 -1.18927 -1.74603 -1.99279 -1.88575 -1.44391 -0.74571 0.08488 0.90041 1.55607 1.93547 1.97123 1.65702 1.04861 0.25404 -0.58565 -1.32135 -1.82246 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... TT......... ..TT......... E6 2.00000 1.74603 1.04861 0.08488 -0.90041 -1.65702 -1.99279 -1.82246 -1.18927 -0.25404 0.74571 1.55607 1.97123 1.88575 1.32135 0.42136 -0.58565 -1.44391 -1.93547 -1.93547 -1.44391 -0.58565 0.42136 1.32135 1.88575 1.97123 1.55607 0.74571 -0.25404 -1.18927 -1.82246 -1.99279 -1.65702 -0.90041 0.08488 1.04861 1.74603 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... TT........... E7 2.00000 1.65702 0.74571 -0.42136 -1.44391 -1.97123 -1.82246 -1.04861 0.08488 1.18927 1.88575 1.93547 1.32135 0.25404 -0.90041 -1.74603 -1.99279 -1.55607 -0.58565 0.58565 1.55607 1.99279 1.74603 0.90041 -0.25404 -1.32135 -1.93547 -1.88575 -1.18927 -0.08488 1.04861 1.82246 1.97123 1.44391 0.42136 -0.74571 -1.65702 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E8 2.00000 1.55607 0.42136 -0.90041 -1.82246 -1.93547 -1.18927 0.08488 1.32135 1.97123 1.74603 0.74571 -0.58565 -1.65702 -1.99279 -1.44391 -0.25404 1.04861 1.88575 1.88575 1.04861 -0.25404 -1.44391 -1.99279 -1.65702 -0.58565 0.74571 1.74603 1.97123 1.32135 0.08488 -1.18927 -1.93547 -1.82246 -0.90041 0.42136 1.55607 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E9 2.00000 1.44391 0.08488 -1.32135 -1.99279 -1.55607 -0.25404 1.18927 1.97123 1.65702 0.42136 -1.04861 -1.93547 -1.74603 -0.58565 0.90041 1.88575 1.82246 0.74571 -0.74571 -1.82246 -1.88575 -0.90041 0.58565 1.74603 1.93547 1.04861 -0.42136 -1.65702 -1.97123 -1.18927 0.25404 1.55607 1.99279 1.32135 -0.08488 -1.44391 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E10 2.00000 1.32135 -0.25404 -1.65702 -1.93547 -0.90041 0.74571 1.88575 1.74603 0.42136 -1.18927 -1.99279 -1.44391 0.08488 1.55607 1.97123 1.04861 -0.58565 -1.82246 -1.82246 -0.58565 1.04861 1.97123 1.55607 0.08488 -1.44391 -1.99279 -1.18927 0.42136 1.74603 1.88575 0.74571 -0.90041 -1.93547 -1.65702 -0.25404 1.32135 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E11 2.00000 1.18927 -0.58565 -1.88575 -1.65702 -0.08488 1.55607 1.93547 0.74571 -1.04861 -1.99279 -1.32135 0.42136 1.82246 1.74603 0.25404 -1.44391 -1.97123 -0.90041 0.90041 1.97123 1.44391 -0.25404 -1.74603 -1.82246 -0.42136 1.32135 1.99279 1.04861 -0.74571 -1.93547 -1.55607 0.08488 1.65702 1.88575 0.58565 -1.18927 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E12 2.00000 1.04861 -0.90041 -1.99279 -1.18927 0.74571 1.97123 1.32135 -0.58565 -1.93547 -1.44391 0.42136 1.88575 1.55607 -0.25404 -1.82246 -1.65702 0.08488 1.74603 1.74603 0.08488 -1.65702 -1.82246 -0.25404 1.55607 1.88575 0.42136 -1.44391 -1.93547 -0.58565 1.32135 1.97123 0.74571 -1.18927 -1.99279 -0.90041 1.04861 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E13 2.00000 0.90041 -1.18927 -1.97123 -0.58565 1.44391 1.88575 0.25404 -1.65702 -1.74603 0.08488 1.82246 1.55607 -0.42136 -1.93547 -1.32135 0.74571 1.99279 1.04861 -1.04861 -1.99279 -0.74571 1.32135 1.93547 0.42136 -1.55607 -1.82246 -0.08488 1.74603 1.65702 -0.25404 -1.88575 -1.44391 0.58565 1.97123 1.18927 -0.90041 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E14 2.00000 0.74571 -1.44391 -1.82246 0.08488 1.88575 1.32135 -0.90041 -1.99279 -0.58565 1.55607 1.74603 -0.25404 -1.93547 -1.18927 1.04861 1.97123 0.42136 -1.65702 -1.65702 0.42136 1.97123 1.04861 -1.18927 -1.93547 -0.25404 1.74603 1.55607 -0.58565 -1.99279 -0.90041 1.32135 1.88575 0.08488 -1.82246 -1.44391 0.74571 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E15 2.00000 0.58565 -1.65702 -1.55607 0.74571 1.99279 0.42136 -1.74603 -1.44391 0.90041 1.97123 0.25404 -1.82246 -1.32135 1.04861 1.93547 0.08488 -1.88575 -1.18927 1.18927 1.88575 -0.08488 -1.93547 -1.04861 1.32135 1.82246 -0.25404 -1.97123 -0.90041 1.44391 1.74603 -0.42136 -1.99279 -0.74571 1.55607 1.65702 -0.58565 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E16 2.00000 0.42136 -1.82246 -1.18927 1.32135 1.74603 -0.58565 -1.99279 -0.25404 1.88575 1.04861 -1.44391 -1.65702 0.74571 1.97123 0.08488 -1.93547 -0.90041 1.55607 1.55607 -0.90041 -1.93547 0.08488 1.97123 0.74571 -1.65702 -1.44391 1.04861 1.88575 -0.25404 -1.99279 -0.58565 1.74603 1.32135 -1.18927 -1.82246 0.42136 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E17 2.00000 0.25404 -1.93547 -0.74571 1.74603 1.18927 -1.44391 -1.55607 1.04861 1.82246 -0.58565 -1.97123 0.08488 1.99279 0.42136 -1.88575 -0.90041 1.65702 1.32135 -1.32135 -1.65702 0.90041 1.88575 -0.42136 -1.99279 -0.08488 1.97123 0.58565 -1.82246 -1.04861 1.55607 1.44391 -1.18927 -1.74603 0.74571 1.93547 -0.25404 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E18 2.00000 0.08488 -1.99279 -0.25404 1.97123 0.42136 -1.93547 -0.58565 1.88575 0.74571 -1.82246 -0.90041 1.74603 1.04861 -1.65702 -1.18927 1.55607 1.32135 -1.44391 -1.44391 1.32135 1.55607 -1.18927 -1.65702 1.04861 1.74603 -0.90041 -1.82246 0.74571 1.88575 -0.58565 -1.93547 0.42136 1.97123 -0.25404 -1.99279 0.08488 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E19 2.00000 -0.08488 -1.99279 0.25404 1.97123 -0.42136 -1.93547 0.58565 1.88575 -0.74571 -1.82246 0.90041 1.74603 -1.04861 -1.65702 1.18927 1.55607 -1.32135 -1.44391 1.44391 1.32135 -1.55607 -1.18927 1.65702 1.04861 -1.74603 -0.90041 1.82246 0.74571 -1.88575 -0.58565 1.93547 0.42136 -1.97123 -0.25404 1.99279 0.08488 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E20 2.00000 -0.25404 -1.93547 0.74571 1.74603 -1.18927 -1.44391 1.55607 1.04861 -1.82246 -0.58565 1.97123 0.08488 -1.99279 0.42136 1.88575 -0.90041 -1.65702 1.32135 1.32135 -1.65702 -0.90041 1.88575 0.42136 -1.99279 0.08488 1.97123 -0.58565 -1.82246 1.04861 1.55607 -1.44391 -1.18927 1.74603 0.74571 -1.93547 -0.25404 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E21 2.00000 -0.42136 -1.82246 1.18927 1.32135 -1.74603 -0.58565 1.99279 -0.25404 -1.88575 1.04861 1.44391 -1.65702 -0.74571 1.97123 -0.08488 -1.93547 0.90041 1.55607 -1.55607 -0.90041 1.93547 0.08488 -1.97123 0.74571 1.65702 -1.44391 -1.04861 1.88575 0.25404 -1.99279 0.58565 1.74603 -1.32135 -1.18927 1.82246 0.42136 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E22 2.00000 -0.58565 -1.65702 1.55607 0.74571 -1.99279 0.42136 1.74603 -1.44391 -0.90041 1.97123 -0.25404 -1.82246 1.32135 1.04861 -1.93547 0.08488 1.88575 -1.18927 -1.18927 1.88575 0.08488 -1.93547 1.04861 1.32135 -1.82246 -0.25404 1.97123 -0.90041 -1.44391 1.74603 0.42136 -1.99279 0.74571 1.55607 -1.65702 -0.58565 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E23 2.00000 -0.74571 -1.44391 1.82246 0.08488 -1.88575 1.32135 0.90041 -1.99279 0.58565 1.55607 -1.74603 -0.25404 1.93547 -1.18927 -1.04861 1.97123 -0.42136 -1.65702 1.65702 0.42136 -1.97123 1.04861 1.18927 -1.93547 0.25404 1.74603 -1.55607 -0.58565 1.99279 -0.90041 -1.32135 1.88575 -0.08488 -1.82246 1.44391 0.74571 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E24 2.00000 -0.90041 -1.18927 1.97123 -0.58565 -1.44391 1.88575 -0.25404 -1.65702 1.74603 0.08488 -1.82246 1.55607 0.42136 -1.93547 1.32135 0.74571 -1.99279 1.04861 1.04861 -1.99279 0.74571 1.32135 -1.93547 0.42136 1.55607 -1.82246 0.08488 1.74603 -1.65702 -0.25404 1.88575 -1.44391 -0.58565 1.97123 -1.18927 -0.90041 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E25 2.00000 -1.04861 -0.90041 1.99279 -1.18927 -0.74571 1.97123 -1.32135 -0.58565 1.93547 -1.44391 -0.42136 1.88575 -1.55607 -0.25404 1.82246 -1.65702 -0.08488 1.74603 -1.74603 0.08488 1.65702 -1.82246 0.25404 1.55607 -1.88575 0.42136 1.44391 -1.93547 0.58565 1.32135 -1.97123 0.74571 1.18927 -1.99279 0.90041 1.04861 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E26 2.00000 -1.18927 -0.58565 1.88575 -1.65702 0.08488 1.55607 -1.93547 0.74571 1.04861 -1.99279 1.32135 0.42136 -1.82246 1.74603 -0.25404 -1.44391 1.97123 -0.90041 -0.90041 1.97123 -1.44391 -0.25404 1.74603 -1.82246 0.42136 1.32135 -1.99279 1.04861 0.74571 -1.93547 1.55607 0.08488 -1.65702 1.88575 -0.58565 -1.18927 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E27 2.00000 -1.32135 -0.25404 1.65702 -1.93547 0.90041 0.74571 -1.88575 1.74603 -0.42136 -1.18927 1.99279 -1.44391 -0.08488 1.55607 -1.97123 1.04861 0.58565 -1.82246 1.82246 -0.58565 -1.04861 1.97123 -1.55607 0.08488 1.44391 -1.99279 1.18927 0.42136 -1.74603 1.88575 -0.74571 -0.90041 1.93547 -1.65702 0.25404 1.32135 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E28 2.00000 -1.44391 0.08488 1.32135 -1.99279 1.55607 -0.25404 -1.18927 1.97123 -1.65702 0.42136 1.04861 -1.93547 1.74603 -0.58565 -0.90041 1.88575 -1.82246 0.74571 0.74571 -1.82246 1.88575 -0.90041 -0.58565 1.74603 -1.93547 1.04861 0.42136 -1.65702 1.97123 -1.18927 -0.25404 1.55607 -1.99279 1.32135 0.08488 -1.44391 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E29 2.00000 -1.55607 0.42136 0.90041 -1.82246 1.93547 -1.18927 -0.08488 1.32135 -1.97123 1.74603 -0.74571 -0.58565 1.65702 -1.99279 1.44391 -0.25404 -1.04861 1.88575 -1.88575 1.04861 0.25404 -1.44391 1.99279 -1.65702 0.58565 0.74571 -1.74603 1.97123 -1.32135 0.08488 1.18927 -1.93547 1.82246 -0.90041 -0.42136 1.55607 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E30 2.00000 -1.65702 0.74571 0.42136 -1.44391 1.97123 -1.82246 1.04861 0.08488 -1.18927 1.88575 -1.93547 1.32135 -0.25404 -0.90041 1.74603 -1.99279 1.55607 -0.58565 -0.58565 1.55607 -1.99279 1.74603 -0.90041 -0.25404 1.32135 -1.93547 1.88575 -1.18927 0.08488 1.04861 -1.82246 1.97123 -1.44391 0.42136 0.74571 -1.65702 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E31 2.00000 -1.74603 1.04861 -0.08488 -0.90041 1.65702 -1.99279 1.82246 -1.18927 0.25404 0.74571 -1.55607 1.97123 -1.88575 1.32135 -0.42136 -0.58565 1.44391 -1.93547 1.93547 -1.44391 0.58565 0.42136 -1.32135 1.88575 -1.97123 1.55607 -0.74571 -0.25404 1.18927 -1.82246 1.99279 -1.65702 0.90041 0.08488 -1.04861 1.74603 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E32 2.00000 -1.82246 1.32135 -0.58565 -0.25404 1.04861 -1.65702 1.97123 -1.93547 1.55607 -0.90041 0.08488 0.74571 -1.44391 1.88575 -1.99279 1.74603 -1.18927 0.42136 0.42136 -1.18927 1.74603 -1.99279 1.88575 -1.44391 0.74571 0.08488 -0.90041 1.55607 -1.93547 1.97123 -1.65702 1.04861 -0.25404 -0.58565 1.32135 -1.82246 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E33 2.00000 -1.88575 1.55607 -1.04861 0.42136 0.25404 -0.90041 1.44391 -1.82246 1.99279 -1.93547 1.65702 -1.18927 0.58565 0.08488 -0.74571 1.32135 -1.74603 1.97123 -1.97123 1.74603 -1.32135 0.74571 -0.08488 -0.58565 1.18927 -1.65702 1.93547 -1.99279 1.82246 -1.44391 0.90041 -0.25404 -0.42136 1.04861 -1.55607 1.88575 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E34 2.00000 -1.93547 1.74603 -1.44391 1.04861 -0.58565 0.08488 0.42136 -0.90041 1.32135 -1.65702 1.88575 -1.99279 1.97123 -1.82246 1.55607 -1.18927 0.74571 -0.25404 -0.25404 0.74571 -1.18927 1.55607 -1.82246 1.97123 -1.99279 1.88575 -1.65702 1.32135 -0.90041 0.42136 0.08488 -0.58565 1.04861 -1.44391 1.74603 -1.93547 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E35 2.00000 -1.97123 1.88575 -1.74603 1.55607 -1.32135 1.04861 -0.74571 0.42136 -0.08488 -0.25404 0.58565 -0.90041 1.18927 -1.44391 1.65702 -1.82246 1.93547 -1.99279 1.99279 -1.93547 1.82246 -1.65702 1.44391 -1.18927 0.90041 -0.58565 0.25404 0.08488 -0.42136 0.74571 -1.04861 1.32135 -1.55607 1.74603 -1.88575 1.97123 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E36 2.00000 -1.99279 1.97123 -1.93547 1.88575 -1.82246 1.74603 -1.65702 1.55607 -1.44391 1.32135 -1.18927 1.04861 -0.90041 0.74571 -0.58565 0.42136 -0.25404 0.08488 0.08488 -0.25404 0.42136 -0.58565 0.74571 -0.90041 1.04861 -1.18927 1.32135 -1.44391 1.55607 -1.65702 1.74603 -1.82246 1.88575 -1.93547 1.97123 -1.99279 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.992794977085 = 2*cos(2*π/74) = 2*cos(π/37) 1.971231820695 = 2*cos(4*π/74) = 2*cos(2*π/37) 1.935465893867 = 2*cos(6*π/74) = 2*cos(3*π/37) 1.885754890922 = 2*cos(8*π/74) = 2*cos(4*π/37) 1.822456980776 = 2*cos(10*π/74) = 2*cos(5*π/37) 1.746028226322 = 2*cos(12*π/74) = 2*cos(6*π/37) 1.657019298488 = 2*cos(14*π/74) = 2*cos(7*π/37) 1.556071508637 = 2*cos(16*π/74) = 2*cos(8*π/37) 1.443912187909 = 2*cos(18*π/74) = 2*cos(9*π/37) 1.321349446780 = 2*cos(20*π/74) = 2*cos(10*π/37) 1.189266352609 = 2*cos(22*π/74) = 2*cos(11*π/37) 1.048614567114 = 2*cos(24*π/74) = 2*cos(12*π/37) 0.900407489635 = 2*cos(26*π/74) = 2*cos(13*π/37) 0.745712955561 = 2*cos(28*π/74) = 2*cos(14*π/37) 0.585645542553 = 2*cos(30*π/74) = 2*cos(15*π/37) 0.421358539991 = 2*cos(32*π/74) = 2*cos(16*π/37) 0.254035639494 = 2*cos(34*π/74) = 2*cos(17*π/37) 0.084882406392 = 2*cos(36*π/74) = 2*cos(18*π/37) 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_74v point group is 148, and the order of the principal axis (C₇₄) is 74. The group has 40 irreducible representations. β The C_74v point group is isomorphic to D_37d, D_37h and D₇₄. γ The C_74v point group is generated by two symmetry elements, C₇₄ 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_74v 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_74v point group was created by Gernot Katzer.

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

	C_72v
	C_73v
C₇₄	C_74v	C_74h D₇₄ D_74h D_74d S₇₄
	C_75v
	C_76v

Character table for the C74v point group

Character table for the C_74v point group