Character Table for Point Group D88d

Character table for the D_88d point group

D88d E 2 S176 2 C88 2 S176^3 2 C44 2 S176^5 2 C88^3 2 S176^7 2 C22 2 S176^9 2 C88^5 2 S16 2 C44^3 2 S176^13 2 C88^7 2 S176^15 2 C11 2 S176^17 2 C88^9 2 S176^19 2 C44^5 2 S176^21 2 C8 2 S176^23 2 C22^3 2 S176^25 2 C88^13 2 S176^27 2 C44^7 2 S176^29 2 C88^15 2 S176^31 2 C11^2 2 S16^3 2 C88^17 2 S176^35 2 C44^9 2 S176^37 2 C88^19 2 S176^39 2 C22^5 2 S176^41 2 C88^21 2 S176^43 2 C4 2 S176^45 2 C88^23 2 S176^47 2 C11^3 2 S176^49 2 C88^25 2 S176^51 2 C44^13 2 S176^53 2 C88^27 2 S16^5 2 C22^7 2 S176^57 2 C88^29 2 S176^59 2 C44^15 2 S176^61 2 C88^31 2 S176^63 2 C11^4 2 S176^65 2 C8^3 2 S176^67 2 C44^17 2 S176^69 2 C88^35 2 S176^71 2 C22^9 2 S176^73 2 C88^37 2 S176^75 2 C44^19 2 S16^7 2 C88^39 2 S176^79 2 C11^5 2 S176^81 2 C88^41 2 S176^83 2 C44^21 2 S176^85 2 C88^43 2 S176^87 C2 88 C2' 88 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 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 -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 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 -1.00000 1.00000 ... ..T ..... ......T ......... ..........T ............. E1 2.00000 1.99873 1.99490 1.98854 1.97964 1.96822 1.95429 1.93787 1.91899 1.89765 1.87390 1.84776 1.81926 1.78845 1.75536 1.72003 1.68251 1.64284 1.60108 1.55728 1.51150 1.46379 1.41421 1.36284 1.30972 1.25494 1.19856 1.14065 1.08128 1.02054 0.95850 0.89523 0.83083 0.76537 0.69893 0.63160 0.56347 0.49461 0.42513 0.35511 0.28463 0.21379 0.14268 0.07138 0.00000 -0.07138 -0.14268 -0.21379 -0.28463 -0.35511 -0.42513 -0.49461 -0.56347 -0.63160 -0.69893 -0.76537 -0.83083 -0.89523 -0.95850 -1.02054 -1.08128 -1.14065 -1.19856 -1.25494 -1.30972 -1.36284 -1.41421 -1.46379 -1.51150 -1.55728 -1.60108 -1.64284 -1.68251 -1.72003 -1.75536 -1.78845 -1.81926 -1.84776 -1.87390 -1.89765 -1.91899 -1.93787 -1.95429 -1.96822 -1.97964 -1.98854 -1.99490 -1.99873 -2.00000 0.00000 0.00000 ... TT. ..... ....TT. ......... ........TT. ............. E2 2.00000 1.99490 1.97964 1.95429 1.91899 1.87390 1.81926 1.75536 1.68251 1.60108 1.51150 1.41421 1.30972 1.19856 1.08128 0.95850 0.83083 0.69893 0.56347 0.42513 0.28463 0.14268 0.00000 -0.14268 -0.28463 -0.42513 -0.56347 -0.69893 -0.83083 -0.95850 -1.08128 -1.19856 -1.30972 -1.41421 -1.51150 -1.60108 -1.68251 -1.75536 -1.81926 -1.87390 -1.91899 -1.95429 -1.97964 -1.99490 -2.00000 -1.99490 -1.97964 -1.95429 -1.91899 -1.87390 -1.81926 -1.75536 -1.68251 -1.60108 -1.51150 -1.41421 -1.30972 -1.19856 -1.08128 -0.95850 -0.83083 -0.69893 -0.56347 -0.42513 -0.28463 -0.14268 0.00000 0.14268 0.28463 0.42513 0.56347 0.69893 0.83083 0.95850 1.08128 1.19856 1.30972 1.41421 1.51150 1.60108 1.68251 1.75536 1.81926 1.87390 1.91899 1.95429 1.97964 1.99490 2.00000 0.00000 0.00000 ... ... TT... ....... ....TT... ........... ........TT... E3 2.00000 1.98854 1.95429 1.89765 1.81926 1.72003 1.60108 1.46379 1.30972 1.14065 0.95850 0.76537 0.56347 0.35511 0.14268 -0.07138 -0.28463 -0.49461 -0.69893 -0.89523 -1.08128 -1.25494 -1.41421 -1.55728 -1.68251 -1.78845 -1.87390 -1.93787 -1.97964 -1.99873 -1.99490 -1.96822 -1.91899 -1.84776 -1.75536 -1.64284 -1.51150 -1.36284 -1.19856 -1.02054 -0.83083 -0.63160 -0.42513 -0.21379 0.00000 0.21379 0.42513 0.63160 0.83083 1.02054 1.19856 1.36284 1.51150 1.64284 1.75536 1.84776 1.91899 1.96822 1.99490 1.99873 1.97964 1.93787 1.87390 1.78845 1.68251 1.55728 1.41421 1.25494 1.08128 0.89523 0.69893 0.49461 0.28463 0.07138 -0.14268 -0.35511 -0.56347 -0.76537 -0.95850 -1.14065 -1.30972 -1.46379 -1.60108 -1.72003 -1.81926 -1.89765 -1.95429 -1.98854 -2.00000 0.00000 0.00000 ... ... ..... TT..... ......... ....TT..... ............. E4 2.00000 1.97964 1.91899 1.81926 1.68251 1.51150 1.30972 1.08128 0.83083 0.56347 0.28463 0.00000 -0.28463 -0.56347 -0.83083 -1.08128 -1.30972 -1.51150 -1.68251 -1.81926 -1.91899 -1.97964 -2.00000 -1.97964 -1.91899 -1.81926 -1.68251 -1.51150 -1.30972 -1.08128 -0.83083 -0.56347 -0.28463 0.00000 0.28463 0.56347 0.83083 1.08128 1.30972 1.51150 1.68251 1.81926 1.91899 1.97964 2.00000 1.97964 1.91899 1.81926 1.68251 1.51150 1.30972 1.08128 0.83083 0.56347 0.28463 0.00000 -0.28463 -0.56347 -0.83083 -1.08128 -1.30972 -1.51150 -1.68251 -1.81926 -1.91899 -1.97964 -2.00000 -1.97964 -1.91899 -1.81926 -1.68251 -1.51150 -1.30972 -1.08128 -0.83083 -0.56347 -0.28463 0.00000 0.28463 0.56347 0.83083 1.08128 1.30972 1.51150 1.68251 1.81926 1.91899 1.97964 2.00000 0.00000 0.00000 ... ... ..... ....... TT....... ........... ....TT....... E5 2.00000 1.96822 1.87390 1.72003 1.51150 1.25494 0.95850 0.63160 0.28463 -0.07138 -0.42513 -0.76537 -1.08128 -1.36284 -1.60108 -1.78845 -1.91899 -1.98854 -1.99490 -1.93787 -1.81926 -1.64284 -1.41421 -1.14065 -0.83083 -0.49461 -0.14268 0.21379 0.56347 0.89523 1.19856 1.46379 1.68251 1.84776 1.95429 1.99873 1.97964 1.89765 1.75536 1.55728 1.30972 1.02054 0.69893 0.35511 0.00000 -0.35511 -0.69893 -1.02054 -1.30972 -1.55728 -1.75536 -1.89765 -1.97964 -1.99873 -1.95429 -1.84776 -1.68251 -1.46379 -1.19856 -0.89523 -0.56347 -0.21379 0.14268 0.49461 0.83083 1.14065 1.41421 1.64284 1.81926 1.93787 1.99490 1.98854 1.91899 1.78845 1.60108 1.36284 1.08128 0.76537 0.42513 0.07138 -0.28463 -0.63160 -0.95850 -1.25494 -1.51150 -1.72003 -1.87390 -1.96822 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... TT......... ............. E6 2.00000 1.95429 1.81926 1.60108 1.30972 0.95850 0.56347 0.14268 -0.28463 -0.69893 -1.08128 -1.41421 -1.68251 -1.87390 -1.97964 -1.99490 -1.91899 -1.75536 -1.51150 -1.19856 -0.83083 -0.42513 0.00000 0.42513 0.83083 1.19856 1.51150 1.75536 1.91899 1.99490 1.97964 1.87390 1.68251 1.41421 1.08128 0.69893 0.28463 -0.14268 -0.56347 -0.95850 -1.30972 -1.60108 -1.81926 -1.95429 -2.00000 -1.95429 -1.81926 -1.60108 -1.30972 -0.95850 -0.56347 -0.14268 0.28463 0.69893 1.08128 1.41421 1.68251 1.87390 1.97964 1.99490 1.91899 1.75536 1.51150 1.19856 0.83083 0.42513 0.00000 -0.42513 -0.83083 -1.19856 -1.51150 -1.75536 -1.91899 -1.99490 -1.97964 -1.87390 -1.68251 -1.41421 -1.08128 -0.69893 -0.28463 0.14268 0.56347 0.95850 1.30972 1.60108 1.81926 1.95429 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... TT........... E7 2.00000 1.93787 1.75536 1.46379 1.08128 0.63160 0.14268 -0.35511 -0.83083 -1.25494 -1.60108 -1.84776 -1.97964 -1.98854 -1.87390 -1.64284 -1.30972 -0.89523 -0.42513 0.07138 0.56347 1.02054 1.41421 1.72003 1.91899 1.99873 1.95429 1.78845 1.51150 1.14065 0.69893 0.21379 -0.28463 -0.76537 -1.19856 -1.55728 -1.81926 -1.96822 -1.99490 -1.89765 -1.68251 -1.36284 -0.95850 -0.49461 0.00000 0.49461 0.95850 1.36284 1.68251 1.89765 1.99490 1.96822 1.81926 1.55728 1.19856 0.76537 0.28463 -0.21379 -0.69893 -1.14065 -1.51150 -1.78845 -1.95429 -1.99873 -1.91899 -1.72003 -1.41421 -1.02054 -0.56347 -0.07138 0.42513 0.89523 1.30972 1.64284 1.87390 1.98854 1.97964 1.84776 1.60108 1.25494 0.83083 0.35511 -0.14268 -0.63160 -1.08128 -1.46379 -1.75536 -1.93787 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E8 2.00000 1.91899 1.68251 1.30972 0.83083 0.28463 -0.28463 -0.83083 -1.30972 -1.68251 -1.91899 -2.00000 -1.91899 -1.68251 -1.30972 -0.83083 -0.28463 0.28463 0.83083 1.30972 1.68251 1.91899 2.00000 1.91899 1.68251 1.30972 0.83083 0.28463 -0.28463 -0.83083 -1.30972 -1.68251 -1.91899 -2.00000 -1.91899 -1.68251 -1.30972 -0.83083 -0.28463 0.28463 0.83083 1.30972 1.68251 1.91899 2.00000 1.91899 1.68251 1.30972 0.83083 0.28463 -0.28463 -0.83083 -1.30972 -1.68251 -1.91899 -2.00000 -1.91899 -1.68251 -1.30972 -0.83083 -0.28463 0.28463 0.83083 1.30972 1.68251 1.91899 2.00000 1.91899 1.68251 1.30972 0.83083 0.28463 -0.28463 -0.83083 -1.30972 -1.68251 -1.91899 -2.00000 -1.91899 -1.68251 -1.30972 -0.83083 -0.28463 0.28463 0.83083 1.30972 1.68251 1.91899 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E9 2.00000 1.89765 1.60108 1.14065 0.56347 -0.07138 -0.69893 -1.25494 -1.68251 -1.93787 -1.99490 -1.84776 -1.51150 -1.02054 -0.42513 0.21379 0.83083 1.36284 1.75536 1.96822 1.97964 1.78845 1.41421 0.89523 0.28463 -0.35511 -0.95850 -1.46379 -1.81926 -1.98854 -1.95429 -1.72003 -1.30972 -0.76537 -0.14268 0.49461 1.08128 1.55728 1.87390 1.99873 1.91899 1.64284 1.19856 0.63160 0.00000 -0.63160 -1.19856 -1.64284 -1.91899 -1.99873 -1.87390 -1.55728 -1.08128 -0.49461 0.14268 0.76537 1.30972 1.72003 1.95429 1.98854 1.81926 1.46379 0.95850 0.35511 -0.28463 -0.89523 -1.41421 -1.78845 -1.97964 -1.96822 -1.75536 -1.36284 -0.83083 -0.21379 0.42513 1.02054 1.51150 1.84776 1.99490 1.93787 1.68251 1.25494 0.69893 0.07138 -0.56347 -1.14065 -1.60108 -1.89765 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E10 2.00000 1.87390 1.51150 0.95850 0.28463 -0.42513 -1.08128 -1.60108 -1.91899 -1.99490 -1.81926 -1.41421 -0.83083 -0.14268 0.56347 1.19856 1.68251 1.95429 1.97964 1.75536 1.30972 0.69893 0.00000 -0.69893 -1.30972 -1.75536 -1.97964 -1.95429 -1.68251 -1.19856 -0.56347 0.14268 0.83083 1.41421 1.81926 1.99490 1.91899 1.60108 1.08128 0.42513 -0.28463 -0.95850 -1.51150 -1.87390 -2.00000 -1.87390 -1.51150 -0.95850 -0.28463 0.42513 1.08128 1.60108 1.91899 1.99490 1.81926 1.41421 0.83083 0.14268 -0.56347 -1.19856 -1.68251 -1.95429 -1.97964 -1.75536 -1.30972 -0.69893 0.00000 0.69893 1.30972 1.75536 1.97964 1.95429 1.68251 1.19856 0.56347 -0.14268 -0.83083 -1.41421 -1.81926 -1.99490 -1.91899 -1.60108 -1.08128 -0.42513 0.28463 0.95850 1.51150 1.87390 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E11 2.00000 1.84776 1.41421 0.76537 0.00000 -0.76537 -1.41421 -1.84776 -2.00000 -1.84776 -1.41421 -0.76537 0.00000 0.76537 1.41421 1.84776 2.00000 1.84776 1.41421 0.76537 0.00000 -0.76537 -1.41421 -1.84776 -2.00000 -1.84776 -1.41421 -0.76537 0.00000 0.76537 1.41421 1.84776 2.00000 1.84776 1.41421 0.76537 0.00000 -0.76537 -1.41421 -1.84776 -2.00000 -1.84776 -1.41421 -0.76537 0.00000 0.76537 1.41421 1.84776 2.00000 1.84776 1.41421 0.76537 0.00000 -0.76537 -1.41421 -1.84776 -2.00000 -1.84776 -1.41421 -0.76537 0.00000 0.76537 1.41421 1.84776 2.00000 1.84776 1.41421 0.76537 0.00000 -0.76537 -1.41421 -1.84776 -2.00000 -1.84776 -1.41421 -0.76537 0.00000 0.76537 1.41421 1.84776 2.00000 1.84776 1.41421 0.76537 0.00000 -0.76537 -1.41421 -1.84776 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E12 2.00000 1.81926 1.30972 0.56347 -0.28463 -1.08128 -1.68251 -1.97964 -1.91899 -1.51150 -0.83083 0.00000 0.83083 1.51150 1.91899 1.97964 1.68251 1.08128 0.28463 -0.56347 -1.30972 -1.81926 -2.00000 -1.81926 -1.30972 -0.56347 0.28463 1.08128 1.68251 1.97964 1.91899 1.51150 0.83083 0.00000 -0.83083 -1.51150 -1.91899 -1.97964 -1.68251 -1.08128 -0.28463 0.56347 1.30972 1.81926 2.00000 1.81926 1.30972 0.56347 -0.28463 -1.08128 -1.68251 -1.97964 -1.91899 -1.51150 -0.83083 0.00000 0.83083 1.51150 1.91899 1.97964 1.68251 1.08128 0.28463 -0.56347 -1.30972 -1.81926 -2.00000 -1.81926 -1.30972 -0.56347 0.28463 1.08128 1.68251 1.97964 1.91899 1.51150 0.83083 0.00000 -0.83083 -1.51150 -1.91899 -1.97964 -1.68251 -1.08128 -0.28463 0.56347 1.30972 1.81926 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. 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E83 2.00000 -1.96822 1.87390 -1.72003 1.51150 -1.25494 0.95850 -0.63160 0.28463 0.07138 -0.42513 0.76537 -1.08128 1.36284 -1.60108 1.78845 -1.91899 1.98854 -1.99490 1.93787 -1.81926 1.64284 -1.41421 1.14065 -0.83083 0.49461 -0.14268 -0.21379 0.56347 -0.89523 1.19856 -1.46379 1.68251 -1.84776 1.95429 -1.99873 1.97964 -1.89765 1.75536 -1.55728 1.30972 -1.02054 0.69893 -0.35511 0.00000 0.35511 -0.69893 1.02054 -1.30972 1.55728 -1.75536 1.89765 -1.97964 1.99873 -1.95429 1.84776 -1.68251 1.46379 -1.19856 0.89523 -0.56347 0.21379 0.14268 -0.49461 0.83083 -1.14065 1.41421 -1.64284 1.81926 -1.93787 1.99490 -1.98854 1.91899 -1.78845 1.60108 -1.36284 1.08128 -0.76537 0.42513 -0.07138 -0.28463 0.63160 -0.95850 1.25494 -1.51150 1.72003 -1.87390 1.96822 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ..TT......... E84 2.00000 -1.97964 1.91899 -1.81926 1.68251 -1.51150 1.30972 -1.08128 0.83083 -0.56347 0.28463 0.00000 -0.28463 0.56347 -0.83083 1.08128 -1.30972 1.51150 -1.68251 1.81926 -1.91899 1.97964 -2.00000 1.97964 -1.91899 1.81926 -1.68251 1.51150 -1.30972 1.08128 -0.83083 0.56347 -0.28463 0.00000 0.28463 -0.56347 0.83083 -1.08128 1.30972 -1.51150 1.68251 -1.81926 1.91899 -1.97964 2.00000 -1.97964 1.91899 -1.81926 1.68251 -1.51150 1.30972 -1.08128 0.83083 -0.56347 0.28463 0.00000 -0.28463 0.56347 -0.83083 1.08128 -1.30972 1.51150 -1.68251 1.81926 -1.91899 1.97964 -2.00000 1.97964 -1.91899 1.81926 -1.68251 1.51150 -1.30972 1.08128 -0.83083 0.56347 -0.28463 0.00000 0.28463 -0.56347 0.83083 -1.08128 1.30972 -1.51150 1.68251 -1.81926 1.91899 -1.97964 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ..TT....... ............. E85 2.00000 -1.98854 1.95429 -1.89765 1.81926 -1.72003 1.60108 -1.46379 1.30972 -1.14065 0.95850 -0.76537 0.56347 -0.35511 0.14268 0.07138 -0.28463 0.49461 -0.69893 0.89523 -1.08128 1.25494 -1.41421 1.55728 -1.68251 1.78845 -1.87390 1.93787 -1.97964 1.99873 -1.99490 1.96822 -1.91899 1.84776 -1.75536 1.64284 -1.51150 1.36284 -1.19856 1.02054 -0.83083 0.63160 -0.42513 0.21379 0.00000 -0.21379 0.42513 -0.63160 0.83083 -1.02054 1.19856 -1.36284 1.51150 -1.64284 1.75536 -1.84776 1.91899 -1.96822 1.99490 -1.99873 1.97964 -1.93787 1.87390 -1.78845 1.68251 -1.55728 1.41421 -1.25494 1.08128 -0.89523 0.69893 -0.49461 0.28463 -0.07138 -0.14268 0.35511 -0.56347 0.76537 -0.95850 1.14065 -1.30972 1.46379 -1.60108 1.72003 -1.81926 1.89765 -1.95429 1.98854 -2.00000 0.00000 0.00000 ... ... ..... ....... ..TT..... ........... ......TT..... E86 2.00000 -1.99490 1.97964 -1.95429 1.91899 -1.87390 1.81926 -1.75536 1.68251 -1.60108 1.51150 -1.41421 1.30972 -1.19856 1.08128 -0.95850 0.83083 -0.69893 0.56347 -0.42513 0.28463 -0.14268 0.00000 0.14268 -0.28463 0.42513 -0.56347 0.69893 -0.83083 0.95850 -1.08128 1.19856 -1.30972 1.41421 -1.51150 1.60108 -1.68251 1.75536 -1.81926 1.87390 -1.91899 1.95429 -1.97964 1.99490 -2.00000 1.99490 -1.97964 1.95429 -1.91899 1.87390 -1.81926 1.75536 -1.68251 1.60108 -1.51150 1.41421 -1.30972 1.19856 -1.08128 0.95850 -0.83083 0.69893 -0.56347 0.42513 -0.28463 0.14268 0.00000 -0.14268 0.28463 -0.42513 0.56347 -0.69893 0.83083 -0.95850 1.08128 -1.19856 1.30972 -1.41421 1.51150 -1.60108 1.68251 -1.75536 1.81926 -1.87390 1.91899 -1.95429 1.97964 -1.99490 2.00000 0.00000 0.00000 ... ... ..... ..TT... ......... ......TT... ............. E87 2.00000 -1.99873 1.99490 -1.98854 1.97964 -1.96822 1.95429 -1.93787 1.91899 -1.89765 1.87390 -1.84776 1.81926 -1.78845 1.75536 -1.72003 1.68251 -1.64284 1.60108 -1.55728 1.51150 -1.46379 1.41421 -1.36284 1.30972 -1.25494 1.19856 -1.14065 1.08128 -1.02054 0.95850 -0.89523 0.83083 -0.76537 0.69893 -0.63160 0.56347 -0.49461 0.42513 -0.35511 0.28463 -0.21379 0.14268 -0.07138 0.00000 0.07138 -0.14268 0.21379 -0.28463 0.35511 -0.42513 0.49461 -0.56347 0.63160 -0.69893 0.76537 -0.83083 0.89523 -0.95850 1.02054 -1.08128 1.14065 -1.19856 1.25494 -1.30972 1.36284 -1.41421 1.46379 -1.51150 1.55728 -1.60108 1.64284 -1.68251 1.72003 -1.75536 1.78845 -1.81926 1.84776 -1.87390 1.89765 -1.91899 1.93787 -1.95429 1.96822 -1.97964 1.98854 -1.99490 1.99873 -2.00000 0.00000 0.00000 TT. ... ..TT. ....... ......TT. ........... ..........TT. Irrational character values: 1.998725651314 = 2*cos(2*π/176) = 2*cos(π/88) 1.994904229221 = 2*cos(4*π/176) = 2*cos(π/44) 1.988540603544 = 2*cos(6*π/176) = 2*cos(3*π/88) 1.979642883762 = 2*cos(8*π/176) = 2*cos(π/22) 1.968222408672 = 2*cos(10*π/176) = 2*cos(5*π/88) 1.954293731942 = 2*cos(12*π/176) = 2*cos(3*π/44) 1.937874603563 = 2*cos(14*π/176) = 2*cos(7*π/88) 1.918985947229 = 2*cos(16*π/176) = 2*cos(π/11) 1.897651833675 = 2*cos(18*π/176) = 2*cos(9*π/88) 1.873899450000 = 2*cos(20*π/176) = 2*cos(5*π/44) 1.847759065023 = 2*cos(22*π/176) = 2*cos(π/8) = √2+√2 1.819263990709 = 2*cos(24*π/176) = 2*cos(3*π/22) 1.788450539719 = 2*cos(26*π/176) = 2*cos(13*π/88) 1.755357979135 = 2*cos(28*π/176) = 2*cos(7*π/44) 1.720028480415 = 2*cos(30*π/176) = 2*cos(15*π/88) 1.682507065662 = 2*cos(32*π/176) = 2*cos(2*π/11) 1.642841550241 = 2*cos(34*π/176) = 2*cos(17*π/88) 1.601082481849 = 2*cos(36*π/176) = 2*cos(9*π/44) 1.557283076100 = 2*cos(38*π/176) = 2*cos(19*π/88) 1.511499148709 = 2*cos(40*π/176) = 2*cos(5*π/22) 1.463789044363 = 2*cos(42*π/176) = 2*cos(21*π/88) 1.414213562373 = 2*cos(44*π/176) = 2*cos(π/4) = √2 1.362835879188 = 2*cos(46*π/176) = 2*cos(23*π/88) 1.309721467891 = 2*cos(48*π/176) = 2*cos(3*π/11) 1.254938014762 = 2*cos(50*π/176) = 2*cos(25*π/88) 1.198555333023 = 2*cos(52*π/176) = 2*cos(13*π/44) 1.140645273870 = 2*cos(54*π/176) = 2*cos(27*π/88) 1.081281634911 = 2*cos(56*π/176) = 2*cos(7*π/22) 1.020540066122 = 2*cos(58*π/176) = 2*cos(29*π/88) 0.958497973440 = 2*cos(60*π/176) = 2*cos(15*π/44) 0.895234420125 = 2*cos(62*π/176) = 2*cos(31*π/88) 0.830830026004 = 2*cos(64*π/176) = 2*cos(4*π/11) 0.765366864730 = 2*cos(66*π/176) = 2*cos(3*π/8) = √2−√2 0.698928359198 = 2*cos(68*π/176) = 2*cos(17*π/44) 0.631599175230 = 2*cos(70*π/176) = 2*cos(35*π/88) 0.563465113683 = 2*cos(72*π/176) = 2*cos(9*π/22) 0.494613001108 = 2*cos(74*π/176) = 2*cos(37*π/88) 0.425130579106 = 2*cos(76*π/176) = 2*cos(19*π/44) 0.355106392509 = 2*cos(78*π/176) = 2*cos(39*π/88) 0.284629676547 = 2*cos(80*π/176) = 2*cos(5*π/11) 0.213790243130 = 2*cos(82*π/176) = 2*cos(41*π/88) 0.142678366398 = 2*cos(84*π/176) = 2*cos(21*π/44) 0.071384667678 = 2*cos(86*π/176) = 2*cos(43*π/88) 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 E79 m E80 l E81 k+m E82 j+l E83 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)} E84 h+j+l {z((x²−y²)²−4x²y²), xyz(x²−y²)} E85 g+i+k+m {xz(x²−3y²), yz(3x²−y²)}, {xz³(x²−3y²), yz³(3x²−y²)} E86 f+h+j+l {z(x²−y²), xyz}, {z³(x²−y²), xyz³} E87 R+d+g+i+k+m {R_x, R_y}, {xz, yz}, {xz³, yz³}, {xz⁵, yz⁵} Notes: α The order of the D_88d point group is 352, and the order of the principal axis (S₁₇₆) is 176. The group has 91 irreducible representations. β The D_88d point group is isomorphic to C_176v and D₁₇₆. γ The D_88d point group is generated by two symmetry elements, S₁₇₆ 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_88d 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, some of the irrational characters cannot be expressed by a closed algebraic form using real numbers only.

This Character Table for the D_88d point group was created by Gernot Katzer.

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

	D_86d
	D_87d
C₈₈ C_88v C_88h D₈₈ D_88h	D_88d	S₈₈
	D_89d
	D_90d

Character table for the D88d point group

Character table for the D_88d point group