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E52 2.00000 -1.99225 1.96905 -1.93058 1.87714 -1.80915 1.72714 -1.63173 1.52367 -1.40379 1.27304 -1.13241 0.98300 -0.82597 0.66254 -0.49397 0.32157 -0.14667 -0.02936 0.20516 -0.37938 0.55065 -0.71765 0.87909 -1.03371 1.18032 -1.31777 1.44501 -1.56104 1.66498 -1.75600 1.83340 -1.89660 1.94508 -1.97849 1.99655 -1.99914 1.98622 -1.95791 1.91441 -1.85607 1.78334 -1.69679 1.59707 -1.48498 1.36137 -1.22721 1.08353 -0.93145 0.77214 -0.60685 0.43686 -0.26348 0.08805 0.00000 ... ... ..... ....... ......... ........... ............. E53 2.00000 -1.99914 1.99655 -1.99225 1.98622 -1.97849 1.96905 -1.95791 1.94508 -1.93058 1.91441 -1.89660 1.87714 -1.85607 1.83340 -1.80915 1.78334 -1.75600 1.72714 -1.69679 1.66498 -1.63173 1.59707 -1.56104 1.52367 -1.48498 1.44501 -1.40379 1.36137 -1.31777 1.27304 -1.22721 1.18032 -1.13241 1.08353 -1.03371 0.98300 -0.93145 0.87909 -0.82597 0.77214 -0.71765 0.66254 -0.60685 0.55065 -0.49397 0.43686 -0.37938 0.32157 -0.26348 0.20516 -0.14667 0.08805 -0.02936 0.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.996552792839 = 2*cos(2*π/107) 1.986223054591 = 2*cos(4*π/107) 1.969046394006 = 2*cos(6*π/107) 1.945082022590 = 2*cos(8*π/107) 1.914412550496 = 2*cos(10*π/107) 1.877143701748 = 2*cos(12*π/107) 1.833403949788 = 2*cos(14*π/107) 1.783344074602 = 2*cos(16*π/107) 1.727136642952 = 2*cos(18*π/107) 1.664975413496 = 2*cos(20*π/107) 1.597074668872 = 2*cos(22*π/107) 1.523668477012 = 2*cos(24*π/107) 1.445009884266 = 2*cos(26*π/107) 1.361370043098 = 2*cos(28*π/107) 1.273037277369 = 2*cos(30*π/107) 1.180316088420 = 2*cos(32*π/107) 1.083526105398 = 2*cos(34*π/107) 0.983000983426 = 2*cos(36*π/107) 0.879087253425 = 2*cos(38*π/107) 0.772143127547 = 2*cos(40*π/107) 0.662537264351 = 2*cos(42*π/107) 0.550647497953 = 2*cos(44*π/107) 0.436859535555 = 2*cos(46*π/107) 0.321565627839 = 2*cos(48*π/107) 0.205163216787 = 2*cos(50*π/107) 0.088053565625 = 2*cos(52*π/107) -0.029359624419 = 2*cos(54*π/107) -0.146671605755 = 2*cos(56*π/107) -0.263477979682 = 2*cos(58*π/107) -0.379376090430 = 2*cos(60*π/107) -0.493966413202 = 2*cos(62*π/107) -0.606853931417 = 2*cos(64*π/107) -0.717649498414 = 2*cos(66*π/107) -0.825971178921 = 2*cos(68*π/107) -0.931445565664 = 2*cos(70*π/107) -1.033709066583 = 2*cos(72*π/107) -1.132409158205 = 2*cos(74*π/107) -1.227205600866 = 2*cos(76*π/107) -1.317771611592 = 2*cos(78*π/107) -1.403794990581 = 2*cos(80*π/107) -1.484979197426 = 2*cos(82*π/107) -1.561044373346 = 2*cos(84*π/107) -1.631728305923 = 2*cos(86*π/107) -1.696787332999 = 2*cos(88*π/107) -1.755997182628 = 2*cos(90*π/107) -1.809153746194 = 2*cos(92*π/107) -1.856073782010 = 2*cos(94*π/107) -1.896595546993 = 2*cos(96*π/107) -1.930579354223 = 2*cos(98*π/107) -1.957908054478 = 2*cos(100*π/107) -1.978487440065 = 2*cos(102*π/107) -1.992246569581 = 2*cos(104*π/107) -1.999138012454 = 2*cos(106*π/107) 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 C107v point group is 214, and the order of the principal axis (C107) is 107. The group has 55 irreducible representations. β The C107v point group is isomorphic to D107. γ The C107v point group is generated by two symmetry elements, C107 and any σv. Also, the group may be generated from any two σv planes. δ The lowest nonvanishing multipole moment in C107v 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.
| C105v | ||
| C106v | ||
| C107 | C107v | C107h D107 D107h D107d |
| C108v | ||
| C109v |
This Character Table for the C107v point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.