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E29 2.00000 -0.39392 -1.84483 1.12063 1.40339 -1.67345 -0.74418 1.96660 -0.03050 -1.95459 0.80045 1.63928 -1.44619 -1.06959 1.86752 0.33394 -1.99907 0.45353 1.82041 -1.17063 -1.35928 1.70608 0.68722 -1.97679 0.09147 1.94076 -0.85597 -1.60357 1.48765 1.01756 -1.88849 -0.27365 1.99628 -0.51273 -1.79431 1.21954 1.31391 -1.73711 -0.62963 1.98513 -0.15236 -1.92512 0.91070 1.56638 -1.52772 -0.96458 1.90769 0.21310 -1.99163 0.57144 1.76653 -1.26731 0.00000 ... ... ..... ....... ......... ........... ............. E30 2.00000 -0.51273 -1.73711 1.40339 1.01756 -1.92512 -0.03050 1.94076 -0.96458 -1.44619 1.70608 0.57144 -1.99907 0.45353 1.76653 -1.35928 -1.06959 1.90769 0.09147 -1.95459 0.91070 1.48765 -1.67345 -0.62963 1.99628 -0.39392 -1.79431 1.31391 1.12063 -1.88849 -0.15236 1.96660 -0.85597 -1.52772 1.63928 0.68722 -1.99163 0.33394 1.82041 -1.26731 -1.17063 1.86752 0.21310 -1.97679 0.80045 1.56638 -1.60357 -0.74418 1.98513 -0.27365 -1.84483 1.21954 0.00000 ... ... ..... ....... ......... ........... ............. E31 2.00000 -0.62963 -1.60357 1.63928 0.57144 -1.99907 0.68722 1.56638 -1.67345 -0.51273 1.99628 -0.74418 -1.52772 1.70608 0.45353 -1.99163 0.80045 1.48765 -1.73711 -0.39392 1.98513 -0.85597 -1.44619 1.76653 0.33394 -1.97679 0.91070 1.40339 -1.79431 -0.27365 1.96660 -0.96458 -1.35928 1.82041 0.21310 -1.95459 1.01756 1.31391 -1.84483 -0.15236 1.94076 -1.06959 -1.26731 1.86752 0.09147 -1.92512 1.12063 1.21954 -1.88849 -0.03050 1.90769 -1.17063 0.00000 ... ... ..... ....... ......... ........... ............. E32 2.00000 -0.74418 -1.44619 1.82041 0.09147 -1.88849 1.31391 0.91070 -1.99163 0.57144 1.56638 -1.73711 -0.27365 1.94076 -1.17063 -1.06959 1.96660 -0.39392 -1.67345 1.63928 0.45353 -1.97679 1.01756 1.21954 -1.92512 0.21310 1.76653 -1.52772 -0.62963 1.99628 -0.85597 -1.35928 1.86752 -0.03050 -1.84483 1.40339 0.80045 -1.99907 0.68722 1.48765 -1.79431 -0.15236 1.90769 -1.26731 -0.96458 1.98513 -0.51273 -1.60357 1.70608 0.33394 -1.95459 1.12063 0.00000 ... ... ..... ....... ......... ........... ............. E33 2.00000 -0.85597 -1.26731 1.94076 -0.39392 -1.60357 1.76653 0.09147 -1.84483 1.48765 0.57144 -1.97679 1.12063 1.01756 -1.99163 0.68722 1.40339 -1.88849 0.21310 1.70608 -1.67345 -0.27365 1.90769 -1.35928 -0.74418 1.99628 -0.96458 -1.17063 1.96660 -0.51273 -1.52772 1.82041 -0.03050 -1.79431 1.56638 0.45353 -1.95459 1.21954 0.91070 -1.99907 0.80045 1.31391 -1.92512 0.33394 1.63928 -1.73711 -0.15236 1.86752 -1.44619 -0.62963 1.98513 -1.06959 0.00000 ... ... ..... ....... ......... ........... ............. E34 2.00000 -0.96458 -1.06959 1.99628 -0.85597 -1.17063 1.98513 -0.74418 -1.26731 1.96660 -0.62963 -1.35928 1.94076 -0.51273 -1.44619 1.90769 -0.39392 -1.52772 1.86752 -0.27365 -1.60357 1.82041 -0.15236 -1.67345 1.76653 -0.03050 -1.73711 1.70608 0.09147 -1.79431 1.63928 0.21310 -1.84483 1.56638 0.33394 -1.88849 1.48765 0.45353 -1.92512 1.40339 0.57144 -1.95459 1.31391 0.68722 -1.97679 1.21954 0.80045 -1.99163 1.12063 0.91070 -1.99907 1.01756 0.00000 ... ... ..... ....... ......... ........... ............. E35 2.00000 -1.06959 -0.85597 1.98513 -1.26731 -0.62963 1.94076 -1.44619 -0.39392 1.86752 -1.60357 -0.15236 1.76653 -1.73711 0.09147 1.63928 -1.84483 0.33394 1.48765 -1.92512 0.57144 1.31391 -1.97679 0.80045 1.12063 -1.99907 1.01756 0.91070 -1.99163 1.21954 0.68722 -1.95459 1.40339 0.45353 -1.88849 1.56638 0.21310 -1.79431 1.70608 -0.03050 -1.67345 1.82041 -0.27365 -1.52772 1.90769 -0.51273 -1.35928 1.96660 -0.74418 -1.17063 1.99628 -0.96458 0.00000 ... ... ..... ....... ......... ........... ............. E36 2.00000 -1.17063 -0.62963 1.90769 -1.60357 -0.03050 1.63928 -1.88849 0.57144 1.21954 -1.99907 1.12063 0.68722 -1.92512 1.56638 0.09147 -1.67345 1.86752 -0.51273 -1.26731 1.99628 -1.06959 -0.74418 1.94076 -1.52772 -0.15236 1.70608 -1.84483 0.45353 1.31391 -1.99163 1.01756 0.80045 -1.95459 1.48765 0.21310 -1.73711 1.82041 -0.39392 -1.35928 1.98513 -0.96458 -0.85597 1.96660 -1.44619 -0.27365 1.76653 -1.79431 0.33394 1.40339 -1.97679 0.91070 0.00000 ... ... ..... ....... ......... ........... ............. E37 2.00000 -1.26731 -0.39392 1.76653 -1.84483 0.57144 1.12063 -1.99163 1.40339 0.21310 -1.67345 1.90769 -0.74418 -0.96458 1.96660 -1.52772 -0.03050 1.56638 -1.95459 0.91070 0.80045 -1.92512 1.63928 -0.15236 -1.44619 1.98513 -1.06959 -0.62963 1.86752 -1.73711 0.33394 1.31391 -1.99907 1.21954 0.45353 -1.79431 1.82041 -0.51273 -1.17063 1.99628 -1.35928 -0.27365 1.70608 -1.88849 0.68722 1.01756 -1.97679 1.48765 0.09147 -1.60357 1.94076 -0.85597 0.00000 ... ... ..... ....... ......... ........... ............. E38 2.00000 -1.35928 -0.15236 1.56638 -1.97679 1.12063 0.45353 -1.73711 1.90769 -0.85597 -0.74418 1.86752 -1.79431 0.57144 1.01756 -1.95459 1.63928 -0.27365 -1.26731 1.99628 -1.44619 -0.03050 1.48765 -1.99163 1.21954 0.33394 -1.67345 1.94076 -0.96458 -0.62963 1.82041 -1.84483 0.68722 0.91070 -1.92512 1.70608 -0.39392 -1.17063 1.98513 -1.52772 0.09147 1.40339 -1.99907 1.31391 0.21310 -1.60357 1.96660 -1.06959 -0.51273 1.76653 -1.88849 0.80045 0.00000 ... ... ..... ....... ......... ........... ............. E39 2.00000 -1.44619 0.09147 1.31391 -1.99163 1.56638 -0.27365 -1.17063 1.96660 -1.67345 0.45353 1.01756 -1.92512 1.76653 -0.62963 -0.85597 1.86752 -1.84483 0.80045 0.68722 -1.79431 1.90769 -0.96458 -0.51273 1.70608 -1.95459 1.12063 0.33394 -1.60357 1.98513 -1.26731 -0.15236 1.48765 -1.99907 1.40339 -0.03050 -1.35928 1.99628 -1.52772 0.21310 1.21954 -1.97679 1.63928 -0.39392 -1.06959 1.94076 -1.73711 0.57144 0.91070 -1.88849 1.82041 -0.74418 0.00000 ... ... ..... ....... ......... ........... ............. E40 2.00000 -1.52772 0.33394 1.01756 -1.88849 1.86752 -0.96458 -0.39392 1.56638 -1.99907 1.48765 -0.27365 -1.06959 1.90769 -1.84483 0.91070 0.45353 -1.60357 1.99628 -1.44619 0.21310 1.12063 -1.92512 1.82041 -0.85597 -0.51273 1.63928 -1.99163 1.40339 -0.15236 -1.17063 1.94076 -1.79431 0.80045 0.57144 -1.67345 1.98513 -1.35928 0.09147 1.21954 -1.95459 1.76653 -0.74418 -0.62963 1.70608 -1.97679 1.31391 -0.03050 -1.26731 1.96660 -1.73711 0.68722 0.00000 ... ... ..... ....... ......... ........... ............. E41 2.00000 -1.60357 0.57144 0.68722 -1.67345 1.99628 -1.52772 0.45353 0.80045 -1.73711 1.98513 -1.44619 0.33394 0.91070 -1.79431 1.96660 -1.35928 0.21310 1.01756 -1.84483 1.94076 -1.26731 0.09147 1.12063 -1.88849 1.90769 -1.17063 -0.03050 1.21954 -1.92512 1.86752 -1.06959 -0.15236 1.31391 -1.95459 1.82041 -0.96458 -0.27365 1.40339 -1.97679 1.76653 -0.85597 -0.39392 1.48765 -1.99163 1.70608 -0.74418 -0.51273 1.56638 -1.99907 1.63928 -0.62963 0.00000 ... ... ..... ....... ......... ........... ............. E42 2.00000 -1.67345 0.80045 0.33394 -1.35928 1.94076 -1.88849 1.21954 -0.15236 -0.96458 1.76653 -1.99163 1.56638 -0.62963 -0.51273 1.48765 -1.97679 1.82041 -1.06959 -0.03050 1.12063 -1.84483 1.96660 -1.44619 0.45353 0.68722 -1.60357 1.99628 -1.73711 0.91070 0.21310 -1.26731 1.90769 -1.92512 1.31391 -0.27365 -0.85597 1.70608 -1.99907 1.63928 -0.74418 -0.39392 1.40339 -1.95459 1.86752 -1.17063 0.09147 1.01756 -1.79431 1.98513 -1.52772 0.57144 0.00000 ... ... ..... ....... ......... ........... ............. E43 2.00000 -1.73711 1.01756 -0.03050 -0.96458 1.70608 -1.99907 1.76653 -1.06959 0.09147 0.91070 -1.67345 1.99628 -1.79431 1.12063 -0.15236 -0.85597 1.63928 -1.99163 1.82041 -1.17063 0.21310 0.80045 -1.60357 1.98513 -1.84483 1.21954 -0.27365 -0.74418 1.56638 -1.97679 1.86752 -1.26731 0.33394 0.68722 -1.52772 1.96660 -1.88849 1.31391 -0.39392 -0.62963 1.48765 -1.95459 1.90769 -1.35928 0.45353 0.57144 -1.44619 1.94076 -1.92512 1.40339 -0.51273 0.00000 ... ... ..... ....... ......... ........... ............. E44 2.00000 -1.79431 1.21954 -0.39392 -0.51273 1.31391 -1.84483 1.99628 -1.73711 1.12063 -0.27365 -0.62963 1.40339 -1.88849 1.98513 -1.67345 1.01756 -0.15236 -0.74418 1.48765 -1.92512 1.96660 -1.60357 0.91070 -0.03050 -0.85597 1.56638 -1.95459 1.94076 -1.52772 0.80045 0.09147 -0.96458 1.63928 -1.97679 1.90769 -1.44619 0.68722 0.21310 -1.06959 1.70608 -1.99163 1.86752 -1.35928 0.57144 0.33394 -1.17063 1.76653 -1.99907 1.82041 -1.26731 0.45353 0.00000 ... ... ..... ....... ......... ........... ............. E45 2.00000 -1.84483 1.40339 -0.74418 -0.03050 0.80045 -1.44619 1.86752 -1.99907 1.82041 -1.35928 0.68722 0.09147 -0.85597 1.48765 -1.88849 1.99628 -1.79431 1.31391 -0.62963 -0.15236 0.91070 -1.52772 1.90769 -1.99163 1.76653 -1.26731 0.57144 0.21310 -0.96458 1.56638 -1.92512 1.98513 -1.73711 1.21954 -0.51273 -0.27365 1.01756 -1.60357 1.94076 -1.97679 1.70608 -1.17063 0.45353 0.33394 -1.06959 1.63928 -1.95459 1.96660 -1.67345 1.12063 -0.39392 0.00000 ... ... ..... ....... ......... ........... ............. E46 2.00000 -1.88849 1.56638 -1.06959 0.45353 0.21310 -0.85597 1.40339 -1.79431 1.98513 -1.95459 1.70608 -1.26731 0.68722 -0.03050 -0.62963 1.21954 -1.67345 1.94076 -1.99163 1.82041 -1.44619 0.91070 -0.27365 -0.39392 1.01756 -1.52772 1.86752 -1.99907 1.90769 -1.60357 1.12063 -0.51273 -0.15236 0.80045 -1.35928 1.76653 -1.97679 1.96660 -1.73711 1.31391 -0.74418 0.09147 0.57144 -1.17063 1.63928 -1.92512 1.99628 -1.84483 1.48765 -0.96458 0.33394 0.00000 ... ... ..... ....... ......... ........... ............. E47 2.00000 -1.92512 1.70608 -1.35928 0.91070 -0.39392 -0.15236 0.68722 -1.17063 1.56638 -1.84483 1.98513 -1.97679 1.82041 -1.52772 1.12063 -0.62963 0.09147 0.45353 -0.96458 1.40339 -1.73711 1.94076 -1.99907 1.90769 -1.67345 1.31391 -0.85597 0.33394 0.21310 -0.74418 1.21954 -1.60357 1.86752 -1.99163 1.96660 -1.79431 1.48765 -1.06959 0.57144 -0.03050 -0.51273 1.01756 -1.44619 1.76653 -1.95459 1.99628 -1.88849 1.63928 -1.26731 0.80045 -0.27365 0.00000 ... ... ..... ....... ......... ........... ............. E48 2.00000 -1.95459 1.82041 -1.60357 1.31391 -0.96458 0.57144 -0.15236 -0.27365 0.68722 -1.06959 1.40339 -1.67345 1.86752 -1.97679 1.99628 -1.92512 1.76653 -1.52772 1.21954 -0.85597 0.45353 -0.03050 -0.39392 0.80045 -1.17063 1.48765 -1.73711 1.90769 -1.99163 1.98513 -1.88849 1.70608 -1.44619 1.12063 -0.74418 0.33394 0.09147 -0.51273 0.91070 -1.26731 1.56638 -1.79431 1.94076 -1.99907 1.96660 -1.84483 1.63928 -1.35928 1.01756 -0.62963 0.21310 0.00000 ... ... ..... ....... ......... ........... ............. E49 2.00000 -1.97679 1.90769 -1.79431 1.63928 -1.44619 1.21954 -0.96458 0.68722 -0.39392 0.09147 0.21310 -0.51273 0.80045 -1.06959 1.31391 -1.52772 1.70608 -1.84483 1.94076 -1.99163 1.99628 -1.95459 1.86752 -1.73711 1.56638 -1.35928 1.12063 -0.85597 0.57144 -0.27365 -0.03050 0.33394 -0.62963 0.91070 -1.17063 1.40339 -1.60357 1.76653 -1.88849 1.96660 -1.99907 1.98513 -1.92512 1.82041 -1.67345 1.48765 -1.26731 1.01756 -0.74418 0.45353 -0.15236 0.00000 ... ... ..... ....... ......... ........... ............. E50 2.00000 -1.99163 1.96660 -1.92512 1.86752 -1.79431 1.70608 -1.60357 1.48765 -1.35928 1.21954 -1.06959 0.91070 -0.74418 0.57144 -0.39392 0.21310 -0.03050 -0.15236 0.33394 -0.51273 0.68722 -0.85597 1.01756 -1.17063 1.31391 -1.44619 1.56638 -1.67345 1.76653 -1.84483 1.90769 -1.95459 1.98513 -1.99907 1.99628 -1.97679 1.94076 -1.88849 1.82041 -1.73711 1.63928 -1.52772 1.40339 -1.26731 1.12063 -0.96458 0.80045 -0.62963 0.45353 -0.27365 0.09147 0.00000 ... ... ..... ....... ......... ........... ............. E51 2.00000 -1.99907 1.99628 -1.99163 1.98513 -1.97679 1.96660 -1.95459 1.94076 -1.92512 1.90769 -1.88849 1.86752 -1.84483 1.82041 -1.79431 1.76653 -1.73711 1.70608 -1.67345 1.63928 -1.60357 1.56638 -1.52772 1.48765 -1.44619 1.40339 -1.35928 1.31391 -1.26731 1.21954 -1.17063 1.12063 -1.06959 1.01756 -0.96458 0.91070 -0.85597 0.80045 -0.74418 0.68722 -0.62963 0.57144 -0.51273 0.45353 -0.39392 0.33394 -0.27365 0.21310 -0.15236 0.09147 -0.03050 0.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.996279934320 = 2*cos(2*π/103) 1.985133576168 = 2*cos(4*π/103) 1.966602390729 = 2*cos(6*π/103) 1.940755315229 = 2*cos(8*π/103) 1.907688502487 = 2*cos(10*π/103) 1.867524963219 = 2*cos(12*π/103) 1.820414108428 = 2*cos(14*π/103) 1.766531193589 = 2*cos(16*π/103) 1.706076666683 = 2*cos(18*π/103) 1.639275422522 = 2*cos(20*π/103) 1.566375966121 = 2*cos(22*π/103) 1.487649488246 = 2*cos(24*π/103) 1.403388856566 = 2*cos(26*π/103) 1.313907526165 = 2*cos(28*π/103) 1.219538373468 = 2*cos(30*π/103) 1.120632457923 = 2*cos(32*π/103) 1.017557716030 = 2*cos(34*π/103) 0.910697592601 = 2*cos(36*π/103) 0.800449614313 = 2*cos(38*π/103) 0.687223910885 = 2*cos(40*π/103) 0.571441689372 = 2*cos(42*π/103) 0.453533667242 = 2*cos(44*π/103) 0.333938470082 = 2*cos(46*π/103) 0.213100999880 = 2*cos(48*π/103) 0.091470779962 = 2*cos(50*π/103) -0.030499717266 = 2*cos(52*π/103) -0.152356753542 = 2*cos(54*π/103) -0.273647012688 = 2*cos(56*π/103) -0.393919286973 = 2*cos(58*π/103) -0.512726155639 = 2*cos(60*π/103) -0.629625649329 = 2*cos(62*π/103) -0.744182894250 = 2*cos(64*π/103) -0.855971729926 = 2*cos(66*π/103) -0.964576294547 = 2*cos(68*π/103) -1.069592571998 = 2*cos(70*π/103) -1.170629894830 = 2*cos(72*π/103) -1.267312397567 = 2*cos(74*π/103) -1.359280414947 = 2*cos(76*π/103) -1.446191819906 = 2*cos(78*π/103) -1.527723296308 = 2*cos(80*π/103) -1.603571541707 = 2*cos(82*π/103) -1.673454395648 = 2*cos(84*π/103) -1.737111889324 = 2*cos(86*π/103) -1.794307212678 = 2*cos(88*π/103) -1.844827595350 = 2*cos(90*π/103) -1.888485098199 = 2*cos(92*π/103) -1.925117312447 = 2*cos(94*π/103) -1.954587963850 = 2*cos(96*π/103) -1.976787419650 = 2*cos(98*π/103) -1.991633096413 = 2*cos(100*π/103) -1.999069767247 = 2*cos(102*π/103) Symmetry of Rotations and Cartesian products A1 p+d+f+g+h+i+j+k+l+m z, z2, z3, z4, z5, z6 A2 R Rz E1 R+p+d+f+g+h+i+j+k+l+m {Rx, Ry}, {x, y}, {xz, yz}, {xz2, yz2}, {xz3, yz3}, {xz4, yz4}, {xz5, yz5} E2 d+f+g+h+i+j+k+l+m {x2−y2, xy}, {z(x2−y2), xyz}, {z2(x2−y2), xyz2}, {z3(x2−y2), xyz3}, {z4(x2−y2), xyz4} E3 f+g+h+i+j+k+l+m {x(x2−3y2), y(3x2−y2)}, {xz(x2−3y2), yz(3x2−y2)}, {xz2(x2−3y2), yz2(3x2−y2)}, {xz3(x2−3y2), yz3(3x2−y2)} E4 g+h+i+j+k+l+m {(x2−y2)2−4x2y2, xy(x2−y2)}, {z((x2−y2)2−4x2y2), xyz(x2−y2)}, {z2((x2−y2)2−4x2y2), xyz2(x2−y2)} E5 h+i+j+k+l+m {x(x2−(5+2√5)y2)(x2−(5−2√5)y2), y((5+2√5)x2−y2)((5−2√5)x2−y2)}, {xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), yz((5+2√5)x2−y2)((5−2√5)x2−y2)} E6 i+j+k+l+m {x2(x2−3y2)2−y2(3x2−y2)2, xy(x2−3y2)(3x2−y2)} E7 j+k+l+m E8 k+l+m E9 l+m E10 m Notes: α The order of the C103v point group is 206, and the order of the principal axis (C103) is 103. The group has 53 irreducible representations. β The C103v point group is isomorphic to D103. γ The C103v point group is generated by two symmetry elements, C103 and any σv. Also, the group may be generated from any two σv planes. δ The lowest nonvanishing multipole moment in C103v is 2 (dipole moment). ε This point group is non-Abelian (some symmetry operations are not commutative). Therefore, the character table contains multi-membered classes and degenerate irreducible representations. ζ 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.
C101v | ||
C102v | ||
C103 | C103v | C103h D103 D103h D103d |
C104v | ||
C105v |
This Character Table for the C103v point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.