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E54 2.00000 -1.99279 1.97123 -1.93547 1.88575 -1.82246 1.74603 -1.65702 1.55607 -1.44391 1.32135 -1.18927 1.04861 -0.90041 0.74571 -0.58565 0.42136 -0.25404 0.08488 0.08488 -0.25404 0.42136 -0.58565 0.74571 -0.90041 1.04861 -1.18927 1.32135 -1.44391 1.55607 -1.65702 1.74603 -1.82246 1.88575 -1.93547 1.97123 -1.99279 2.00000 -1.99279 1.97123 -1.93547 1.88575 -1.82246 1.74603 -1.65702 1.55607 -1.44391 1.32135 -1.18927 1.04861 -0.90041 0.74571 -0.58565 0.42136 -0.25404 0.08488 0.00000 ... ... ..... ....... ......... ........... ............. E55 2.00000 -1.99920 1.99680 -1.99279 1.98720 -1.98001 1.97123 -1.96088 1.94895 -1.93547 1.92043 -1.90385 1.88575 -1.86614 1.84504 -1.82246 1.79841 -1.77293 1.74603 -1.71773 1.68805 -1.65702 1.62466 -1.59100 1.55607 -1.51989 1.48250 -1.44391 1.40417 -1.36331 1.32135 -1.27833 1.23429 -1.18927 1.14329 -1.09639 1.04861 -1.00000 0.95058 -0.90041 0.84951 -0.79793 0.74571 -0.69290 0.63953 -0.58565 0.53129 -0.47652 0.42136 -0.36586 0.31007 -0.25404 0.19779 -0.14140 0.08488 -0.02830 0.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.996796698525 = 2*cos(2*π/111) 1.987197055239 = 2*cos(4*π/111) 1.971231820695 = 2*cos(6*π/111) = 2*cos(2*π/37) 1.948952136352 = 2*cos(8*π/111) 1.920429370755 = 2*cos(10*π/111) 1.885754890922 = 2*cos(12*π/111) = 2*cos(4*π/37) 1.845039769665 = 2*cos(14*π/111) 1.798414429792 = 2*cos(16*π/111) 1.746028226322 = 2*cos(18*π/111) = 2*cos(6*π/37) 1.688048968060 = 2*cos(20*π/111) 1.624662380048 = 2*cos(22*π/111) 1.556071508637 = 2*cos(24*π/111) = 2*cos(8*π/37) 1.482496071067 = 2*cos(26*π/111) 1.404171751645 = 2*cos(28*π/111) 1.321349446780 = 2*cos(30*π/111) = 2*cos(10*π/37) 1.234294461283 = 2*cos(32*π/111) 1.143285658517 = 2*cos(34*π/111) 1.048614567114 = 2*cos(36*π/111) = 2*cos(12*π/37) 0.950584447122 = 2*cos(38*π/111) 0.849509318568 = 2*cos(40*π/111) 0.745712955561 = 2*cos(42*π/111) = 2*cos(14*π/37) 0.639527849142 = 2*cos(44*π/111) 0.531294142222 = 2*cos(46*π/111) 0.421358539991 = 2*cos(48*π/111) = 2*cos(16*π/37) 0.310073199328 = 2*cos(50*π/111) 0.197794600728 = 2*cos(52*π/111) 0.084882406392 = 2*cos(54*π/111) = 2*cos(18*π/37) -0.028301691882 = 2*cos(56*π/111) -0.141395131304 = 2*cos(58*π/111) -0.254035639494 = 2*cos(60*π/111) = 2*cos(20*π/37) -0.365862394945 = 2*cos(62*π/111) -0.476517182846 = 2*cos(64*π/111) -0.585645542553 = 2*cos(66*π/111) = 2*cos(22*π/37) -0.692897903029 = 2*cos(68*π/111) -0.797930702631 = 2*cos(70*π/111) -0.900407489635 = 2*cos(72*π/111) = 2*cos(24*π/37) -1.096389208889 = 2*cos(76*π/111) -1.189266352609 = 2*cos(78*π/111) = 2*cos(26*π/37) -1.278333917666 = 2*cos(80*π/111) -1.363306593799 = 2*cos(82*π/111) -1.443912187909 = 2*cos(84*π/111) = 2*cos(28*π/37) -1.519892495977 = 2*cos(86*π/111) -1.591004130171 = 2*cos(88*π/111) -1.657019298488 = 2*cos(90*π/111) = 2*cos(30*π/37) -1.717726534441 = 2*cos(92*π/111) -1.772931374452 = 2*cos(94*π/111) -1.822456980776 = 2*cos(96*π/111) = 2*cos(32*π/37) -1.866144707965 = 2*cos(98*π/111) -1.903854611058 = 2*cos(100*π/111) -1.935465893867 = 2*cos(102*π/111) = 2*cos(34*π/37) -1.960877295923 = 2*cos(104*π/111) -1.980007416844 = 2*cos(106*π/111) -1.992794977085 = 2*cos(108*π/111) = 2*cos(36*π/37) -1.999199014237 = 2*cos(110*π/111) 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 C111v point group is 222, and the order of the principal axis (C111) is 111. The group has 57 irreducible representations. β The C111v point group is isomorphic to D111. γ The C111v point group is generated by two symmetry elements, C111 and any σv. Also, the group may be generated from two σv planes (some pairs will yield smaller groups, though; choosing a minimum angle is safe). δ The lowest nonvanishing multipole moment in C111v 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 much less than half the order of the principal axis. η The point group corresponds to a polygon inconstructible by the classical means of ruler and compass. Yet it becomes constructible if angle trisection is allowed, e.g., with neusis construction or origami. This is because the order of the principal axis is given by a product of any number of different Pierpont primes (...,5,7,13,17,19,37,73,97,109,163,...) times arbitrary powers of two and three. All characters of this group can be expressed using complex numbers, elementary arithmetic operations, square roots and third roots.
| C109v | ||
| C110v | ||
| C111 | C111v | C111h D111 D111h D111d |
| C112v | ||
| C113v |
This Character Table for the C111v point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.