Character Table for Point Group D37d

Character table for the D_37d point group

D37d 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' i 2 S74 2 S74^3 2 S74^5 2 S74^7 2 S74^9 2 S74^11 2 S74^13 2 S74^15 2 S74^17 2 S74^19 2 S74^21 2 S74^23 2 S74^25 2 S74^27 2 S74^29 2 S74^31 2 S74^33 2 S74^35 37 sd <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> A1g 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 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 A2g 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 -1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 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 ... ..... ....... ......... ........... ............. E1g 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 -1.99279 -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 TT. ... ..TT. ....... ......TT. ........... ..........TT. E2g 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 1.97123 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... E3g 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 -1.93547 -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 ... ... ..... ....... ..TT..... ........... ......TT..... E4g 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 1.88575 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....... E5g 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 -1.82246 -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 ... ... ..... ....... ......... ........... ..TT......... E6g 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 1.74603 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........... E7g 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 -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 ... ... ..... ....... ......... ........... ............. E8g 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 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 ... ... ..... ....... ......... ........... ............. E9g 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 -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 ... ... ..... ....... ......... ........... ............. E10g 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 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 ... ... ..... ....... ......... ........... ............. E11g 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 -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 ... ... ..... ....... ......... ........... ............. E12g 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 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 ... ... ..... ....... ......... ........... ............. E13g 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 -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 ... ... ..... ....... ......... ........... ............. E14g 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 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 ... ... ..... ....... ......... ........... ............. E15g 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 -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 ... ... ..... ....... ......... ........... ............. E16g 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 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 ... ... ..... ....... ......... ........... ............. E17g 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 -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 ... ... ..... ....... ......... ........... ............. E18g 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 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 ... ... ..... ....... ......... ........... ............. A1u 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 ... ... ..... ....... ......... ........... ............. A2u 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -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 ............. E1u 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 1.99279 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 ... TT. ..... ....TT. ......... ........TT. ............. E2u 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 -1.97123 -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... ............. E3u 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 1.93547 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 ... ... ..... TT..... ......... ....TT..... ............. E4u 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 -1.88575 -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....... ............. E5u 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 1.82246 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 ... ... ..... ....... ......... TT......... ............. E6u 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 -1.74603 -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 ... ... ..... ....... ......... ........... ............. E7u 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 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 ... ... ..... ....... ......... ........... ............. E8u 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 -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 ... ... ..... ....... ......... ........... ............. E9u 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 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 ... ... ..... ....... ......... ........... ............. E10u 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 -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 ... ... ..... ....... ......... ........... ............. E11u 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 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 ... ... ..... ....... ......... ........... ............. E12u 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 -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 ... ... ..... ....... ......... ........... ............. E13u 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 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 ... ... ..... ....... ......... ........... ............. E14u 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 -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 ... ... ..... ....... ......... ........... ............. E15u 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 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 ... ... ..... ....... ......... ........... ............. E16u 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 -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 ... ... ..... ....... ......... ........... ............. E17u 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 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 ... ... ..... ....... ......... ........... ............. E18u 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 -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 ... ... ..... ....... ......... ........... ............. 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 A1g d+g+i+k+m z², z⁴, z⁶ A2g R R_z E1g R+d+g+i+k+m {R_x, R_y}, {xz, yz}, {xz³, yz³}, {xz⁵, yz⁵} E2g d+g+i+k+m {x²−y², xy}, {z²(x²−y²), xyz²}, {z⁴(x²−y²), xyz⁴} E3g g+i+k+m {xz(x²−3y²), yz(3x²−y²)}, {xz³(x²−3y²), yz³(3x²−y²)} E4g g+i+k+m {(x²−y²)²−4x²y², xy(x²−y²)}, {z²((x²−y²)²−4x²y²), xyz²(x²−y²)} E5g 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)} E6g i+k+m {x²(x²−3y²)²−y²(3x²−y²)², xy(x²−3y²)(3x²−y²)} E7g k+m E8g k+m E9g m E10g m A2u p+f+h+j+l z, z³, z⁵ E1u p+f+h+j+l {x, y}, {xz², yz²}, {xz⁴, yz⁴} E2u f+h+j+l {z(x²−y²), xyz}, {z³(x²−y²), xyz³} E3u f+h+j+l {x(x²−3y²), y(3x²−y²)}, {xz²(x²−3y²), yz²(3x²−y²)} E4u h+j+l {z((x²−y²)²−4x²y²), xyz(x²−y²)} E5u 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)} E6u j+l E7u j+l E8u l E9u l Notes: α The order of the D_37d point group is 148, and the order of the principal axis (S₇₄) is 74. The group has 40 irreducible representations. β The D_37d point group is isomorphic to D_37h, C_74v and D₇₄. γ The D_37d point group is generated by two symmetry elements, S₇₄ and either a perpendicular C₂^′ or a vertical σ_d. Also, the group may be generated from any C₂^′ plus any σ_d plane. The canonical choice, however, is to use redundant generators: C₃₇, C₂^′ and i. 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. Reversely, the single set of symmetry planes denoted σ_d contains the yz plane but not the xz plane. ε The lowest nonvanishing multipole moment in D_37d 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_37d point group was created by Gernot Katzer.

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

	D_35d
	D_36d
C₃₇ C_37v C_37h D₃₇ D_37h	D_37d
	D_38d
	D_39d

Character table for the D37d point group

Character table for the D_37d point group