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E31 2.00000 -1.03917 -0.92013 1.99534 -1.15336 -0.79680 1.98137 -1.26218 -0.66976 1.95817 -1.36511 -0.53959 1.92583 -1.46167 -0.40691 1.88452 -1.55142 -0.27233 1.83442 -1.63394 -0.13648 1.77577 -1.70884 0.00000 1.70884 -1.77577 0.13648 1.63394 -1.83442 0.27233 1.55142 -1.88452 0.40691 1.46167 -1.92583 0.53959 1.36511 -1.95817 0.66976 1.26218 -1.98137 0.79680 1.15336 -1.99534 0.92013 1.03917 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E32 2.00000 -1.15336 -0.66976 1.92583 -1.55142 -0.13648 1.70884 -1.83442 0.40691 1.36511 -1.98137 0.92013 0.92013 -1.98137 1.36511 0.40691 -1.83442 1.70884 -0.13648 -1.55142 1.92583 -0.66976 -1.15336 2.00000 -1.15336 -0.66976 1.92583 -1.55142 -0.13648 1.70884 -1.83442 0.40691 1.36511 -1.98137 0.92013 0.92013 -1.98137 1.36511 0.40691 -1.83442 1.70884 -0.13648 -1.55142 1.92583 -0.66976 -1.15336 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E33 2.00000 -1.26218 -0.40691 1.77577 -1.83442 0.53959 1.15336 -1.99534 1.36511 0.27233 -1.70884 1.88452 -0.66976 -1.03917 1.98137 -1.46167 -0.13648 1.63394 -1.92583 0.79680 0.92013 -1.95817 1.55142 0.00000 -1.55142 1.95817 -0.92013 -0.79680 1.92583 -1.63394 0.13648 1.46167 -1.98137 1.03917 0.66976 -1.88452 1.70884 -0.27233 -1.36511 1.99534 -1.15336 -0.53959 1.83442 -1.77577 0.40691 1.26218 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E34 2.00000 -1.36511 -0.13648 1.55142 -1.98137 1.15336 0.40691 -1.70884 1.92583 -0.92013 -0.66976 1.83442 -1.83442 0.66976 0.92013 -1.92583 1.70884 -0.40691 -1.15336 1.98137 -1.55142 0.13648 1.36511 -2.00000 1.36511 0.13648 -1.55142 1.98137 -1.15336 -0.40691 1.70884 -1.92583 0.92013 0.66976 -1.83442 1.83442 -0.66976 -0.92013 1.92583 -1.70884 0.40691 1.15336 -1.98137 1.55142 -0.13648 -1.36511 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E35 2.00000 -1.46167 0.13648 1.26218 -1.98137 1.63394 -0.40691 -1.03917 1.92583 -1.77577 0.66976 0.79680 -1.83442 1.88452 -0.92013 -0.53959 1.70884 -1.95817 1.15336 0.27233 -1.55142 1.99534 -1.36511 0.00000 1.36511 -1.99534 1.55142 -0.27233 -1.15336 1.95817 -1.70884 0.53959 0.92013 -1.88452 1.83442 -0.79680 -0.66976 1.77577 -1.92583 1.03917 0.40691 -1.63394 1.98137 -1.26218 -0.13648 1.46167 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E36 2.00000 -1.55142 0.40691 0.92013 -1.83442 1.92583 -1.15336 -0.13648 1.36511 -1.98137 1.70884 -0.66976 -0.66976 1.70884 -1.98137 1.36511 -0.13648 -1.15336 1.92583 -1.83442 0.92013 0.40691 -1.55142 2.00000 -1.55142 0.40691 0.92013 -1.83442 1.92583 -1.15336 -0.13648 1.36511 -1.98137 1.70884 -0.66976 -0.66976 1.70884 -1.98137 1.36511 -0.13648 -1.15336 1.92583 -1.83442 0.92013 0.40691 -1.55142 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E37 2.00000 -1.63394 0.66976 0.53959 -1.55142 1.99534 -1.70884 0.79680 0.40691 -1.46167 1.98137 -1.77577 0.92013 0.27233 -1.36511 1.95817 -1.83442 1.03917 0.13648 -1.26218 1.92583 -1.88452 1.15336 0.00000 -1.15336 1.88452 -1.92583 1.26218 -0.13648 -1.03917 1.83442 -1.95817 1.36511 -0.27233 -0.92013 1.77577 -1.98137 1.46167 -0.40691 -0.79680 1.70884 -1.99534 1.55142 -0.53959 -0.66976 1.63394 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E38 2.00000 -1.70884 0.92013 0.13648 -1.15336 1.83442 -1.98137 1.55142 -0.66976 -0.40691 1.36511 -1.92583 1.92583 -1.36511 0.40691 0.66976 -1.55142 1.98137 -1.83442 1.15336 -0.13648 -0.92013 1.70884 -2.00000 1.70884 -0.92013 -0.13648 1.15336 -1.83442 1.98137 -1.55142 0.66976 0.40691 -1.36511 1.92583 -1.92583 1.36511 -0.40691 -0.66976 1.55142 -1.98137 1.83442 -1.15336 0.13648 0.92013 -1.70884 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E39 2.00000 -1.77577 1.15336 -0.27233 -0.66976 1.46167 -1.92583 1.95817 -1.55142 0.79680 0.13648 -1.03917 1.70884 -1.99534 1.83442 -1.26218 0.40691 0.53959 -1.36511 1.88452 -1.98137 1.63394 -0.92013 0.00000 0.92013 -1.63394 1.98137 -1.88452 1.36511 -0.53959 -0.40691 1.26218 -1.83442 1.99534 -1.70884 1.03917 -0.13648 -0.79680 1.55142 -1.95817 1.92583 -1.46167 0.66976 0.27233 -1.15336 1.77577 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E40 2.00000 -1.83442 1.36511 -0.66976 -0.13648 0.92013 -1.55142 1.92583 -1.98137 1.70884 -1.15336 0.40691 0.40691 -1.15336 1.70884 -1.98137 1.92583 -1.55142 0.92013 -0.13648 -0.66976 1.36511 -1.83442 2.00000 -1.83442 1.36511 -0.66976 -0.13648 0.92013 -1.55142 1.92583 -1.98137 1.70884 -1.15336 0.40691 0.40691 -1.15336 1.70884 -1.98137 1.92583 -1.55142 0.92013 -0.13648 -0.66976 1.36511 -1.83442 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E41 2.00000 -1.88452 1.55142 -1.03917 0.40691 0.27233 -0.92013 1.46167 -1.83442 1.99534 -1.92583 1.63394 -1.15336 0.53959 0.13648 -0.79680 1.36511 -1.77577 1.98137 -1.95817 1.70884 -1.26218 0.66976 0.00000 -0.66976 1.26218 -1.70884 1.95817 -1.98137 1.77577 -1.36511 0.79680 -0.13648 -0.53959 1.15336 -1.63394 1.92583 -1.99534 1.83442 -1.46167 0.92013 -0.27233 -0.40691 1.03917 -1.55142 1.88452 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E42 2.00000 -1.92583 1.70884 -1.36511 0.92013 -0.40691 -0.13648 0.66976 -1.15336 1.55142 -1.83442 1.98137 -1.98137 1.83442 -1.55142 1.15336 -0.66976 0.13648 0.40691 -0.92013 1.36511 -1.70884 1.92583 -2.00000 1.92583 -1.70884 1.36511 -0.92013 0.40691 0.13648 -0.66976 1.15336 -1.55142 1.83442 -1.98137 1.98137 -1.83442 1.55142 -1.15336 0.66976 -0.13648 -0.40691 0.92013 -1.36511 1.70884 -1.92583 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E43 2.00000 -1.95817 1.83442 -1.63394 1.36511 -1.03917 0.66976 -0.27233 -0.13648 0.53959 -0.92013 1.26218 -1.55142 1.77577 -1.92583 1.99534 -1.98137 1.88452 -1.70884 1.46167 -1.15336 0.79680 -0.40691 0.00000 0.40691 -0.79680 1.15336 -1.46167 1.70884 -1.88452 1.98137 -1.99534 1.92583 -1.77577 1.55142 -1.26218 0.92013 -0.53959 0.13648 0.27233 -0.66976 1.03917 -1.36511 1.63394 -1.83442 1.95817 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E44 2.00000 -1.98137 1.92583 -1.83442 1.70884 -1.55142 1.36511 -1.15336 0.92013 -0.66976 0.40691 -0.13648 -0.13648 0.40691 -0.66976 0.92013 -1.15336 1.36511 -1.55142 1.70884 -1.83442 1.92583 -1.98137 2.00000 -1.98137 1.92583 -1.83442 1.70884 -1.55142 1.36511 -1.15336 0.92013 -0.66976 0.40691 -0.13648 -0.13648 0.40691 -0.66976 0.92013 -1.15336 1.36511 -1.55142 1.70884 -1.83442 1.92583 -1.98137 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E45 2.00000 -1.99534 1.98137 -1.95817 1.92583 -1.88452 1.83442 -1.77577 1.70884 -1.63394 1.55142 -1.46167 1.36511 -1.26218 1.15336 -1.03917 0.92013 -0.79680 0.66976 -0.53959 0.40691 -0.27233 0.13648 0.00000 -0.13648 0.27233 -0.40691 0.53959 -0.66976 0.79680 -0.92013 1.03917 -1.15336 1.26218 -1.36511 1.46167 -1.55142 1.63394 -1.70884 1.77577 -1.83442 1.88452 -1.92583 1.95817 -1.98137 1.99534 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.995337538381 = 2*cos(2*π/92) = 2*cos(π/46) 1.981371892073 = 2*cos(4*π/92) = 2*cos(π/23) 1.958168175365 = 2*cos(6*π/92) = 2*cos(3*π/46) 1.925834574696 = 2*cos(8*π/92) = 2*cos(2*π/23) 1.884521844238 = 2*cos(10*π/92) = 2*cos(5*π/46) 1.834422603011 = 2*cos(12*π/92) = 2*cos(3*π/23) 1.775770436805 = 2*cos(14*π/92) = 2*cos(7*π/46) 1.708838809093 = 2*cos(16*π/92) = 2*cos(4*π/23) 1.633939786021 = 2*cos(18*π/92) = 2*cos(9*π/46) 1.551422581409 = 2*cos(20*π/92) = 2*cos(5*π/23) 1.461671928556 = 2*cos(22*π/92) = 2*cos(11*π/46) 1.365106286437 = 2*cos(24*π/92) = 2*cos(6*π/23) 1.262175888652 = 2*cos(26*π/92) = 2*cos(13*π/46) 1.153360644230 = 2*cos(28*π/92) = 2*cos(7*π/23) 1.039167900071 = 2*cos(30*π/92) = 2*cos(15*π/46) 0.920130075462 = 2*cos(32*π/92) = 2*cos(8*π/23) 0.796802179692 = 2*cos(34*π/92) = 2*cos(17*π/46) 0.669759224342 = 2*cos(36*π/92) = 2*cos(9*π/23) 0.539593542314 = 2*cos(38*π/92) = 2*cos(19*π/46) 0.406912026105 = 2*cos(40*π/92) = 2*cos(10*π/23) 0.272333298192 = 2*cos(42*π/92) = 2*cos(21*π/46) 0.136484826729 = 2*cos(44*π/92) = 2*cos(11*π/23) Symmetry of Rotations and Cartesian products A1 d+g+i z2, z4, z6 A2 R+p+f+h z, Rz, z3, z5 E1 R+p+d+f+g+h+i {x, y}, {Rx, Ry}, {xz, yz}, {xz2, yz2}, {xz3, yz3}, {xz4, yz4}, {xz5, yz5} E2 d+f+g+h+i {x2–y2, xy}, {z(x2–y2), xyz}, {z2(x2–y2), xyz2}, {z3(x2–y2), xyz3}, {z4(x2–y2), xyz4} E3 f+g+h+i {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 {(x2–y2)2–2x2y2, xy(x2–y2)}, {z((x2–y2)2–4x2y2), xyz(x2–y2)}, {z2((x2–y2)2–4x2y2), xyz2(x2–y2)} E5 h+i {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)}, {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)} E6 i {x2(x2–3y2)2–y2(3x2–y2)2, xy(x2–3y2)(3x2–y2)} Notes: 1 The order of the D92 point group is 184, and the order of the principal axis is 92. 2 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. 3 This point group is non-Abelian (some symmetry operations are not commutative). Therefore, the character table contains multi-membered classes and degenerate irreducible representations. 4 The point group is chiral, as it does not contain any mirroring operation. 5 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.
D91 | ||
C92 C92v C92h | D92 | D92h D92d S92 |
D93 |
This Character Table for the D92 point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.