Character Table for Point Group D41h

Character table for the D_41h point group

D41h E 2 C41 2 C41^2 2 C41^3 2 C41^4 2 C41^5 2 C41^6 2 C41^7 2 C41^8 2 C41^9 2 C41^10 2 C41^11 2 C41^12 2 C41^13 2 C41^14 2 C41^15 2 C41^16 2 C41^17 2 C41^18 2 C41^19 2 C41^20 41 C2' sh 2 S41 2 S41^3 2 S41^5 2 S41^7 2 S41^9 2 S41^11 2 S41^13 2 S41^15 2 S41^17 2 S41^19 2 S41^21 2 S41^23 2 S41^25 2 S41^27 2 S41^29 2 S41^31 2 S41^33 2 S41^35 2 S41^37 2 S41^39 41 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 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 -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 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 -1.00000 -1.00000 -1.00000 1.00000 ... ..T ..... ......T ......... ..........T ............. E1' 2.00000 1.97656 1.90679 1.79233 1.63586 1.44104 1.21245 0.95544 0.67603 0.38078 0.07661 -0.22937 -0.52996 -0.81814 -1.08714 -1.33065 -1.54298 -1.71914 -1.85500 -1.94739 -1.99413 0.00000 2.00000 1.97656 1.79233 1.44104 0.95544 0.38078 -0.22937 -0.81814 -1.33065 -1.71914 -1.94739 -1.99413 -1.85500 -1.54298 -1.08714 -0.52996 0.07661 0.67603 1.21245 1.63586 1.90679 0.00000 ... TT. ..... ....TT. ......... ........TT. ............. E1" 2.00000 1.97656 1.90679 1.79233 1.63586 1.44104 1.21245 0.95544 0.67603 0.38078 0.07661 -0.22937 -0.52996 -0.81814 -1.08714 -1.33065 -1.54298 -1.71914 -1.85500 -1.94739 -1.99413 0.00000 -2.00000 -1.97656 -1.79233 -1.44104 -0.95544 -0.38078 0.22937 0.81814 1.33065 1.71914 1.94739 1.99413 1.85500 1.54298 1.08714 0.52996 -0.07661 -0.67603 -1.21245 -1.63586 -1.90679 0.00000 TT. ... ..TT. ....... ......TT. ........... ..........TT. E2' 2.00000 1.90679 1.63586 1.21245 0.67603 0.07661 -0.52996 -1.08714 -1.54298 -1.85500 -1.99413 -1.94739 -1.71914 -1.33065 -0.81814 -0.22937 0.38078 0.95544 1.44104 1.79233 1.97656 0.00000 2.00000 1.90679 1.21245 0.07661 -1.08714 -1.85500 -1.94739 -1.33065 -0.22937 0.95544 1.79233 1.97656 1.44104 0.38078 -0.81814 -1.71914 -1.99413 -1.54298 -0.52996 0.67603 1.63586 0.00000 ... ... TT... ....... ....TT... ........... ........TT... E2" 2.00000 1.90679 1.63586 1.21245 0.67603 0.07661 -0.52996 -1.08714 -1.54298 -1.85500 -1.99413 -1.94739 -1.71914 -1.33065 -0.81814 -0.22937 0.38078 0.95544 1.44104 1.79233 1.97656 0.00000 -2.00000 -1.90679 -1.21245 -0.07661 1.08714 1.85500 1.94739 1.33065 0.22937 -0.95544 -1.79233 -1.97656 -1.44104 -0.38078 0.81814 1.71914 1.99413 1.54298 0.52996 -0.67603 -1.63586 0.00000 ... ... ..... ..TT... ......... ......TT... ............. E3' 2.00000 1.79233 1.21245 0.38078 -0.52996 -1.33065 -1.85500 -1.99413 -1.71914 -1.08714 -0.22937 0.67603 1.44104 1.90679 1.97656 1.63586 0.95544 0.07661 -0.81814 -1.54298 -1.94739 0.00000 2.00000 1.79233 0.38078 -1.33065 -1.99413 -1.08714 0.67603 1.90679 1.63586 0.07661 -1.54298 -1.94739 -0.81814 0.95544 1.97656 1.44104 -0.22937 -1.71914 -1.85500 -0.52996 1.21245 0.00000 ... ... ..... TT..... ......... ....TT..... ............. E3" 2.00000 1.79233 1.21245 0.38078 -0.52996 -1.33065 -1.85500 -1.99413 -1.71914 -1.08714 -0.22937 0.67603 1.44104 1.90679 1.97656 1.63586 0.95544 0.07661 -0.81814 -1.54298 -1.94739 0.00000 -2.00000 -1.79233 -0.38078 1.33065 1.99413 1.08714 -0.67603 -1.90679 -1.63586 -0.07661 1.54298 1.94739 0.81814 -0.95544 -1.97656 -1.44104 0.22937 1.71914 1.85500 0.52996 -1.21245 0.00000 ... ... ..... ....... ..TT..... ........... ......TT..... E4' 2.00000 1.63586 0.67603 -0.52996 -1.54298 -1.99413 -1.71914 -0.81814 0.38078 1.44104 1.97656 1.79233 0.95544 -0.22937 -1.33065 -1.94739 -1.85500 -1.08714 0.07661 1.21245 1.90679 0.00000 2.00000 1.63586 -0.52996 -1.99413 -0.81814 1.44104 1.79233 -0.22937 -1.94739 -1.08714 1.21245 1.90679 0.07661 -1.85500 -1.33065 0.95544 1.97656 0.38078 -1.71914 -1.54298 0.67603 0.00000 ... ... ..... ....... TT....... ........... ....TT....... E4" 2.00000 1.63586 0.67603 -0.52996 -1.54298 -1.99413 -1.71914 -0.81814 0.38078 1.44104 1.97656 1.79233 0.95544 -0.22937 -1.33065 -1.94739 -1.85500 -1.08714 0.07661 1.21245 1.90679 0.00000 -2.00000 -1.63586 0.52996 1.99413 0.81814 -1.44104 -1.79233 0.22937 1.94739 1.08714 -1.21245 -1.90679 -0.07661 1.85500 1.33065 -0.95544 -1.97656 -0.38078 1.71914 1.54298 -0.67603 0.00000 ... ... ..... ....... ......... ..TT....... ............. E5' 2.00000 1.44104 0.07661 -1.33065 -1.99413 -1.54298 -0.22937 1.21245 1.97656 1.63586 0.38078 -1.08714 -1.94739 -1.71914 -0.52996 0.95544 1.90679 1.79233 0.67603 -0.81814 -1.85500 0.00000 2.00000 1.44104 -1.33065 -1.54298 1.21245 1.63586 -1.08714 -1.71914 0.95544 1.79233 -0.81814 -1.85500 0.67603 1.90679 -0.52996 -1.94739 0.38078 1.97656 -0.22937 -1.99413 0.07661 0.00000 ... ... ..... ....... ......... TT......... ............. E5" 2.00000 1.44104 0.07661 -1.33065 -1.99413 -1.54298 -0.22937 1.21245 1.97656 1.63586 0.38078 -1.08714 -1.94739 -1.71914 -0.52996 0.95544 1.90679 1.79233 0.67603 -0.81814 -1.85500 0.00000 -2.00000 -1.44104 1.33065 1.54298 -1.21245 -1.63586 1.08714 1.71914 -0.95544 -1.79233 0.81814 1.85500 -0.67603 -1.90679 0.52996 1.94739 -0.38078 -1.97656 0.22937 1.99413 -0.07661 0.00000 ... ... ..... ....... ......... ........... ..TT......... E6' 2.00000 1.21245 -0.52996 -1.85500 -1.71914 -0.22937 1.44104 1.97656 0.95544 -0.81814 -1.94739 -1.54298 0.07661 1.63586 1.90679 0.67603 -1.08714 -1.99413 -1.33065 0.38078 1.79233 0.00000 2.00000 1.21245 -1.85500 -0.22937 1.97656 -0.81814 -1.54298 1.63586 0.67603 -1.99413 0.38078 1.79233 -1.33065 -1.08714 1.90679 0.07661 -1.94739 0.95544 1.44104 -1.71914 -0.52996 0.00000 ... ... ..... ....... ......... ........... TT........... E6" 2.00000 1.21245 -0.52996 -1.85500 -1.71914 -0.22937 1.44104 1.97656 0.95544 -0.81814 -1.94739 -1.54298 0.07661 1.63586 1.90679 0.67603 -1.08714 -1.99413 -1.33065 0.38078 1.79233 0.00000 -2.00000 -1.21245 1.85500 0.22937 -1.97656 0.81814 1.54298 -1.63586 -0.67603 1.99413 -0.38078 -1.79233 1.33065 1.08714 -1.90679 -0.07661 1.94739 -0.95544 -1.44104 1.71914 0.52996 0.00000 ... ... ..... ....... ......... ........... ............. E7' 2.00000 0.95544 -1.08714 -1.99413 -0.81814 1.21245 1.97656 0.67603 -1.33065 -1.94739 -0.52996 1.44104 1.90679 0.38078 -1.54298 -1.85500 -0.22937 1.63586 1.79233 0.07661 -1.71914 0.00000 2.00000 0.95544 -1.99413 1.21245 0.67603 -1.94739 1.44104 0.38078 -1.85500 1.63586 0.07661 -1.71914 1.79233 -0.22937 -1.54298 1.90679 -0.52996 -1.33065 1.97656 -0.81814 -1.08714 0.00000 ... ... ..... ....... ......... ........... ............. E7" 2.00000 0.95544 -1.08714 -1.99413 -0.81814 1.21245 1.97656 0.67603 -1.33065 -1.94739 -0.52996 1.44104 1.90679 0.38078 -1.54298 -1.85500 -0.22937 1.63586 1.79233 0.07661 -1.71914 0.00000 -2.00000 -0.95544 1.99413 -1.21245 -0.67603 1.94739 -1.44104 -0.38078 1.85500 -1.63586 -0.07661 1.71914 -1.79233 0.22937 1.54298 -1.90679 0.52996 1.33065 -1.97656 0.81814 1.08714 0.00000 ... ... ..... ....... ......... ........... ............. E8' 2.00000 0.67603 -1.54298 -1.71914 0.38078 1.97656 0.95544 -1.33065 -1.85500 0.07661 1.90679 1.21245 -1.08714 -1.94739 -0.22937 1.79233 1.44104 -0.81814 -1.99413 -0.52996 1.63586 0.00000 2.00000 0.67603 -1.71914 1.97656 -1.33065 0.07661 1.21245 -1.94739 1.79233 -0.81814 -0.52996 1.63586 -1.99413 1.44104 -0.22937 -1.08714 1.90679 -1.85500 0.95544 0.38078 -1.54298 0.00000 ... ... ..... ....... ......... ........... ............. E8" 2.00000 0.67603 -1.54298 -1.71914 0.38078 1.97656 0.95544 -1.33065 -1.85500 0.07661 1.90679 1.21245 -1.08714 -1.94739 -0.22937 1.79233 1.44104 -0.81814 -1.99413 -0.52996 1.63586 0.00000 -2.00000 -0.67603 1.71914 -1.97656 1.33065 -0.07661 -1.21245 1.94739 -1.79233 0.81814 0.52996 -1.63586 1.99413 -1.44104 0.22937 1.08714 -1.90679 1.85500 -0.95544 -0.38078 1.54298 0.00000 ... ... ..... ....... ......... ........... ............. E9' 2.00000 0.38078 -1.85500 -1.08714 1.44104 1.63586 -0.81814 -1.94739 0.07661 1.97656 0.67603 -1.71914 -1.33065 1.21245 1.79233 -0.52996 -1.99413 -0.22937 1.90679 0.95544 -1.54298 0.00000 2.00000 0.38078 -1.08714 1.63586 -1.94739 1.97656 -1.71914 1.21245 -0.52996 -0.22937 0.95544 -1.54298 1.90679 -1.99413 1.79233 -1.33065 0.67603 0.07661 -0.81814 1.44104 -1.85500 0.00000 ... ... ..... ....... ......... ........... ............. E9" 2.00000 0.38078 -1.85500 -1.08714 1.44104 1.63586 -0.81814 -1.94739 0.07661 1.97656 0.67603 -1.71914 -1.33065 1.21245 1.79233 -0.52996 -1.99413 -0.22937 1.90679 0.95544 -1.54298 0.00000 -2.00000 -0.38078 1.08714 -1.63586 1.94739 -1.97656 1.71914 -1.21245 0.52996 0.22937 -0.95544 1.54298 -1.90679 1.99413 -1.79233 1.33065 -0.67603 -0.07661 0.81814 -1.44104 1.85500 0.00000 ... ... ..... ....... ......... ........... ............. E10' 2.00000 0.07661 -1.99413 -0.22937 1.97656 0.38078 -1.94739 -0.52996 1.90679 0.67603 -1.85500 -0.81814 1.79233 0.95544 -1.71914 -1.08714 1.63586 1.21245 -1.54298 -1.33065 1.44104 0.00000 2.00000 0.07661 -0.22937 0.38078 -0.52996 0.67603 -0.81814 0.95544 -1.08714 1.21245 -1.33065 1.44104 -1.54298 1.63586 -1.71914 1.79233 -1.85500 1.90679 -1.94739 1.97656 -1.99413 0.00000 ... ... ..... ....... ......... ........... ............. E10" 2.00000 0.07661 -1.99413 -0.22937 1.97656 0.38078 -1.94739 -0.52996 1.90679 0.67603 -1.85500 -0.81814 1.79233 0.95544 -1.71914 -1.08714 1.63586 1.21245 -1.54298 -1.33065 1.44104 0.00000 -2.00000 -0.07661 0.22937 -0.38078 0.52996 -0.67603 0.81814 -0.95544 1.08714 -1.21245 1.33065 -1.44104 1.54298 -1.63586 1.71914 -1.79233 1.85500 -1.90679 1.94739 -1.97656 1.99413 0.00000 ... ... ..... ....... ......... ........... ............. E11' 2.00000 -0.22937 -1.94739 0.67603 1.79233 -1.08714 -1.54298 1.44104 1.21245 -1.71914 -0.81814 1.90679 0.38078 -1.99413 0.07661 1.97656 -0.52996 -1.85500 0.95544 1.63586 -1.33065 0.00000 2.00000 -0.22937 0.67603 -1.08714 1.44104 -1.71914 1.90679 -1.99413 1.97656 -1.85500 1.63586 -1.33065 0.95544 -0.52996 0.07661 0.38078 -0.81814 1.21245 -1.54298 1.79233 -1.94739 0.00000 ... ... ..... ....... ......... ........... ............. E11" 2.00000 -0.22937 -1.94739 0.67603 1.79233 -1.08714 -1.54298 1.44104 1.21245 -1.71914 -0.81814 1.90679 0.38078 -1.99413 0.07661 1.97656 -0.52996 -1.85500 0.95544 1.63586 -1.33065 0.00000 -2.00000 0.22937 -0.67603 1.08714 -1.44104 1.71914 -1.90679 1.99413 -1.97656 1.85500 -1.63586 1.33065 -0.95544 0.52996 -0.07661 -0.38078 0.81814 -1.21245 1.54298 -1.79233 1.94739 0.00000 ... ... ..... ....... ......... ........... ............. E12' 2.00000 -0.52996 -1.71914 1.44104 0.95544 -1.94739 0.07661 1.90679 -1.08714 -1.33065 1.79233 0.38078 -1.99413 0.67603 1.63586 -1.54298 -0.81814 1.97656 -0.22937 -1.85500 1.21245 0.00000 2.00000 -0.52996 1.44104 -1.94739 1.90679 -1.33065 0.38078 0.67603 -1.54298 1.97656 -1.85500 1.21245 -0.22937 -0.81814 1.63586 -1.99413 1.79233 -1.08714 0.07661 0.95544 -1.71914 0.00000 ... ... ..... ....... ......... ........... ............. E12" 2.00000 -0.52996 -1.71914 1.44104 0.95544 -1.94739 0.07661 1.90679 -1.08714 -1.33065 1.79233 0.38078 -1.99413 0.67603 1.63586 -1.54298 -0.81814 1.97656 -0.22937 -1.85500 1.21245 0.00000 -2.00000 0.52996 -1.44104 1.94739 -1.90679 1.33065 -0.38078 -0.67603 1.54298 -1.97656 1.85500 -1.21245 0.22937 0.81814 -1.63586 1.99413 -1.79233 1.08714 -0.07661 -0.95544 1.71914 0.00000 ... ... ..... ....... ......... ........... ............. E13' 2.00000 -0.81814 -1.33065 1.90679 -0.22937 -1.71914 1.63586 0.38078 -1.94739 1.21245 0.95544 -1.99413 0.67603 1.44104 -1.85500 0.07661 1.79233 -1.54298 -0.52996 1.97656 -1.08714 0.00000 2.00000 -0.81814 1.90679 -1.71914 0.38078 1.21245 -1.99413 1.44104 0.07661 -1.54298 1.97656 -1.08714 -0.52996 1.79233 -1.85500 0.67603 0.95544 -1.94739 1.63586 -0.22937 -1.33065 0.00000 ... ... ..... ....... ......... ........... ............. E13" 2.00000 -0.81814 -1.33065 1.90679 -0.22937 -1.71914 1.63586 0.38078 -1.94739 1.21245 0.95544 -1.99413 0.67603 1.44104 -1.85500 0.07661 1.79233 -1.54298 -0.52996 1.97656 -1.08714 0.00000 -2.00000 0.81814 -1.90679 1.71914 -0.38078 -1.21245 1.99413 -1.44104 -0.07661 1.54298 -1.97656 1.08714 0.52996 -1.79233 1.85500 -0.67603 -0.95544 1.94739 -1.63586 0.22937 1.33065 0.00000 ... ... ..... ....... ......... ........... ............. E14' 2.00000 -1.08714 -0.81814 1.97656 -1.33065 -0.52996 1.90679 -1.54298 -0.22937 1.79233 -1.71914 0.07661 1.63586 -1.85500 0.38078 1.44104 -1.94739 0.67603 1.21245 -1.99413 0.95544 0.00000 2.00000 -1.08714 1.97656 -0.52996 -1.54298 1.79233 0.07661 -1.85500 1.44104 0.67603 -1.99413 0.95544 1.21245 -1.94739 0.38078 1.63586 -1.71914 -0.22937 1.90679 -1.33065 -0.81814 0.00000 ... ... ..... ....... ......... ........... ............. E14" 2.00000 -1.08714 -0.81814 1.97656 -1.33065 -0.52996 1.90679 -1.54298 -0.22937 1.79233 -1.71914 0.07661 1.63586 -1.85500 0.38078 1.44104 -1.94739 0.67603 1.21245 -1.99413 0.95544 0.00000 -2.00000 1.08714 -1.97656 0.52996 1.54298 -1.79233 -0.07661 1.85500 -1.44104 -0.67603 1.99413 -0.95544 -1.21245 1.94739 -0.38078 -1.63586 1.71914 0.22937 -1.90679 1.33065 0.81814 0.00000 ... ... ..... ....... ......... ........... ............. E15' 2.00000 -1.33065 -0.22937 1.63586 -1.94739 0.95544 0.67603 -1.85500 1.79233 -0.52996 -1.08714 1.97656 -1.54298 0.07661 1.44104 -1.99413 1.21245 0.38078 -1.71914 1.90679 -0.81814 0.00000 2.00000 -1.33065 1.63586 0.95544 -1.85500 -0.52996 1.97656 0.07661 -1.99413 0.38078 1.90679 -0.81814 -1.71914 1.21245 1.44104 -1.54298 -1.08714 1.79233 0.67603 -1.94739 -0.22937 0.00000 ... ... ..... ....... ......... ........... ............. E15" 2.00000 -1.33065 -0.22937 1.63586 -1.94739 0.95544 0.67603 -1.85500 1.79233 -0.52996 -1.08714 1.97656 -1.54298 0.07661 1.44104 -1.99413 1.21245 0.38078 -1.71914 1.90679 -0.81814 0.00000 -2.00000 1.33065 -1.63586 -0.95544 1.85500 0.52996 -1.97656 -0.07661 1.99413 -0.38078 -1.90679 0.81814 1.71914 -1.21245 -1.44104 1.54298 1.08714 -1.79233 -0.67603 1.94739 0.22937 0.00000 ... ... ..... ....... ......... ........... ............. E16' 2.00000 -1.54298 0.38078 0.95544 -1.85500 1.90679 -1.08714 -0.22937 1.44104 -1.99413 1.63586 -0.52996 -0.81814 1.79233 -1.94739 1.21245 0.07661 -1.33065 1.97656 -1.71914 0.67603 0.00000 2.00000 -1.54298 0.95544 1.90679 -0.22937 -1.99413 -0.52996 1.79233 1.21245 -1.33065 -1.71914 0.67603 1.97656 0.07661 -1.94739 -0.81814 1.63586 1.44104 -1.08714 -1.85500 0.38078 0.00000 ... ... ..... ....... ......... ........... ............. E16" 2.00000 -1.54298 0.38078 0.95544 -1.85500 1.90679 -1.08714 -0.22937 1.44104 -1.99413 1.63586 -0.52996 -0.81814 1.79233 -1.94739 1.21245 0.07661 -1.33065 1.97656 -1.71914 0.67603 0.00000 -2.00000 1.54298 -0.95544 -1.90679 0.22937 1.99413 0.52996 -1.79233 -1.21245 1.33065 1.71914 -0.67603 -1.97656 -0.07661 1.94739 0.81814 -1.63586 -1.44104 1.08714 1.85500 -0.38078 0.00000 ... ... ..... ....... ......... ........... ............. E17' 2.00000 -1.71914 0.95544 0.07661 -1.08714 1.79233 -1.99413 1.63586 -0.81814 -0.22937 1.21245 -1.85500 1.97656 -1.54298 0.67603 0.38078 -1.33065 1.90679 -1.94739 1.44104 -0.52996 0.00000 2.00000 -1.71914 0.07661 1.79233 1.63586 -0.22937 -1.85500 -1.54298 0.38078 1.90679 1.44104 -0.52996 -1.94739 -1.33065 0.67603 1.97656 1.21245 -0.81814 -1.99413 -1.08714 0.95544 0.00000 ... ... ..... ....... ......... ........... ............. E17" 2.00000 -1.71914 0.95544 0.07661 -1.08714 1.79233 -1.99413 1.63586 -0.81814 -0.22937 1.21245 -1.85500 1.97656 -1.54298 0.67603 0.38078 -1.33065 1.90679 -1.94739 1.44104 -0.52996 0.00000 -2.00000 1.71914 -0.07661 -1.79233 -1.63586 0.22937 1.85500 1.54298 -0.38078 -1.90679 -1.44104 0.52996 1.94739 1.33065 -0.67603 -1.97656 -1.21245 0.81814 1.99413 1.08714 -0.95544 0.00000 ... ... ..... ....... ......... ........... ............. E18' 2.00000 -1.85500 1.44104 -0.81814 0.07661 0.67603 -1.33065 1.79233 -1.99413 1.90679 -1.54298 0.95544 -0.22937 -0.52996 1.21245 -1.71914 1.97656 -1.94739 1.63586 -1.08714 0.38078 0.00000 2.00000 -1.85500 -0.81814 0.67603 1.79233 1.90679 0.95544 -0.52996 -1.71914 -1.94739 -1.08714 0.38078 1.63586 1.97656 1.21245 -0.22937 -1.54298 -1.99413 -1.33065 0.07661 1.44104 0.00000 ... ... ..... ....... ......... ........... ............. E18" 2.00000 -1.85500 1.44104 -0.81814 0.07661 0.67603 -1.33065 1.79233 -1.99413 1.90679 -1.54298 0.95544 -0.22937 -0.52996 1.21245 -1.71914 1.97656 -1.94739 1.63586 -1.08714 0.38078 0.00000 -2.00000 1.85500 0.81814 -0.67603 -1.79233 -1.90679 -0.95544 0.52996 1.71914 1.94739 1.08714 -0.38078 -1.63586 -1.97656 -1.21245 0.22937 1.54298 1.99413 1.33065 -0.07661 -1.44104 0.00000 ... ... ..... ....... ......... ........... ............. E19' 2.00000 -1.94739 1.79233 -1.54298 1.21245 -0.81814 0.38078 0.07661 -0.52996 0.95544 -1.33065 1.63586 -1.85500 1.97656 -1.99413 1.90679 -1.71914 1.44104 -1.08714 0.67603 -0.22937 0.00000 2.00000 -1.94739 -1.54298 -0.81814 0.07661 0.95544 1.63586 1.97656 1.90679 1.44104 0.67603 -0.22937 -1.08714 -1.71914 -1.99413 -1.85500 -1.33065 -0.52996 0.38078 1.21245 1.79233 0.00000 ... ... ..... ....... ......... ........... ............. E19" 2.00000 -1.94739 1.79233 -1.54298 1.21245 -0.81814 0.38078 0.07661 -0.52996 0.95544 -1.33065 1.63586 -1.85500 1.97656 -1.99413 1.90679 -1.71914 1.44104 -1.08714 0.67603 -0.22937 0.00000 -2.00000 1.94739 1.54298 0.81814 -0.07661 -0.95544 -1.63586 -1.97656 -1.90679 -1.44104 -0.67603 0.22937 1.08714 1.71914 1.99413 1.85500 1.33065 0.52996 -0.38078 -1.21245 -1.79233 0.00000 ... ... ..... ....... ......... ........... ............. E20' 2.00000 -1.99413 1.97656 -1.94739 1.90679 -1.85500 1.79233 -1.71914 1.63586 -1.54298 1.44104 -1.33065 1.21245 -1.08714 0.95544 -0.81814 0.67603 -0.52996 0.38078 -0.22937 0.07661 0.00000 2.00000 -1.99413 -1.94739 -1.85500 -1.71914 -1.54298 -1.33065 -1.08714 -0.81814 -0.52996 -0.22937 0.07661 0.38078 0.67603 0.95544 1.21245 1.44104 1.63586 1.79233 1.90679 1.97656 0.00000 ... ... ..... ....... ......... ........... ............. E20" 2.00000 -1.99413 1.97656 -1.94739 1.90679 -1.85500 1.79233 -1.71914 1.63586 -1.54298 1.44104 -1.33065 1.21245 -1.08714 0.95544 -0.81814 0.67603 -0.52996 0.38078 -0.22937 0.07661 0.00000 -2.00000 1.99413 1.94739 1.85500 1.71914 1.54298 1.33065 1.08714 0.81814 0.52996 0.22937 -0.07661 -0.38078 -0.67603 -0.95544 -1.21245 -1.44104 -1.63586 -1.79233 -1.90679 -1.97656 0.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.994131602367 = 2*cos(2*π/82) = 2*cos(π/41) 1.976560847561 = 2*cos(4*π/82) = 2*cos(2*π/41) 1.947390847756 = 2*cos(6*π/82) = 2*cos(3*π/41) 1.906792784110 = 2*cos(8*π/82) = 2*cos(4*π/41) 1.855004902204 = 2*cos(10*π/82) = 2*cos(5*π/41) 1.792331113922 = 2*cos(12*π/82) = 2*cos(6*π/41) 1.719139213974 = 2*cos(14*π/82) = 2*cos(7*π/41) 1.635858721533 = 2*cos(16*π/82) = 2*cos(8*π/41) 1.542978359644 = 2*cos(18*π/82) = 2*cos(9*π/41) 1.441043187202 = 2*cos(20*π/82) = 2*cos(10*π/41) 1.330651400331 = 2*cos(22*π/82) = 2*cos(11*π/41) 1.212450821933 = 2*cos(24*π/82) = 2*cos(12*π/41) 1.087135100002 = 2*cos(26*π/82) = 2*cos(13*π/41) 0.955439637025 = 2*cos(28*π/82) = 2*cos(14*π/41) 0.818137274343 = 2*cos(30*π/82) = 2*cos(15*π/41) 0.676033756817 = 2*cos(32*π/82) = 2*cos(16*π/41) 0.529963004393 = 2*cos(34*π/82) = 2*cos(17*π/41) 0.380782218329 = 2*cos(36*π/82) = 2*cos(18*π/41) 0.229366850797 = 2*cos(38*π/82) = 2*cos(19*π/41) 0.076605467380 = 2*cos(40*π/82) = 2*cos(20*π/41) 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_41h point group is 164, and the order of the principal axis (S₄₁) is 82. The group has 44 irreducible representations. β The D_41h point group is isomorphic to D_41d, C_82v and D₈₂. γ The D_41h point group is generated by two symmetry elements, S₄₁ 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: C₄₁, 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_41h 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. For this group, however, none of the irrational characters can be expressed by a closed algebraic form using real numbers only.

This Character Table for the D_41h point group was created by Gernot Katzer.

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

	D_39h
	D_40h
C₄₁ C_41v C_41h D₄₁	D_41h	D_41d
	D_42h
	D_43h

Character table for the D41h point group

Character table for the D_41h point group