Character Table for Point Group D37h

Character table for the D_37h point group

D37h E 2 C37 2 C37^2 2 C37^3 2 C37^4 2 C37^5 2 C37^6 2 C37^7 2 C37^8 2 C37^9 2 C37^10 2 C37^11 2 C37^12 2 C37^13 2 C37^14 2 C37^15 2 C37^16 2 C37^17 2 C37^18 37 C2' sh 2 S37 2 S37^3 2 S37^5 2 S37^7 2 S37^9 2 S37^11 2 S37^13 2 S37^15 2 S37^17 2 S37^19 2 S37^21 2 S37^23 2 S37^25 2 S37^27 2 S37^29 2 S37^31 2 S37^33 2 S37^35 37 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 1.00000 1.00000 1.00000 1.00000 1.00000 ... ... ....T ....... ........T ........... ............T 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 ... ... ..... ....... ......... ........... ............. 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 ... ..... ....... ......... ........... ............. 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 ..... ......T ......... ..........T ............. E1' 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 0.00000 2.00000 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 0.00000 ... TT. ..... ....TT. ......... ........TT. ............. E1" 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 0.00000 -2.00000 -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 0.00000 TT. ... ..TT. ....... ......TT. ........... ..........TT. E2' 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 0.00000 2.00000 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 0.00000 ... ... TT... ....... ....TT... ........... ........TT... E2" 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 0.00000 -2.00000 -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 0.00000 ... ... ..... ..TT... ......... ......TT... ............. E3' 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 0.00000 2.00000 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 0.00000 ... ... ..... TT..... ......... ....TT..... ............. E3" 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 0.00000 -2.00000 -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 0.00000 ... ... ..... ....... ..TT..... ........... ......TT..... E4' 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 0.00000 2.00000 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 0.00000 ... ... ..... ....... TT....... ........... ....TT....... E4" 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 0.00000 -2.00000 -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 0.00000 ... ... ..... ....... ......... ..TT....... ............. E5' 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 0.00000 2.00000 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 0.00000 ... ... ..... ....... ......... TT......... ............. E5" 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 0.00000 -2.00000 -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 0.00000 ... ... ..... ....... ......... ........... ..TT......... E6' 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 0.00000 2.00000 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 0.00000 ... ... ..... ....... ......... ........... TT........... E6" 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 0.00000 -2.00000 -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 0.00000 ... ... ..... ....... ......... ........... ............. E7' 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 0.00000 2.00000 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 0.00000 ... ... ..... ....... ......... ........... ............. E7" 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 0.00000 -2.00000 -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 0.00000 ... ... ..... ....... ......... ........... ............. E8' 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 0.00000 2.00000 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 0.00000 ... ... ..... ....... ......... ........... ............. E8" 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 0.00000 -2.00000 -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 0.00000 ... ... ..... ....... ......... ........... ............. E9' 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 0.00000 2.00000 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 0.00000 ... ... ..... ....... ......... ........... ............. E9" 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 0.00000 -2.00000 -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 0.00000 ... ... ..... ....... ......... ........... ............. E10' 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 0.00000 2.00000 -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 0.00000 ... ... ..... ....... ......... ........... ............. E10" 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 0.00000 -2.00000 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 0.00000 ... ... ..... ....... ......... ........... ............. E11' 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 0.00000 2.00000 -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 0.00000 ... ... ..... ....... ......... ........... ............. E11" 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 0.00000 -2.00000 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 0.00000 ... ... ..... ....... ......... ........... ............. E12' 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 0.00000 2.00000 -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 0.00000 ... ... ..... ....... ......... ........... ............. E12" 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 0.00000 -2.00000 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 0.00000 ... ... ..... ....... ......... ........... ............. E13' 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 0.00000 2.00000 -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 0.00000 ... ... ..... ....... ......... ........... ............. E13" 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 0.00000 -2.00000 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 0.00000 ... ... ..... ....... ......... ........... ............. E14' 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 0.00000 2.00000 -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 0.00000 ... ... ..... ....... ......... ........... ............. E14" 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 0.00000 -2.00000 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 0.00000 ... ... ..... ....... ......... ........... ............. E15' 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 0.00000 2.00000 -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 0.00000 ... ... ..... ....... ......... ........... ............. E15" 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 0.00000 -2.00000 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 0.00000 ... ... ..... ....... ......... ........... ............. E16' 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 0.00000 2.00000 -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 0.00000 ... ... ..... ....... ......... ........... ............. E16" 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 0.00000 -2.00000 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 0.00000 ... ... ..... ....... ......... ........... ............. E17' 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 0.00000 2.00000 -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 0.00000 ... ... ..... ....... ......... ........... ............. E17" 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 0.00000 -2.00000 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 0.00000 ... ... ..... ....... ......... ........... ............. E18' 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 0.00000 2.00000 -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 0.00000 ... ... ..... ....... ......... ........... ............. E18" 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 0.00000 -2.00000 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 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' d+g+i+k+m z², z⁴, z⁶ A2' R R_z A2" p+f+h+j+l z, z³, z⁵ E1' p+f+h+j+l {x, y}, {xz², yz²}, {xz⁴, yz⁴} E1" R+d+g+i+k+m {R_x, R_y}, {xz, yz}, {xz³, yz³}, {xz⁵, yz⁵} E2' d+g+i+k+m {x²−y², xy}, {z²(x²−y²), xyz²}, {z⁴(x²−y²), xyz⁴} E2" f+h+j+l {z(x²−y²), xyz}, {z³(x²−y²), xyz³} E3' f+h+j+l {x(x²−3y²), y(3x²−y²)}, {xz²(x²−3y²), yz²(3x²−y²)} E3" g+i+k+m {xz(x²−3y²), yz(3x²−y²)}, {xz³(x²−3y²), yz³(3x²−y²)} E4' g+i+k+m {(x²−y²)²−4x²y², xy(x²−y²)}, {z²((x²−y²)²−4x²y²), xyz²(x²−y²)} E4" h+j+l {z((x²−y²)²−4x²y²), xyz(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)} E5" 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)} E6' i+k+m {x²(x²−3y²)²−y²(3x²−y²)², xy(x²−3y²)(3x²−y²)} E6" j+l E7' j+l E7" k+m E8' k+m E8" l E9' l E9" m E10' m Notes: α The order of the D_37h point group is 148, and the order of the principal axis (S₃₇) is 74. The group has 40 irreducible representations. β The D_37h point group is isomorphic to D_37d, C_74v and D₇₄. γ The D_37h point group is generated by two symmetry elements, S₃₇ and either a perpendicular C₂^′ or a vertical σ_v. Also, the group may be generated from any two σ_v planes, or any σ_v and a non-coplanar C₂^′. The canonical choice, however, is to use redundant generators: C₃₇, C₂^′ and σ_h. δ The group contains one set of C₂^′ symmetry axes perpendicular to the principal (z) axis. The x axis (but not the y axis) is a member of that set. Similarly, the single set of symmetry planes denoted σ_d contains the xz plane but not the yz plane. ε The lowest nonvanishing multipole moment in D_37h is 4 (quadrupole 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 just 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 D_37h point group was created by Gernot Katzer.

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

	D_35h
	D_36h
C₃₇ C_37v C_37h D₃₇	D_37h	D_37d
	D_38h
	D_39h

Character table for the D37h point group

Character table for the D_37h point group