S116 E 2 S116 2 C58 2 S116^3 2 C29 2 S116^5 2 C58^3 2 S116^7 2 C29^2 2 S116^9 2 C58^5 2 S116^11 2 C29^3 2 S116^13 2 C58^7 2 S116^15 2 C29^4 2 S116^17 2 C58^9 2 S116^19 2 C29^5 2 S116^21 2 C58^11 2 S116^23 2 C29^6 2 S116^25 2 C58^13 2 S116^27 2 C29^7 2 S4 2 C58^15 2 S116^31 2 C29^8 2 S116^33 2 C58^17 2 S116^35 2 C29^9 2 S116^37 2 C58^19 2 S116^39 2 C29^10 2 S116^41 2 C58^21 2 S116^43 2 C29^11 2 S116^45 2 C58^23 2 S116^47 2 C29^12 2 S116^49 2 C58^25 2 S116^51 2 C29^13 2 S116^53 2 C58^27 2 S116^55 2 C29^14 2 S116^57 C2 <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> A 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 ..T ... ....T ....... ........T ........... ............T B 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 ... ..T ..... ......T ......... ..........T ............. E1 * 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 ... TT. ..... ....TT. ......... ........TT. ............. E2 * 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 ... ... TT... ....... ....TT... ........... ........TT... 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E9 * 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 ... ... ..... ....... ......... ........... ............. 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E11 * 2.00000 1.65538 0.74028 -0.42994 -1.45199 -1.97365 -1.81515 -1.03111 0.10828 1.21035 1.89531 1.92710 1.29477 0.21624 -0.93682 -1.76702 -1.98828 -1.52432 -0.53506 0.63860 1.59219 1.99707 1.71371 0.83978 -0.32356 -1.37540 -1.95324 -1.85795 -1.12237 0.00000 1.12237 1.85795 1.95324 1.37540 0.32356 -0.83978 -1.71371 -1.99707 -1.59219 -0.63860 0.53506 1.52432 1.98828 1.76702 0.93682 -0.21624 -1.29477 -1.92710 -1.89531 -1.21035 -0.10828 1.03111 1.81515 1.97365 1.45199 0.42994 -0.74028 -1.65538 -2.00000 ... ... ..... ....... ......... ........... ............. E12 * 2.00000 1.59219 0.53506 -0.74028 -1.71371 -1.98828 -1.45199 -0.32356 0.93682 1.81515 1.95324 1.29477 0.10828 -1.12237 -1.89531 -1.89531 -1.12237 0.10828 1.29477 1.95324 1.81515 0.93682 -0.32356 -1.45199 -1.98828 -1.71371 -0.74028 0.53506 1.59219 2.00000 1.59219 0.53506 -0.74028 -1.71371 -1.98828 -1.45199 -0.32356 0.93682 1.81515 1.95324 1.29477 0.10828 -1.12237 -1.89531 -1.89531 -1.12237 0.10828 1.29477 1.95324 1.81515 0.93682 -0.32356 -1.45199 -1.98828 -1.71371 -0.74028 0.53506 1.59219 2.00000 ... ... ..... ....... ......... ........... ............. 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E14 * 2.00000 1.45199 0.10828 -1.29477 -1.98828 -1.59219 -0.32356 1.12237 1.95324 1.71371 0.53506 -0.93682 -1.89531 -1.81515 -0.74028 0.74028 1.81515 1.89531 0.93682 -0.53506 -1.71371 -1.95324 -1.12237 0.32356 1.59219 1.98828 1.29477 -0.10828 -1.45199 -2.00000 -1.45199 -0.10828 1.29477 1.98828 1.59219 0.32356 -1.12237 -1.95324 -1.71371 -0.53506 0.93682 1.89531 1.81515 0.74028 -0.74028 -1.81515 -1.89531 -0.93682 0.53506 1.71371 1.95324 1.12237 -0.32356 -1.59219 -1.98828 -1.29477 0.10828 1.45199 2.00000 ... ... ..... ....... ......... ........... ............. 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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 ... ... ..... ....... ......... ........... ............. 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 ... ... ..... ....... ......... ........... ............. 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 ... ... ..... ....... ......... ........... ............. 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 ... ... ..... ....... ......... ........... ..TT......... 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 ... ... ..... ....... ......... ..TT....... ............. 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 ... ... ..... ....... ..TT..... ........... ......TT..... 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 ... ... ..... ..TT... ......... ......TT... ............. 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 TT. ... ..TT. ....... ......TT. ........... ..........TT. 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) 1.210348430387 = 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) 0.936816881399 = 2*cos(40*π/116) = 2*cos(10*π/29) 0.839778203120 = 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) 0.323563993105 = 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 A R+d+g+i Rz, z2, z4, z6 B p+f+h z, z3, z5 E1 p+f+h {x, y}, {xz2, yz2}, {xz4, yz4} E2 d+g+i {x2–y2, xy}, {z2(x2–y2), xyz2}, {z4(x2–y2), xyz4} E3 f+h {x(x2–3y2), y(3x2–y2)}, {xz2(x2–3y2), yz2(3x2–y2)} E4 g+i {(x2–y2)2–2x2y2, xy(x2–y2)}, {z2((x2–y2)2–4x2y2), xyz2(x2–y2)} E5 h {x(x2–(5+2√5)y2)(x2–(5–2√5)y2), y(y2–(5+2√5)x2)(y2–(5–2√5)x2)(y2–(5–2√5)x2)} E6 i {x2(x2–3y2)2–y2(3x2–y2)2, xy(x2–3y2)(3x2–y2)} E53 i {xz(x2–(5+2√5)y2)(x2–(5–2√5)y2), yz(y2–(5+2√5)x2)(y2–(5–2√5)x2)(y2–(5–2√5)x2)} E54 h {z((x2–y2)2–4x2y2), xyz(x2–y2)} E55 g+i {xz(x2–3y2), yz(3x2–y2)}, {xz3(x2–3y2), yz3(3x2–y2)} E56 f+h {z(x2–y2), xyz}, {z3(x2–y2), xyz3} E57 R+d+g+i {Rx, Ry}, {xz, yz}, {xz3, yz3}, {xz5, yz5} Notes: α The order of the S116 point group is 116, and the order of the principal axis is 116. β The S116 point group could also be named C116i, because the S116 axis is identical to a roto-inversion axis of order 116. γ This is an Abelian point group (the commutative law holds between all symmetry operations). The S116 group is Abelian because it contains only one symmetry element, all the powers of which necessarily commute (sufficient condition). In Abelian groups, all symmetry operations form a class of their own, and all irreducible representations are one-dimensional. δ Because the group is Abelian and the maximum order of rotation is >2, some irreducible representations have complex characters. These 114 cases have been combined into 57 two-dimensional representations that are no longer irreducible but have real-valued characters. Accordingly, 57 pairs of left and right rotations have been combined into one two-membered pseudo-class each. ε The 57 reducible "E" representations almost behave like true irreducible representations. Their norm, however, is twice the group order. Therefore, they have been marked with an asterisk in the table. This is essential when trying to decompose a reducible representation into "irreducible" ones using the familiar projection formula. 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 less than half the order of the principal axis (and often much less). For this group, however, none of the characters can be expressed by a closed algebraic form using real numbers only.
S114 | ||
C116 C116v C116h D116 D116h D116d | S116 | |
S118 |
This Character Table for the S116 point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.