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E48 2.00000 -1.44021 0.07419 1.33335 -1.99450 1.53913 -0.22217 -1.21916 1.97801 -1.62959 0.36893 1.09825 -1.95064 1.71107 -0.51365 -0.97130 1.91253 -1.78313 0.65555 0.83900 -1.86389 1.84538 -0.79384 -0.70209 1.80499 -1.89747 0.92776 0.56131 -1.73616 1.93912 -1.05657 -0.41744 1.65777 -1.97009 1.17957 0.27127 -1.57025 1.99022 -1.29607 -0.12361 1.47409 -1.99939 1.40544 -0.02474 -1.36982 1.99755 -1.50707 0.17294 1.25800 -1.98472 1.60041 -0.32020 -1.13926 1.96096 -1.68493 0.46569 1.01425 -1.92641 1.76019 -0.60862 -0.88365 1.88126 -1.82575 0.74819 0.00000 ... ... ..... ....... ......... ........... ............. E49 2.00000 -1.50707 0.27127 1.09825 -1.92641 1.80499 -0.79384 -0.60862 1.71107 -1.97009 1.25800 0.07419 -1.36982 1.99022 -1.62959 0.46569 0.92776 -1.86389 1.88126 -0.97130 -0.41744 1.60041 -1.99450 1.40544 -0.12361 -1.21916 1.96096 -1.73616 0.65555 0.74819 -1.78313 1.93912 -1.13926 -0.22217 1.47409 -1.99939 1.53913 -0.32020 -1.05657 1.91253 -1.82575 0.83900 0.56131 -1.68493 1.97801 -1.29607 -0.02474 1.33335 -1.98472 1.65777 -0.51365 -0.88365 1.84538 -1.89747 1.01425 0.36893 -1.57025 1.99755 -1.44021 0.17294 1.17957 -1.95064 1.76019 -0.70209 0.00000 ... ... ..... ....... ......... ........... ............. E50 2.00000 -1.57025 0.46569 0.83900 -1.78313 1.96096 -1.29607 0.07419 1.17957 -1.92641 1.84538 -0.97130 -0.32020 1.47409 -1.99450 1.65777 -0.60862 -0.70209 1.71107 -1.98472 1.40544 -0.22217 -1.05657 1.88126 -1.89747 1.09825 0.17294 -1.36982 1.97801 -1.73616 0.74819 0.56131 -1.62959 1.99755 -1.50707 0.36893 0.92776 -1.82575 1.93912 -1.21916 -0.02474 1.25800 -1.95064 1.80499 -0.88365 -0.41744 1.53913 -1.99939 1.60041 -0.51365 -0.79384 1.76019 -1.97009 1.33335 -0.12361 -1.13926 1.91253 -1.86389 1.01425 0.27127 -1.44021 1.99022 -1.68493 0.65555 0.00000 ... ... ..... ....... ......... ........... ............. E51 2.00000 -1.62959 0.65555 0.56131 -1.57025 1.99755 -1.68493 0.74819 0.46569 -1.50707 1.99022 -1.73616 0.83900 0.36893 -1.44021 1.97801 -1.78313 0.92776 0.27127 -1.36982 1.96096 -1.82575 1.01425 0.17294 -1.29607 1.93912 -1.86389 1.09825 0.07419 -1.21916 1.91253 -1.89747 1.17957 -0.02474 -1.13926 1.88126 -1.92641 1.25800 -0.12361 -1.05657 1.84538 -1.95064 1.33335 -0.22217 -0.97130 1.80499 -1.97009 1.40544 -0.32020 -0.88365 1.76019 -1.98472 1.47409 -0.41744 -0.79384 1.71107 -1.99450 1.53913 -0.51365 -0.70209 1.65777 -1.99939 1.60041 -0.60862 0.00000 ... ... ..... ....... ......... ........... ............. E52 2.00000 -1.68493 0.83900 0.27127 -1.29607 1.91253 -1.92641 1.33335 -0.32020 -0.79384 1.65777 -1.99939 1.71107 -0.88365 -0.22217 1.25800 -1.89747 1.93912 -1.36982 0.36893 0.74819 -1.62959 1.99755 -1.73616 0.92776 0.17294 -1.21916 1.88126 -1.95064 1.40544 -0.41744 -0.70209 1.60041 -1.99450 1.76019 -0.97130 -0.12361 1.17957 -1.86389 1.96096 -1.44021 0.46569 0.65555 -1.57025 1.99022 -1.78313 1.01425 0.07419 -1.13926 1.84538 -1.97009 1.47409 -0.51365 -0.60862 1.53913 -1.98472 1.80499 -1.05657 -0.02474 1.09825 -1.82575 1.97801 -1.50707 0.56131 0.00000 ... ... ..... ....... ......... ........... ............. E53 2.00000 -1.73616 1.01425 -0.02474 -0.97130 1.71107 -1.99939 1.76019 -1.05657 0.07419 0.92776 -1.68493 1.99755 -1.78313 1.09825 -0.12361 -0.88365 1.65777 -1.99450 1.80499 -1.13926 0.17294 0.83900 -1.62959 1.99022 -1.82575 1.17957 -0.22217 -0.79384 1.60041 -1.98472 1.84538 -1.21916 0.27127 0.74819 -1.57025 1.97801 -1.86389 1.25800 -0.32020 -0.70209 1.53913 -1.97009 1.88126 -1.29607 0.36893 0.65555 -1.50707 1.96096 -1.89747 1.33335 -0.41744 -0.60862 1.47409 -1.95064 1.91253 -1.36982 0.46569 0.56131 -1.44021 1.93912 -1.92641 1.40544 -0.51365 0.00000 ... ... ..... ....... ......... ........... ............. E54 2.00000 -1.78313 1.17957 -0.32020 -0.60862 1.40544 -1.89747 1.97801 -1.62959 0.92776 -0.02474 -0.88365 1.60041 -1.97009 1.91253 -1.44021 0.65555 0.27127 -1.13926 1.76019 -1.99939 1.80499 -1.21916 0.36893 0.56131 -1.36982 1.88126 -1.98472 1.65777 -0.97130 0.07419 0.83900 -1.57025 1.96096 -1.92641 1.47409 -0.70209 -0.22217 1.09825 -1.73616 1.99755 -1.82575 1.25800 -0.41744 -0.51365 1.33335 -1.86389 1.99022 -1.68493 1.01425 -0.12361 -0.79384 1.53913 -1.95064 1.93912 -1.50707 0.74819 0.17294 -1.05657 1.71107 -1.99450 1.84538 -1.29607 0.46569 0.00000 ... ... ..... ....... ......... ........... ............. E55 2.00000 -1.82575 1.33335 -0.60862 -0.22217 1.01425 -1.62959 1.96096 -1.95064 1.60041 -0.97130 0.17294 0.65555 -1.36982 1.84538 -1.99939 1.80499 -1.29607 0.56131 0.27127 -1.05657 1.65777 -1.97009 1.93912 -1.57025 0.92776 -0.12361 -0.70209 1.40544 -1.86389 1.99755 -1.78313 1.25800 -0.51365 -0.32020 1.09825 -1.68493 1.97801 -1.92641 1.53913 -0.88365 0.07419 0.74819 -1.44021 1.88126 -1.99450 1.76019 -1.21916 0.46569 0.36893 -1.13926 1.71107 -1.98472 1.91253 -1.50707 0.83900 -0.02474 -0.79384 1.47409 -1.89747 1.99022 -1.73616 1.17957 -0.41744 0.00000 ... ... ..... ....... ......... ........... ............. E56 2.00000 -1.86389 1.47409 -0.88365 0.17294 0.56131 -1.21916 1.71107 -1.97009 1.96096 -1.68493 1.17957 -0.51365 -0.22217 0.92776 -1.50707 1.88126 -1.99939 1.84538 -1.44021 0.83900 -0.12361 -0.60862 1.25800 -1.73616 1.97801 -1.95064 1.65777 -1.13926 0.46569 0.27127 -0.97130 1.53913 -1.89747 1.99755 -1.82575 1.40544 -0.79384 0.07419 0.65555 -1.29607 1.76019 -1.98472 1.93912 -1.62959 1.09825 -0.41744 -0.32020 1.01425 -1.57025 1.91253 -1.99450 1.80499 -1.36982 0.74819 -0.02474 -0.70209 1.33335 -1.78313 1.99022 -1.92641 1.60041 -1.05657 0.36893 0.00000 ... ... ..... ....... ......... ........... ............. E57 2.00000 -1.89747 1.60041 -1.13926 0.56131 0.07419 -0.70209 1.25800 -1.68493 1.93912 -1.99450 1.84538 -1.50707 1.01425 -0.41744 -0.22217 0.83900 -1.36982 1.76019 -1.97009 1.97801 -1.78313 1.40544 -0.88365 0.27127 0.36893 -0.97130 1.47409 -1.82575 1.99022 -1.95064 1.71107 -1.29607 0.74819 -0.12361 -0.51365 1.09825 -1.57025 1.88126 -1.99939 1.91253 -1.62959 1.17957 -0.60862 -0.02474 0.65555 -1.21916 1.65777 -1.92641 1.99755 -1.86389 1.53913 -1.05657 0.46569 0.17294 -0.79384 1.33335 -1.73616 1.96096 -1.98472 1.80499 -1.44021 0.92776 -0.32020 0.00000 ... ... ..... ....... ......... ........... ............. E58 2.00000 -1.92641 1.71107 -1.36982 0.92776 -0.41744 -0.12361 0.65555 -1.13926 1.53913 -1.82575 1.97801 -1.98472 1.84538 -1.57025 1.17957 -0.70209 0.17294 0.36893 -0.88365 1.33335 -1.68493 1.91253 -1.99939 1.93912 -1.73616 1.40544 -0.97130 0.46569 0.07419 -0.60862 1.09825 -1.50707 1.80499 -1.97009 1.99022 -1.86389 1.60041 -1.21916 0.74819 -0.22217 -0.32020 0.83900 -1.29607 1.65777 -1.89747 1.99755 -1.95064 1.76019 -1.44021 1.01425 -0.51365 -0.02474 0.56131 -1.05657 1.47409 -1.78313 1.96096 -1.99450 1.88126 -1.62959 1.25800 -0.79384 0.27127 0.00000 ... ... ..... ....... ......... ........... ............. E59 2.00000 -1.95064 1.80499 -1.57025 1.25800 -0.88365 0.46569 -0.02474 -0.41744 0.83900 -1.21916 1.53913 -1.78313 1.93912 -1.99939 1.96096 -1.82575 1.60041 -1.29607 0.92776 -0.51365 0.07419 0.36893 -0.79384 1.17957 -1.50707 1.76019 -1.92641 1.99755 -1.97009 1.84538 -1.62959 1.33335 -0.97130 0.56131 -0.12361 -0.32020 0.74819 -1.13926 1.47409 -1.73616 1.91253 -1.99450 1.97801 -1.86389 1.65777 -1.36982 1.01425 -0.60862 0.17294 0.27127 -0.70209 1.09825 -1.44021 1.71107 -1.89747 1.99022 -1.98472 1.88126 -1.68493 1.40544 -1.05657 0.65555 -0.22217 0.00000 ... ... ..... ....... ......... ........... ............. E60 2.00000 -1.97009 1.88126 -1.73616 1.53913 -1.29607 1.01425 -0.70209 0.36893 -0.02474 -0.32020 0.65555 -0.97130 1.25800 -1.50707 1.71107 -1.86389 1.96096 -1.99939 1.97801 -1.89747 1.76019 -1.57025 1.33335 -1.05657 0.74819 -0.41744 0.07419 0.27127 -0.60862 0.92776 -1.21916 1.47409 -1.68493 1.84538 -1.95064 1.99755 -1.98472 1.91253 -1.78313 1.60041 -1.36982 1.09825 -0.79384 0.46569 -0.12361 -0.22217 0.56131 -0.88365 1.17957 -1.44021 1.65777 -1.82575 1.93912 -1.99450 1.99022 -1.92641 1.80499 -1.62959 1.40544 -1.13926 0.83900 -0.51365 0.17294 0.00000 ... ... ..... ....... ......... ........... ............. E61 2.00000 -1.98472 1.93912 -1.86389 1.76019 -1.62959 1.47409 -1.29607 1.09825 -0.88365 0.65555 -0.41744 0.17294 0.07419 -0.32020 0.56131 -0.79384 1.01425 -1.21916 1.40544 -1.57025 1.71107 -1.82575 1.91253 -1.97009 1.99755 -1.99450 1.96096 -1.89747 1.80499 -1.68493 1.53913 -1.36982 1.17957 -0.97130 0.74819 -0.51365 0.27127 -0.02474 -0.22217 0.46569 -0.70209 0.92776 -1.13926 1.33335 -1.50707 1.65777 -1.78313 1.88126 -1.95064 1.99022 -1.99939 1.97801 -1.92641 1.84538 -1.73616 1.60041 -1.44021 1.25800 -1.05657 0.83900 -0.60862 0.36893 -0.12361 0.00000 ... ... ..... ....... ......... ........... ............. E62 2.00000 -1.99450 1.97801 -1.95064 1.91253 -1.86389 1.80499 -1.73616 1.65777 -1.57025 1.47409 -1.36982 1.25800 -1.13926 1.01425 -0.88365 0.74819 -0.60862 0.46569 -0.32020 0.17294 -0.02474 -0.12361 0.27127 -0.41744 0.56131 -0.70209 0.83900 -0.97130 1.09825 -1.21916 1.33335 -1.44021 1.53913 -1.62959 1.71107 -1.78313 1.84538 -1.89747 1.93912 -1.97009 1.99022 -1.99939 1.99755 -1.98472 1.96096 -1.92641 1.88126 -1.82575 1.76019 -1.68493 1.60041 -1.50707 1.40544 -1.29607 1.17957 -1.05657 0.92776 -0.79384 0.65555 -0.51365 0.36893 -0.22217 0.07419 0.00000 ... ... ..... ....... ......... ........... ............. E63 2.00000 -1.99939 1.99755 -1.99450 1.99022 -1.98472 1.97801 -1.97009 1.96096 -1.95064 1.93912 -1.92641 1.91253 -1.89747 1.88126 -1.86389 1.84538 -1.82575 1.80499 -1.78313 1.76019 -1.73616 1.71107 -1.68493 1.65777 -1.62959 1.60041 -1.57025 1.53913 -1.50707 1.47409 -1.44021 1.40544 -1.36982 1.33335 -1.29607 1.25800 -1.21916 1.17957 -1.13926 1.09825 -1.05657 1.01425 -0.97130 0.92776 -0.88365 0.83900 -0.79384 0.74819 -0.70209 0.65555 -0.60862 0.56131 -0.51365 0.46569 -0.41744 0.36893 -0.32020 0.27127 -0.22217 0.17294 -0.12361 0.07419 -0.02474 0.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.997552832429 = 2*cos(2*π/127) 1.990217318343 = 2*cos(4*π/127) 1.978011408976 = 2*cos(6*π/127) 1.960964974233 = 2*cos(8*π/127) 1.939119729596 = 2*cos(10*π/127) 1.912529134040 = 2*cos(12*π/127) 1.881258259208 = 2*cos(14*π/127) 1.845383630170 = 2*cos(16*π/127) 1.804993038155 = 2*cos(18*π/127) 1.760185325710 = 2*cos(20*π/127) 1.711070144817 = 2*cos(22*π/127) 1.657767688553 = 2*cos(24*π/127) 1.600408396960 = 2*cos(26*π/127) 1.539132637837 = 2*cos(28*π/127) 1.474090363235 = 2*cos(30*π/127) 1.405440742498 = 2*cos(32*π/127) 1.333351772753 = 2*cos(34*π/127) 1.257999867787 = 2*cos(36*π/127) 1.179569426341 = 2*cos(38*π/127) 1.098252380846 = 2*cos(40*π/127) 1.014247727740 = 2*cos(42*π/127) 0.927761040484 = 2*cos(44*π/127) 0.839003966496 = 2*cos(46*π/127) 0.748193709210 = 2*cos(48*π/127) 0.655552496540 = 2*cos(50*π/127) 0.561307037060 = 2*cos(52*π/127) 0.465687965201 = 2*cos(54*π/127) 0.368929276856 = 2*cos(56*π/127) 0.271267756747 = 2*cos(58*π/127) 0.172942398981 = 2*cos(60*π/127) 0.074193822185 = 2*cos(62*π/127) -0.024736319327 = 2*cos(64*π/127) -0.123605926920 = 2*cos(66*π/127) -0.222173050098 = 2*cos(68*π/127) -0.320196478592 = 2*cos(70*π/127) -0.417436332647 = 2*cos(72*π/127) -0.513654650045 = 2*cos(74*π/127) -0.608615968441 = 2*cos(76*π/127) -0.702087901576 = 2*cos(78*π/127) -0.793841707966 = 2*cos(80*π/127) -0.883652850671 = 2*cos(82*π/127) -0.971301546775 = 2*cos(84*π/127) -1.056573305232 = 2*cos(86*π/127) -1.139259451760 = 2*cos(88*π/127) -1.219157639501 = 2*cos(90*π/127) -1.296072344203 = 2*cos(92*π/127) -1.369815342694 = 2*cos(94*π/127) -1.440206173499 = 2*cos(96*π/127) -1.507072578460 = 2*cos(98*π/127) -1.570250924280 = 2*cos(100*π/127) -1.629586602958 = 2*cos(102*π/127) -1.684934410147 = 2*cos(104*π/127) -1.736158900487 = 2*cos(106*π/127) -1.783134719067 = 2*cos(108*π/127) -1.825746908186 = 2*cos(110*π/127) -1.863891188679 = 2*cos(112*π/127) -1.897474215098 = 2*cos(114*π/127) -1.926413804149 = 2*cos(116*π/127) -1.950639135810 = 2*cos(118*π/127) -1.970090926635 = 2*cos(120*π/127) -1.984721574830 = 2*cos(122*π/127) -1.994495276750 = 2*cos(124*π/127) -1.999388114506 = 2*cos(126*π/127) Symmetry of Rotations and Cartesian products A1 d+g+i+k+m z2, z4, z6 A2 R+p+f+h+j+l Rz, z, z3, z5 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 D127 point group is 254, and the order of the principal axis (C127) is 127. The group has 65 irreducible representations. β The D127 point group is isomorphic to C127v. γ The D127 point group is generated by two symmetry elements, C127 and a perpendicular C2′. Also, the group may be generated from any two C2′ axes. δ The lowest nonvanishing multipole moment in D127 is 4 (quadrupole 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 The point group is chiral, as it does not contain any mirroring operation. 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.
| D125 | ||
| D126 | ||
| C127 C127v C127h | D127 | D127h D127d |
| D128 | ||
| D136 |
This Character Table for the D127 point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.