Character Table for Point Group D86d

Character table for the D_86d point group

D86d E 2 S172 2 C86 2 S172^3 2 C43 2 S172^5 2 C86^3 2 S172^7 2 C43^2 2 S172^9 2 C86^5 2 S172^11 2 C43^3 2 S172^13 2 C86^7 2 S172^15 2 C43^4 2 S172^17 2 C86^9 2 S172^19 2 C43^5 2 S172^21 2 C86^11 2 S172^23 2 C43^6 2 S172^25 2 C86^13 2 S172^27 2 C43^7 2 S172^29 2 C86^15 2 S172^31 2 C43^8 2 S172^33 2 C86^17 2 S172^35 2 C43^9 2 S172^37 2 C86^19 2 S172^39 2 C43^10 2 S172^41 2 C86^21 2 S4 2 C43^11 2 S172^45 2 C86^23 2 S172^47 2 C43^12 2 S172^49 2 C86^25 2 S172^51 2 C43^13 2 S172^53 2 C86^27 2 S172^55 2 C43^14 2 S172^57 2 C86^29 2 S172^59 2 C43^15 2 S172^61 2 C86^31 2 S172^63 2 C43^16 2 S172^65 2 C86^33 2 S172^67 2 C43^17 2 S172^69 2 C86^35 2 S172^71 2 C43^18 2 S172^73 2 C86^37 2 S172^75 2 C43^19 2 S172^77 2 C86^39 2 S172^79 2 C43^20 2 S172^81 2 C86^41 2 S172^83 2 C43^21 2 S172^85 C2 86 C2' 86 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 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 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 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 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 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 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -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 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -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 1.99867 1.99466 1.98800 1.97869 1.96673 1.95215 1.93497 1.91520 1.89288 1.86803 1.84069 1.81090 1.77868 1.74410 1.70719 1.66800 1.62658 1.58299 1.53729 1.48954 1.43981 1.38815 1.33464 1.27935 1.22235 1.16372 1.10354 1.04188 0.97884 0.91448 0.84891 0.78221 0.71446 0.64576 0.57620 0.50587 0.43486 0.36327 0.29120 0.21874 0.14599 0.07304 0.00000 -0.07304 -0.14599 -0.21874 -0.29120 -0.36327 -0.43486 -0.50587 -0.57620 -0.64576 -0.71446 -0.78221 -0.84891 -0.91448 -0.97884 -1.04188 -1.10354 -1.16372 -1.22235 -1.27935 -1.33464 -1.38815 -1.43981 -1.48954 -1.53729 -1.58299 -1.62658 -1.66800 -1.70719 -1.74410 -1.77868 -1.81090 -1.84069 -1.86803 -1.89288 -1.91520 -1.93497 -1.95215 -1.96673 -1.97869 -1.98800 -1.99466 -1.99867 -2.00000 0.00000 0.00000 ... TT. ..... ....TT. ......... ........TT. ............. E2 2.00000 1.99466 1.97869 1.95215 1.91520 1.86803 1.81090 1.74410 1.66800 1.58299 1.48954 1.38815 1.27935 1.16372 1.04188 0.91448 0.78221 0.64576 0.50587 0.36327 0.21874 0.07304 -0.07304 -0.21874 -0.36327 -0.50587 -0.64576 -0.78221 -0.91448 -1.04188 -1.16372 -1.27935 -1.38815 -1.48954 -1.58299 -1.66800 -1.74410 -1.81090 -1.86803 -1.91520 -1.95215 -1.97869 -1.99466 -2.00000 -1.99466 -1.97869 -1.95215 -1.91520 -1.86803 -1.81090 -1.74410 -1.66800 -1.58299 -1.48954 -1.38815 -1.27935 -1.16372 -1.04188 -0.91448 -0.78221 -0.64576 -0.50587 -0.36327 -0.21874 -0.07304 0.07304 0.21874 0.36327 0.50587 0.64576 0.78221 0.91448 1.04188 1.16372 1.27935 1.38815 1.48954 1.58299 1.66800 1.74410 1.81090 1.86803 1.91520 1.95215 1.97869 1.99466 2.00000 0.00000 0.00000 ... ... TT... ....... ....TT... ........... ........TT... E3 2.00000 1.98800 1.95215 1.89288 1.81090 1.70719 1.58299 1.43981 1.27935 1.10354 0.91448 0.71446 0.50587 0.29120 0.07304 -0.14599 -0.36327 -0.57620 -0.78221 -0.97884 -1.16372 -1.33464 -1.48954 -1.62658 -1.74410 -1.84069 -1.91520 -1.96673 -1.99466 -1.99867 -1.97869 -1.93497 -1.86803 -1.77868 -1.66800 -1.53729 -1.38815 -1.22235 -1.04188 -0.84891 -0.64576 -0.43486 -0.21874 0.00000 0.21874 0.43486 0.64576 0.84891 1.04188 1.22235 1.38815 1.53729 1.66800 1.77868 1.86803 1.93497 1.97869 1.99867 1.99466 1.96673 1.91520 1.84069 1.74410 1.62658 1.48954 1.33464 1.16372 0.97884 0.78221 0.57620 0.36327 0.14599 -0.07304 -0.29120 -0.50587 -0.71446 -0.91448 -1.10354 -1.27935 -1.43981 -1.58299 -1.70719 -1.81090 -1.89288 -1.95215 -1.98800 -2.00000 0.00000 0.00000 ... ... ..... TT..... ......... ....TT..... ............. E4 2.00000 1.97869 1.91520 1.81090 1.66800 1.48954 1.27935 1.04188 0.78221 0.50587 0.21874 -0.07304 -0.36327 -0.64576 -0.91448 -1.16372 -1.38815 -1.58299 -1.74410 -1.86803 -1.95215 -1.99466 -1.99466 -1.95215 -1.86803 -1.74410 -1.58299 -1.38815 -1.16372 -0.91448 -0.64576 -0.36327 -0.07304 0.21874 0.50587 0.78221 1.04188 1.27935 1.48954 1.66800 1.81090 1.91520 1.97869 2.00000 1.97869 1.91520 1.81090 1.66800 1.48954 1.27935 1.04188 0.78221 0.50587 0.21874 -0.07304 -0.36327 -0.64576 -0.91448 -1.16372 -1.38815 -1.58299 -1.74410 -1.86803 -1.95215 -1.99466 -1.99466 -1.95215 -1.86803 -1.74410 -1.58299 -1.38815 -1.16372 -0.91448 -0.64576 -0.36327 -0.07304 0.21874 0.50587 0.78221 1.04188 1.27935 1.48954 1.66800 1.81090 1.91520 1.97869 2.00000 0.00000 0.00000 ... ... ..... ....... TT....... ........... ....TT....... E5 2.00000 1.96673 1.86803 1.70719 1.48954 1.22235 0.91448 0.57620 0.21874 -0.14599 -0.50587 -0.84891 -1.16372 -1.43981 -1.66800 -1.84069 -1.95215 -1.99867 -1.97869 -1.89288 -1.74410 -1.53729 -1.27935 -0.97884 -0.64576 -0.29120 0.07304 0.43486 0.78221 1.10354 1.38815 1.62658 1.81090 1.93497 1.99466 1.98800 1.91520 1.77868 1.58299 1.33464 1.04188 0.71446 0.36327 0.00000 -0.36327 -0.71446 -1.04188 -1.33464 -1.58299 -1.77868 -1.91520 -1.98800 -1.99466 -1.93497 -1.81090 -1.62658 -1.38815 -1.10354 -0.78221 -0.43486 -0.07304 0.29120 0.64576 0.97884 1.27935 1.53729 1.74410 1.89288 1.97869 1.99867 1.95215 1.84069 1.66800 1.43981 1.16372 0.84891 0.50587 0.14599 -0.21874 -0.57620 -0.91448 -1.22235 -1.48954 -1.70719 -1.86803 -1.96673 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... TT......... ............. E6 2.00000 1.95215 1.81090 1.58299 1.27935 0.91448 0.50587 0.07304 -0.36327 -0.78221 -1.16372 -1.48954 -1.74410 -1.91520 -1.99466 -1.97869 -1.86803 -1.66800 -1.38815 -1.04188 -0.64576 -0.21874 0.21874 0.64576 1.04188 1.38815 1.66800 1.86803 1.97869 1.99466 1.91520 1.74410 1.48954 1.16372 0.78221 0.36327 -0.07304 -0.50587 -0.91448 -1.27935 -1.58299 -1.81090 -1.95215 -2.00000 -1.95215 -1.81090 -1.58299 -1.27935 -0.91448 -0.50587 -0.07304 0.36327 0.78221 1.16372 1.48954 1.74410 1.91520 1.99466 1.97869 1.86803 1.66800 1.38815 1.04188 0.64576 0.21874 -0.21874 -0.64576 -1.04188 -1.38815 -1.66800 -1.86803 -1.97869 -1.99466 -1.91520 -1.74410 -1.48954 -1.16372 -0.78221 -0.36327 0.07304 0.50587 0.91448 1.27935 1.58299 1.81090 1.95215 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... TT........... E7 2.00000 1.93497 1.74410 1.43981 1.04188 0.57620 0.07304 -0.43486 -0.91448 -1.33464 -1.66800 -1.89288 -1.99466 -1.96673 -1.81090 -1.53729 -1.16372 -0.71446 -0.21874 0.29120 0.78221 1.22235 1.58299 1.84069 1.97869 1.98800 1.86803 1.62658 1.27935 0.84891 0.36327 -0.14599 -0.64576 -1.10354 -1.48954 -1.77868 -1.95215 -1.99867 -1.91520 -1.70719 -1.38815 -0.97884 -0.50587 0.00000 0.50587 0.97884 1.38815 1.70719 1.91520 1.99867 1.95215 1.77868 1.48954 1.10354 0.64576 0.14599 -0.36327 -0.84891 -1.27935 -1.62658 -1.86803 -1.98800 -1.97869 -1.84069 -1.58299 -1.22235 -0.78221 -0.29120 0.21874 0.71446 1.16372 1.53729 1.81090 1.96673 1.99466 1.89288 1.66800 1.33464 0.91448 0.43486 -0.07304 -0.57620 -1.04188 -1.43981 -1.74410 -1.93497 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E8 2.00000 1.91520 1.66800 1.27935 0.78221 0.21874 -0.36327 -0.91448 -1.38815 -1.74410 -1.95215 -1.99466 -1.86803 -1.58299 -1.16372 -0.64576 -0.07304 0.50587 1.04188 1.48954 1.81090 1.97869 1.97869 1.81090 1.48954 1.04188 0.50587 -0.07304 -0.64576 -1.16372 -1.58299 -1.86803 -1.99466 -1.95215 -1.74410 -1.38815 -0.91448 -0.36327 0.21874 0.78221 1.27935 1.66800 1.91520 2.00000 1.91520 1.66800 1.27935 0.78221 0.21874 -0.36327 -0.91448 -1.38815 -1.74410 -1.95215 -1.99466 -1.86803 -1.58299 -1.16372 -0.64576 -0.07304 0.50587 1.04188 1.48954 1.81090 1.97869 1.97869 1.81090 1.48954 1.04188 0.50587 -0.07304 -0.64576 -1.16372 -1.58299 -1.86803 -1.99466 -1.95215 -1.74410 -1.38815 -0.91448 -0.36327 0.21874 0.78221 1.27935 1.66800 1.91520 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E9 2.00000 1.89288 1.58299 1.10354 0.50587 -0.14599 -0.78221 -1.33464 -1.74410 -1.96673 -1.97869 -1.77868 -1.38815 -0.84891 -0.21874 0.43486 1.04188 1.53729 1.86803 1.99867 1.91520 1.62658 1.16372 0.57620 -0.07304 -0.71446 -1.27935 -1.70719 -1.95215 -1.98800 -1.81090 -1.43981 -0.91448 -0.29120 0.36327 0.97884 1.48954 1.84069 1.99466 1.93497 1.66800 1.22235 0.64576 0.00000 -0.64576 -1.22235 -1.66800 -1.93497 -1.99466 -1.84069 -1.48954 -0.97884 -0.36327 0.29120 0.91448 1.43981 1.81090 1.98800 1.95215 1.70719 1.27935 0.71446 0.07304 -0.57620 -1.16372 -1.62658 -1.91520 -1.99867 -1.86803 -1.53729 -1.04188 -0.43486 0.21874 0.84891 1.38815 1.77868 1.97869 1.96673 1.74410 1.33464 0.78221 0.14599 -0.50587 -1.10354 -1.58299 -1.89288 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E10 2.00000 1.86803 1.48954 0.91448 0.21874 -0.50587 -1.16372 -1.66800 -1.95215 -1.97869 -1.74410 -1.27935 -0.64576 0.07304 0.78221 1.38815 1.81090 1.99466 1.91520 1.58299 1.04188 0.36327 -0.36327 -1.04188 -1.58299 -1.91520 -1.99466 -1.81090 -1.38815 -0.78221 -0.07304 0.64576 1.27935 1.74410 1.97869 1.95215 1.66800 1.16372 0.50587 -0.21874 -0.91448 -1.48954 -1.86803 -2.00000 -1.86803 -1.48954 -0.91448 -0.21874 0.50587 1.16372 1.66800 1.95215 1.97869 1.74410 1.27935 0.64576 -0.07304 -0.78221 -1.38815 -1.81090 -1.99466 -1.91520 -1.58299 -1.04188 -0.36327 0.36327 1.04188 1.58299 1.91520 1.99466 1.81090 1.38815 0.78221 0.07304 -0.64576 -1.27935 -1.74410 -1.97869 -1.95215 -1.66800 -1.16372 -0.50587 0.21874 0.91448 1.48954 1.86803 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E11 2.00000 1.84069 1.38815 0.71446 -0.07304 -0.84891 -1.48954 -1.89288 -1.99466 -1.77868 -1.27935 -0.57620 0.21874 0.97884 1.58299 1.93497 1.97869 1.70719 1.16372 0.43486 -0.36327 -1.10354 -1.66800 -1.96673 -1.95215 -1.62658 -1.04188 -0.29120 0.50587 1.22235 1.74410 1.98800 1.91520 1.53729 0.91448 0.14599 -0.64576 -1.33464 -1.81090 -1.99867 -1.86803 -1.43981 -0.78221 0.00000 0.78221 1.43981 1.86803 1.99867 1.81090 1.33464 0.64576 -0.14599 -0.91448 -1.53729 -1.91520 -1.98800 -1.74410 -1.22235 -0.50587 0.29120 1.04188 1.62658 1.95215 1.96673 1.66800 1.10354 0.36327 -0.43486 -1.16372 -1.70719 -1.97869 -1.93497 -1.58299 -0.97884 -0.21874 0.57620 1.27935 1.77868 1.99466 1.89288 1.48954 0.84891 0.07304 -0.71446 -1.38815 -1.84069 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E12 2.00000 1.81090 1.27935 0.50587 -0.36327 -1.16372 -1.74410 -1.99466 -1.86803 -1.38815 -0.64576 0.21874 1.04188 1.66800 1.97869 1.91520 1.48954 0.78221 -0.07304 -0.91448 -1.58299 -1.95215 -1.95215 -1.58299 -0.91448 -0.07304 0.78221 1.48954 1.91520 1.97869 1.66800 1.04188 0.21874 -0.64576 -1.38815 -1.86803 -1.99466 -1.74410 -1.16372 -0.36327 0.50587 1.27935 1.81090 2.00000 1.81090 1.27935 0.50587 -0.36327 -1.16372 -1.74410 -1.99466 -1.86803 -1.38815 -0.64576 0.21874 1.04188 1.66800 1.97869 1.91520 1.48954 0.78221 -0.07304 -0.91448 -1.58299 -1.95215 -1.95215 -1.58299 -0.91448 -0.07304 0.78221 1.48954 1.91520 1.97869 1.66800 1.04188 0.21874 -0.64576 -1.38815 -1.86803 -1.99466 -1.74410 -1.16372 -0.36327 0.50587 1.27935 1.81090 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E13 2.00000 1.77868 1.16372 0.29120 -0.64576 -1.43981 -1.91520 -1.96673 -1.58299 -0.84891 0.07304 0.97884 1.66800 1.98800 1.86803 1.33464 0.50587 -0.43486 -1.27935 -1.84069 -1.99466 -1.70719 -1.04188 -0.14599 0.78221 1.53729 1.95215 1.93497 1.48954 0.71446 -0.21874 -1.10354 -1.74410 -1.99867 -1.81090 -1.22235 -0.36327 0.57620 1.38815 1.89288 1.97869 1.62658 0.91448 0.00000 -0.91448 -1.62658 -1.97869 -1.89288 -1.38815 -0.57620 0.36327 1.22235 1.81090 1.99867 1.74410 1.10354 0.21874 -0.71446 -1.48954 -1.93497 -1.95215 -1.53729 -0.78221 0.14599 1.04188 1.70719 1.99466 1.84069 1.27935 0.43486 -0.50587 -1.33464 -1.86803 -1.98800 -1.66800 -0.97884 -0.07304 0.84891 1.58299 1.96673 1.91520 1.43981 0.64576 -0.29120 -1.16372 -1.77868 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. 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E84 2.00000 -1.99466 1.97869 -1.95215 1.91520 -1.86803 1.81090 -1.74410 1.66800 -1.58299 1.48954 -1.38815 1.27935 -1.16372 1.04188 -0.91448 0.78221 -0.64576 0.50587 -0.36327 0.21874 -0.07304 -0.07304 0.21874 -0.36327 0.50587 -0.64576 0.78221 -0.91448 1.04188 -1.16372 1.27935 -1.38815 1.48954 -1.58299 1.66800 -1.74410 1.81090 -1.86803 1.91520 -1.95215 1.97869 -1.99466 2.00000 -1.99466 1.97869 -1.95215 1.91520 -1.86803 1.81090 -1.74410 1.66800 -1.58299 1.48954 -1.38815 1.27935 -1.16372 1.04188 -0.91448 0.78221 -0.64576 0.50587 -0.36327 0.21874 -0.07304 -0.07304 0.21874 -0.36327 0.50587 -0.64576 0.78221 -0.91448 1.04188 -1.16372 1.27935 -1.38815 1.48954 -1.58299 1.66800 -1.74410 1.81090 -1.86803 1.91520 -1.95215 1.97869 -1.99466 2.00000 0.00000 0.00000 ... ... ..... ..TT... ......... ......TT... ............. E85 2.00000 -1.99867 1.99466 -1.98800 1.97869 -1.96673 1.95215 -1.93497 1.91520 -1.89288 1.86803 -1.84069 1.81090 -1.77868 1.74410 -1.70719 1.66800 -1.62658 1.58299 -1.53729 1.48954 -1.43981 1.38815 -1.33464 1.27935 -1.22235 1.16372 -1.10354 1.04188 -0.97884 0.91448 -0.84891 0.78221 -0.71446 0.64576 -0.57620 0.50587 -0.43486 0.36327 -0.29120 0.21874 -0.14599 0.07304 0.00000 -0.07304 0.14599 -0.21874 0.29120 -0.36327 0.43486 -0.50587 0.57620 -0.64576 0.71446 -0.78221 0.84891 -0.91448 0.97884 -1.04188 1.10354 -1.16372 1.22235 -1.27935 1.33464 -1.38815 1.43981 -1.48954 1.53729 -1.58299 1.62658 -1.66800 1.70719 -1.74410 1.77868 -1.81090 1.84069 -1.86803 1.89288 -1.91520 1.93497 -1.95215 1.96673 -1.97869 1.98800 -1.99466 1.99867 -2.00000 0.00000 0.00000 TT. ... ..TT. ....... ......TT. ........... ..........TT. Irrational character values: 1.998665696740 = 2*cos(2*π/172) = 2*cos(π/86) 1.994664567327 = 2*cos(4*π/172) = 2*cos(π/43) 1.988001950480 = 2*cos(6*π/172) = 2*cos(3*π/86) 1.978686736150 = 2*cos(8*π/172) = 2*cos(2*π/43) 1.966731353659 = 2*cos(10*π/172) = 2*cos(5*π/86) 1.952151755112 = 2*cos(12*π/172) = 2*cos(3*π/43) 1.934967394115 = 2*cos(14*π/172) = 2*cos(7*π/86) 1.915201199817 = 2*cos(16*π/172) = 2*cos(4*π/43) 1.892879546315 = 2*cos(18*π/172) = 2*cos(9*π/86) 1.868032217465 = 2*cos(20*π/172) = 2*cos(5*π/43) 1.840692367138 = 2*cos(22*π/172) = 2*cos(11*π/86) 1.810896474986 = 2*cos(24*π/172) = 2*cos(6*π/43) 1.778684297765 = 2*cos(26*π/172) = 2*cos(13*π/86) 1.744098816288 = 2*cos(28*π/172) = 2*cos(7*π/43) 1.707186178075 = 2*cos(30*π/172) = 2*cos(15*π/86) 1.667995635780 = 2*cos(32*π/172) = 2*cos(8*π/43) 1.626579481471 = 2*cos(34*π/172) = 2*cos(17*π/86) 1.582992976859 = 2*cos(36*π/172) = 2*cos(9*π/43) 1.537294279557 = 2*cos(38*π/172) = 2*cos(19*π/86) 1.489544365488 = 2*cos(40*π/172) = 2*cos(10*π/43) 1.439806947516 = 2*cos(42*π/172) = 2*cos(21*π/86) 1.388148390441 = 2*cos(44*π/172) = 2*cos(11*π/43) 1.334637622444 = 2*cos(46*π/172) = 2*cos(23*π/86) 1.279346043118 = 2*cos(48*π/172) = 2*cos(12*π/43) 1.222347428196 = 2*cos(50*π/172) = 2*cos(25*π/86) 1.163717831116 = 2*cos(52*π/172) = 2*cos(13*π/43) 1.103535481541 = 2*cos(54*π/172) = 2*cos(27*π/86) 1.041880680976 = 2*cos(56*π/172) = 2*cos(14*π/43) 0.978835695622 = 2*cos(58*π/172) = 2*cos(29*π/86) 0.914484646609 = 2*cos(60*π/172) = 2*cos(15*π/43) 0.848913397752 = 2*cos(62*π/172) = 2*cos(31*π/86) 0.782209440980 = 2*cos(64*π/172) = 2*cos(16*π/43) 0.714461779602 = 2*cos(66*π/172) = 2*cos(33*π/86) 0.645760809543 = 2*cos(68*π/172) = 2*cos(17*π/43) 0.576198198730 = 2*cos(70*π/172) = 2*cos(35*π/86) 0.505866764783 = 2*cos(72*π/172) = 2*cos(18*π/43) 0.434860351163 = 2*cos(74*π/172) = 2*cos(37*π/86) 0.363273701959 = 2*cos(76*π/172) = 2*cos(19*π/43) 0.291202335470 = 2*cos(78*π/172) = 2*cos(39*π/86) 0.218742416756 = 2*cos(80*π/172) = 2*cos(20*π/43) 0.145990629322 = 2*cos(82*π/172) = 2*cos(41*π/86) 0.073044046115 = 2*cos(84*π/172) = 2*cos(21*π/43) Symmetry of Rotations and Cartesian products A1 d+g+i+k+m z², z⁴, z⁶ A2 R R_z B2 p+f+h+j+l z, z³, z⁵ E1 p+f+h+j+l {x, y}, {xz², yz²}, {xz⁴, yz⁴} E2 d+g+i+k+m {x²−y², xy}, {z²(x²−y²), xyz²}, {z⁴(x²−y²), xyz⁴} E3 f+h+j+l {x(x²−3y²), y(3x²−y²)}, {xz²(x²−3y²), yz²(3x²−y²)} E4 g+i+k+m {(x²−y²)²−4x²y², xy(x²−y²)}, {z²((x²−y²)²−4x²y²), xyz²(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)} E6 i+k+m {x²(x²−3y²)²−y²(3x²−y²)², xy(x²−3y²)(3x²−y²)} E7 j+l E8 k+m E9 l E10 m E77 m E78 l E79 k+m E80 j+l E81 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)} E82 h+j+l {z((x²−y²)²−4x²y²), xyz(x²−y²)} E83 g+i+k+m {xz(x²−3y²), yz(3x²−y²)}, {xz³(x²−3y²), yz³(3x²−y²)} E84 f+h+j+l {z(x²−y²), xyz}, {z³(x²−y²), xyz³} E85 R+d+g+i+k+m {R_x, R_y}, {xz, yz}, {xz³, yz³}, {xz⁵, yz⁵} Notes: α The order of the D_86d point group is 344, and the order of the principal axis (S₁₇₂) is 172. The group has 89 irreducible representations. β The D_86d point group is isomorphic to C_172v and D₁₇₂. γ The D_86d point group is generated by two symmetry elements, S₁₇₂ and either a perpendicular C₂^′ or a vertical σ_d. Also, the group may be generated from a C₂^′ plus a σ_d (some pairs will yield smaller groups, though; choosing a minimum angle is safe). δ The group contains one set of twofold symmetry axes (C₂^′) perpendicular to the principal (z) axis. Both x and y axes are members of that set. ε The single σ_d set of symmetry planes contains neither the xz nor the yz planes; but it contains the median plane (x+y)z. ζ The lowest nonvanishing multipole moment in D_86d is 4 (quadrupole moment). η This point group is non-Abelian (some symmetry operations are not commutative). Therefore, the character table contains multi-membered classes and degenerate irreducible representations. 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, none of the irrational characters can be expressed by a closed algebraic form using real numbers only.

This Character Table for the D_86d point group was created by Gernot Katzer.

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

	D_84d
	D_85d
C₈₆ C_86v C_86h D₈₆ D_86h	D_86d	S₈₆
	D_87d
	D_88d

Character table for the D86d point group

Character table for the D_86d point group