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E49 2.00000 -1.76702 1.12237 -0.21624 -0.74028 1.52432 -1.95324 1.92710 -1.45199 0.63860 0.32356 -1.21035 1.81515 -1.99707 1.71371 -1.03111 0.10828 0.83978 -1.59219 1.97365 -1.89531 1.37540 -0.53506 -0.42994 1.29477 -1.85795 1.98828 -1.65538 0.93682 0.00000 -0.93682 1.65538 -1.98828 1.85795 -1.29477 0.42994 0.53506 -1.37540 1.89531 -1.97365 1.59219 -0.83978 -0.10828 1.03111 -1.71371 1.99707 -1.81515 1.21035 -0.32356 -0.63860 1.45199 -1.92710 1.95324 -1.52432 0.74028 0.21624 -1.12237 1.76702 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E50 2.00000 -1.81515 1.29477 -0.53506 -0.32356 1.12237 -1.71371 1.98828 -1.89531 1.45199 -0.74028 -0.10828 0.93682 -1.59219 1.95324 -1.95324 1.59219 -0.93682 0.10828 0.74028 -1.45199 1.89531 -1.98828 1.71371 -1.12237 0.32356 0.53506 -1.29477 1.81515 -2.00000 1.81515 -1.29477 0.53506 0.32356 -1.12237 1.71371 -1.98828 1.89531 -1.45199 0.74028 0.10828 -0.93682 1.59219 -1.95324 1.95324 -1.59219 0.93682 -0.10828 -0.74028 1.45199 -1.89531 1.98828 -1.71371 1.12237 -0.32356 -0.53506 1.29477 -1.81515 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E51 2.00000 -1.85795 1.45199 -0.83978 0.10828 0.63860 -1.29477 1.76702 -1.98828 1.92710 -1.59219 1.03111 -0.32356 -0.42994 1.12237 -1.65538 1.95324 -1.97365 1.71371 -1.21035 0.53506 0.21624 -0.93682 1.52432 -1.89531 1.99707 -1.81515 1.37540 -0.74028 0.00000 0.74028 -1.37540 1.81515 -1.99707 1.89531 -1.52432 0.93682 -0.21624 -0.53506 1.21035 -1.71371 1.97365 -1.95324 1.65538 -1.12237 0.42994 0.32356 -1.03111 1.59219 -1.92710 1.98828 -1.76702 1.29477 -0.63860 -0.10828 0.83978 -1.45199 1.85795 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E52 2.00000 -1.89531 1.59219 -1.12237 0.53506 0.10828 -0.74028 1.29477 -1.71371 1.95324 -1.98828 1.81515 -1.45199 0.93682 -0.32356 -0.32356 0.93682 -1.45199 1.81515 -1.98828 1.95324 -1.71371 1.29477 -0.74028 0.10828 0.53506 -1.12237 1.59219 -1.89531 2.00000 -1.89531 1.59219 -1.12237 0.53506 0.10828 -0.74028 1.29477 -1.71371 1.95324 -1.98828 1.81515 -1.45199 0.93682 -0.32356 -0.32356 0.93682 -1.45199 1.81515 -1.98828 1.95324 -1.71371 1.29477 -0.74028 0.10828 0.53506 -1.12237 1.59219 -1.89531 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E53 2.00000 -1.92710 1.71371 -1.37540 0.93682 -0.42994 -0.10828 0.63860 -1.12237 1.52432 -1.81515 1.97365 -1.98828 1.85795 -1.59219 1.21035 -0.74028 0.21624 0.32356 -0.83978 1.29477 -1.65538 1.89531 -1.99707 1.95324 -1.76702 1.45199 -1.03111 0.53506 0.00000 -0.53506 1.03111 -1.45199 1.76702 -1.95324 1.99707 -1.89531 1.65538 -1.29477 0.83978 -0.32356 -0.21624 0.74028 -1.21035 1.59219 -1.85795 1.98828 -1.97365 1.81515 -1.52432 1.12237 -0.63860 0.10828 0.42994 -0.93682 1.37540 -1.71371 1.92710 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E54 2.00000 -1.95324 1.81515 -1.59219 1.29477 -0.93682 0.53506 -0.10828 -0.32356 0.74028 -1.12237 1.45199 -1.71371 1.89531 -1.98828 1.98828 -1.89531 1.71371 -1.45199 1.12237 -0.74028 0.32356 0.10828 -0.53506 0.93682 -1.29477 1.59219 -1.81515 1.95324 -2.00000 1.95324 -1.81515 1.59219 -1.29477 0.93682 -0.53506 0.10828 0.32356 -0.74028 1.12237 -1.45199 1.71371 -1.89531 1.98828 -1.98828 1.89531 -1.71371 1.45199 -1.12237 0.74028 -0.32356 -0.10828 0.53506 -0.93682 1.29477 -1.59219 1.81515 -1.95324 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E55 2.00000 -1.97365 1.89531 -1.76702 1.59219 -1.37540 1.12237 -0.83978 0.53506 -0.21624 -0.10828 0.42994 -0.74028 1.03111 -1.29477 1.52432 -1.71371 1.85795 -1.95324 1.99707 -1.98828 1.92710 -1.81515 1.65538 -1.45199 1.21035 -0.93682 0.63860 -0.32356 0.00000 0.32356 -0.63860 0.93682 -1.21035 1.45199 -1.65538 1.81515 -1.92710 1.98828 -1.99707 1.95324 -1.85795 1.71371 -1.52432 1.29477 -1.03111 0.74028 -0.42994 0.10828 0.21624 -0.53506 0.83978 -1.12237 1.37540 -1.59219 1.76702 -1.89531 1.97365 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E56 2.00000 -1.98828 1.95324 -1.89531 1.81515 -1.71371 1.59219 -1.45199 1.29477 -1.12237 0.93682 -0.74028 0.53506 -0.32356 0.10828 0.10828 -0.32356 0.53506 -0.74028 0.93682 -1.12237 1.29477 -1.45199 1.59219 -1.71371 1.81515 -1.89531 1.95324 -1.98828 2.00000 -1.98828 1.95324 -1.89531 1.81515 -1.71371 1.59219 -1.45199 1.29477 -1.12237 0.93682 -0.74028 0.53506 -0.32356 0.10828 0.10828 -0.32356 0.53506 -0.74028 0.93682 -1.12237 1.29477 -1.45199 1.59219 -1.71371 1.81515 -1.89531 1.95324 -1.98828 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E57 2.00000 -1.99707 1.98828 -1.97365 1.95324 -1.92710 1.89531 -1.85795 1.81515 -1.76702 1.71371 -1.65538 1.59219 -1.52432 1.45199 -1.37540 1.29477 -1.21035 1.12237 -1.03111 0.93682 -0.83978 0.74028 -0.63860 0.53506 -0.42994 0.32356 -0.21624 0.10828 0.00000 -0.10828 0.21624 -0.32356 0.42994 -0.53506 0.63860 -0.74028 0.83978 -0.93682 1.03111 -1.12237 1.21035 -1.29477 1.37540 -1.45199 1.52432 -1.59219 1.65538 -1.71371 1.76702 -1.81515 1.85795 -1.89531 1.92710 -1.95324 1.97365 -1.98828 1.99707 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.997066827702 = 2*cos(2*π/116) = 2*cos(π/58) 1.988275914309 = 2*cos(4*π/116) = 2*cos(π/29) 1.973653045083 = 2*cos(6*π/116) = 2*cos(3*π/58) 1.953241111420 = 2*cos(8*π/116) = 2*cos(2*π/29) 1.927099985038 = 2*cos(10*π/116) = 2*cos(5*π/58) 1.895306342366 = 2*cos(12*π/116) = 2*cos(3*π/29) 1.857953439634 = 2*cos(14*π/116) = 2*cos(7*π/58) 1.815150839342 = 2*cos(16*π/116) = 2*cos(4*π/29) 1.767024088892 = 2*cos(18*π/116) = 2*cos(9*π/58) 1.713714352335 = 2*cos(20*π/116) = 2*cos(5*π/29) 1.655377996314 = 2*cos(22*π/116) = 2*cos(11*π/58) 1.592186131411 = 2*cos(24*π/116) = 2*cos(6*π/29) 1.524324110255 = 2*cos(26*π/116) = 2*cos(13*π/58) 1.451990983846 = 2*cos(28*π/116) = 2*cos(7*π/29) 1.375398917707 = 2*cos(30*π/116) = 2*cos(15*π/58) 1.294772569564 = 2*cos(32*π/116) = 2*cos(8*π/29) 1.210348430388 = 2*cos(34*π/116) = 2*cos(17*π/58) 1.122374130725 = 2*cos(36*π/116) = 2*cos(9*π/29) 1.031107714354 = 2*cos(38*π/116) = 2*cos(19*π/58) 0.936816881400 = 2*cos(40*π/116) = 2*cos(10*π/29) 0.839778203121 = 2*cos(42*π/116) = 2*cos(21*π/58) 0.740276310680 = 2*cos(44*π/116) = 2*cos(11*π/29) 0.638603060272 = 2*cos(46*π/116) = 2*cos(23*π/58) 0.535056677058 = 2*cos(48*π/116) = 2*cos(12*π/29) 0.429940880422 = 2*cos(50*π/116) = 2*cos(25*π/58) 0.323563993106 = 2*cos(52*π/116) = 2*cos(13*π/29) 0.216238036848 = 2*cos(54*π/116) = 2*cos(27*π/58) 0.108277817171 = 2*cos(56*π/116) = 2*cos(14*π/29) 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 D116 point group is 232, and the order of the principal axis (C116) is 116. The group has 61 irreducible representations. β The D116 point group is isomorphic to D58d and C116v. γ The D116 point group is generated by two symmetry elements, C116 and a perpendicular C2′ (or, non-canonically, C2″). Also, the group may be generated from a C2′ plus a C2″ (some pairs will yield smaller groups, though; choosing a minimum angle is safe). δ There are two different sets of twofold symmetry axes perpendicular to the principal axis (z axis in standard orientation). By convention, the set denoted as C2′ contains both the x and y axes. ε The lowest nonvanishing multipole moment in D116 is 4 (quadrupole moment). ζ This point group is non-Abelian (some symmetry operations are not commutative). Therefore, the character table contains multi-membered classes and degenerate irreducible representations. η The point group is chiral, as it does not contain any mirroring operation. 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, none of the irrational characters can be expressed by a closed algebraic form using real numbers only.
| D114 | ||
| D115 | ||
| C116 C116v C116h | D116 | D116h D116d S116 |
| D117 | ||
| D118 |
This Character Table for the D116 point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.