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E31 * 2.00000 -0.88458 -1.21752 1.96157 -0.51764 -1.50368 1.84776 -0.13081 -1.73205 1.66294 0.26105 -1.89386 1.41421 0.64288 -1.98289 1.11114 1.00000 -1.99572 0.76537 1.31869 -1.93185 0.39018 1.58671 -1.79375 0.00000 1.79375 -1.58671 -0.39018 1.93185 -1.31869 -0.76537 1.99572 -1.00000 -1.11114 1.98289 -0.64288 -1.41421 1.89386 -0.26105 -1.66294 1.73205 0.13081 -1.84776 1.50368 0.51764 -1.96157 1.21752 0.88458 -2.00000 ... ... ..... ....... ......... ........... ............. E32 * 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 ... ... ..... ....... ......... ........... ............. E33 * 2.00000 -1.11114 -0.76537 1.96157 -1.41421 -0.39018 1.84776 -1.66294 0.00000 1.66294 -1.84776 0.39018 1.41421 -1.96157 0.76537 1.11114 -2.00000 1.11114 0.76537 -1.96157 1.41421 0.39018 -1.84776 1.66294 0.00000 -1.66294 1.84776 -0.39018 -1.41421 1.96157 -0.76537 -1.11114 2.00000 -1.11114 -0.76537 1.96157 -1.41421 -0.39018 1.84776 -1.66294 0.00000 1.66294 -1.84776 0.39018 1.41421 -1.96157 0.76537 1.11114 -2.00000 ... ... ..... ....... ......... ........... ............. E34 * 2.00000 -1.21752 -0.51764 1.84776 -1.73205 0.26105 1.41421 -1.98289 1.00000 0.76537 -1.93185 1.58671 0.00000 -1.58671 1.93185 -0.76537 -1.00000 1.98289 -1.41421 -0.26105 1.73205 -1.84776 0.51764 1.21752 -2.00000 1.21752 0.51764 -1.84776 1.73205 -0.26105 -1.41421 1.98289 -1.00000 -0.76537 1.93185 -1.58671 0.00000 1.58671 -1.93185 0.76537 1.00000 -1.98289 1.41421 0.26105 -1.73205 1.84776 -0.51764 -1.21752 2.00000 ... ... ..... ....... ......... ........... ............. E35 * 2.00000 -1.31869 -0.26105 1.66294 -1.93185 0.88458 0.76537 -1.89386 1.73205 -0.39018 -1.21752 1.99572 -1.41421 -0.13081 1.58671 -1.96157 1.00000 0.64288 -1.84776 1.79375 -0.51764 -1.11114 1.98289 -1.50368 0.00000 1.50368 -1.98289 1.11114 0.51764 -1.79375 1.84776 -0.64288 -1.00000 1.96157 -1.58671 0.13081 1.41421 -1.99572 1.21752 0.39018 -1.73205 1.89386 -0.76537 -0.88458 1.93185 -1.66294 0.26105 1.31869 -2.00000 ... ... ..... ....... ......... ........... ............. E36 * 2.00000 -1.41421 0.00000 1.41421 -2.00000 1.41421 0.00000 -1.41421 2.00000 -1.41421 0.00000 1.41421 -2.00000 1.41421 0.00000 -1.41421 2.00000 -1.41421 0.00000 1.41421 -2.00000 1.41421 0.00000 -1.41421 2.00000 -1.41421 0.00000 1.41421 -2.00000 1.41421 0.00000 -1.41421 2.00000 -1.41421 0.00000 1.41421 -2.00000 1.41421 0.00000 -1.41421 2.00000 -1.41421 0.00000 1.41421 -2.00000 1.41421 0.00000 -1.41421 2.00000 ... ... ..... ....... ......... ........... ............. E37 * 2.00000 -1.50368 0.26105 1.11114 -1.93185 1.79375 -0.76537 -0.64288 1.73205 -1.96157 1.21752 0.13081 -1.41421 1.99572 -1.58671 0.39018 1.00000 -1.89386 1.84776 -0.88458 -0.51764 1.66294 -1.98289 1.31869 0.00000 -1.31869 1.98289 -1.66294 0.51764 0.88458 -1.84776 1.89386 -1.00000 -0.39018 1.58671 -1.99572 1.41421 -0.13081 -1.21752 1.96157 -1.73205 0.64288 0.76537 -1.79375 1.93185 -1.11114 -0.26105 1.50368 -2.00000 ... ... ..... ....... ......... ........... ............. E38 * 2.00000 -1.58671 0.51764 0.76537 -1.73205 1.98289 -1.41421 0.26105 1.00000 -1.84776 1.93185 -1.21752 0.00000 1.21752 -1.93185 1.84776 -1.00000 -0.26105 1.41421 -1.98289 1.73205 -0.76537 -0.51764 1.58671 -2.00000 1.58671 -0.51764 -0.76537 1.73205 -1.98289 1.41421 -0.26105 -1.00000 1.84776 -1.93185 1.21752 0.00000 -1.21752 1.93185 -1.84776 1.00000 0.26105 -1.41421 1.98289 -1.73205 0.76537 0.51764 -1.58671 2.00000 ... ... ..... ....... ......... ........... ............. E39 * 2.00000 -1.66294 0.76537 0.39018 -1.41421 1.96157 -1.84776 1.11114 0.00000 -1.11114 1.84776 -1.96157 1.41421 -0.39018 -0.76537 1.66294 -2.00000 1.66294 -0.76537 -0.39018 1.41421 -1.96157 1.84776 -1.11114 0.00000 1.11114 -1.84776 1.96157 -1.41421 0.39018 0.76537 -1.66294 2.00000 -1.66294 0.76537 0.39018 -1.41421 1.96157 -1.84776 1.11114 0.00000 -1.11114 1.84776 -1.96157 1.41421 -0.39018 -0.76537 1.66294 -2.00000 ... ... ..... ....... ......... ........... ............. E40 * 2.00000 -1.73205 1.00000 0.00000 -1.00000 1.73205 -2.00000 1.73205 -1.00000 0.00000 1.00000 -1.73205 2.00000 -1.73205 1.00000 0.00000 -1.00000 1.73205 -2.00000 1.73205 -1.00000 0.00000 1.00000 -1.73205 2.00000 -1.73205 1.00000 0.00000 -1.00000 1.73205 -2.00000 1.73205 -1.00000 0.00000 1.00000 -1.73205 2.00000 -1.73205 1.00000 0.00000 -1.00000 1.73205 -2.00000 1.73205 -1.00000 0.00000 1.00000 -1.73205 2.00000 ... ... ..... ....... ......... ........... ............. E41 * 2.00000 -1.79375 1.21752 -0.39018 -0.51764 1.31869 -1.84776 1.99572 -1.73205 1.11114 -0.26105 -0.64288 1.41421 -1.89386 1.98289 -1.66294 1.00000 -0.13081 -0.76537 1.50368 -1.93185 1.96157 -1.58671 0.88458 0.00000 -0.88458 1.58671 -1.96157 1.93185 -1.50368 0.76537 0.13081 -1.00000 1.66294 -1.98289 1.89386 -1.41421 0.64288 0.26105 -1.11114 1.73205 -1.99572 1.84776 -1.31869 0.51764 0.39018 -1.21752 1.79375 -2.00000 ... ... ..... ....... ......... ........... ............. E42 * 2.00000 -1.84776 1.41421 -0.76537 0.00000 0.76537 -1.41421 1.84776 -2.00000 1.84776 -1.41421 0.76537 0.00000 -0.76537 1.41421 -1.84776 2.00000 -1.84776 1.41421 -0.76537 0.00000 0.76537 -1.41421 1.84776 -2.00000 1.84776 -1.41421 0.76537 0.00000 -0.76537 1.41421 -1.84776 2.00000 -1.84776 1.41421 -0.76537 0.00000 0.76537 -1.41421 1.84776 -2.00000 1.84776 -1.41421 0.76537 0.00000 -0.76537 1.41421 -1.84776 2.00000 ... ... ..... ....... ......... ........... ............. E43 * 2.00000 -1.89386 1.58671 -1.11114 0.51764 0.13081 -0.76537 1.31869 -1.73205 1.96157 -1.98289 1.79375 -1.41421 0.88458 -0.26105 -0.39018 1.00000 -1.50368 1.84776 -1.99572 1.93185 -1.66294 1.21752 -0.64288 0.00000 0.64288 -1.21752 1.66294 -1.93185 1.99572 -1.84776 1.50368 -1.00000 0.39018 0.26105 -0.88458 1.41421 -1.79375 1.98289 -1.96157 1.73205 -1.31869 0.76537 -0.13081 -0.51764 1.11114 -1.58671 1.89386 -2.00000 ... ... ..... ....... ......... ........... ..TT......... E44 * 2.00000 -1.93185 1.73205 -1.41421 1.00000 -0.51764 0.00000 0.51764 -1.00000 1.41421 -1.73205 1.93185 -2.00000 1.93185 -1.73205 1.41421 -1.00000 0.51764 0.00000 -0.51764 1.00000 -1.41421 1.73205 -1.93185 2.00000 -1.93185 1.73205 -1.41421 1.00000 -0.51764 0.00000 0.51764 -1.00000 1.41421 -1.73205 1.93185 -2.00000 1.93185 -1.73205 1.41421 -1.00000 0.51764 0.00000 -0.51764 1.00000 -1.41421 1.73205 -1.93185 2.00000 ... ... ..... ....... ......... ..TT....... ............. E45 * 2.00000 -1.96157 1.84776 -1.66294 1.41421 -1.11114 0.76537 -0.39018 0.00000 0.39018 -0.76537 1.11114 -1.41421 1.66294 -1.84776 1.96157 -2.00000 1.96157 -1.84776 1.66294 -1.41421 1.11114 -0.76537 0.39018 0.00000 -0.39018 0.76537 -1.11114 1.41421 -1.66294 1.84776 -1.96157 2.00000 -1.96157 1.84776 -1.66294 1.41421 -1.11114 0.76537 -0.39018 0.00000 0.39018 -0.76537 1.11114 -1.41421 1.66294 -1.84776 1.96157 -2.00000 ... ... ..... ....... ..TT..... ........... ......TT..... E46 * 2.00000 -1.98289 1.93185 -1.84776 1.73205 -1.58671 1.41421 -1.21752 1.00000 -0.76537 0.51764 -0.26105 0.00000 0.26105 -0.51764 0.76537 -1.00000 1.21752 -1.41421 1.58671 -1.73205 1.84776 -1.93185 1.98289 -2.00000 1.98289 -1.93185 1.84776 -1.73205 1.58671 -1.41421 1.21752 -1.00000 0.76537 -0.51764 0.26105 0.00000 -0.26105 0.51764 -0.76537 1.00000 -1.21752 1.41421 -1.58671 1.73205 -1.84776 1.93185 -1.98289 2.00000 ... ... ..... ..TT... ......... ......TT... ............. E47 * 2.00000 -1.99572 1.98289 -1.96157 1.93185 -1.89386 1.84776 -1.79375 1.73205 -1.66294 1.58671 -1.50368 1.41421 -1.31869 1.21752 -1.11114 1.00000 -0.88458 0.76537 -0.64288 0.51764 -0.39018 0.26105 -0.13081 0.00000 0.13081 -0.26105 0.39018 -0.51764 0.64288 -0.76537 0.88458 -1.00000 1.11114 -1.21752 1.31869 -1.41421 1.50368 -1.58671 1.66294 -1.73205 1.79375 -1.84776 1.89386 -1.93185 1.96157 -1.98289 1.99572 -2.00000 TT. ... ..TT. ....... ......TT. ........... ..........TT. Irrational character values: 1.995717846477 = 2*cos(2*π/96) = 2*cos(π/48) = √2+√2+√2+√3 1.982889722748 = 2*cos(4*π/96) = 2*cos(π/24) = √2+√2+√3 1.961570560806 = 2*cos(6*π/96) = 2*cos(π/16) = √2+√2+√2 1.931851652578 = 2*cos(8*π/96) = 2*cos(π/12) = √2+√3 1.893860258990 = 2*cos(10*π/96) = 2*cos(5*π/48) = √2+√2+√2-√3 1.847759065023 = 2*cos(12*π/96) = 2*cos(π/8) = √2+√2 1.793745483065 = 2*cos(14*π/96) = 2*cos(7*π/48) = √2+√2-√2-√3 1.732050807569 = 2*cos(16*π/96) = 2*cos(π/6) = √3 1.662939224605 = 2*cos(18*π/96) = 2*cos(3*π/16) = √2+√2-√2 1.586706680582 = 2*cos(20*π/96) = 2*cos(5*π/24) = √2+√2-√3 1.503679614958 = 2*cos(22*π/96) = 2*cos(11*π/48) = √2+√2-√2+√3 1.414213562373 = 2*cos(24*π/96) = 2*cos(π/4) = √2 1.318691630200 = 2*cos(26*π/96) = 2*cos(13*π/48) = √2-√2-√2+√3 1.217522858017 = 2*cos(28*π/96) = 2*cos(7*π/24) = √2-√2-√3 1.111140466039 = 2*cos(30*π/96) = 2*cos(5*π/16) = √2-√2-√2 0.884577380438 = 2*cos(34*π/96) = 2*cos(17*π/48) = √2-√2-√2-√3 0.765366864730 = 2*cos(36*π/96) = 2*cos(3*π/8) = √2-√2 0.642878930606 = 2*cos(38*π/96) = 2*cos(19*π/48) = √2-√2+√2-√3 0.517638090205 = 2*cos(40*π/96) = 2*cos(5*π/12) = √2-√3 0.390180644032 = 2*cos(42*π/96) = 2*cos(7*π/16) = √2-√2+√2 0.261052384440 = 2*cos(44*π/96) = 2*cos(11*π/24) = √2-√2+√3 0.130806258460 = 2*cos(46*π/96) = 2*cos(23*π/48) = √2-√2+√2+√3 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)} E43 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)} E44 h {z((x2–y2)2–4x2y2), xyz(x2–y2)} E45 g+i {xz(x2–3y2), yz(3x2–y2)}, {xz3(x2–3y2), yz3(3x2–y2)} E46 f+h {z(x2–y2), xyz}, {z3(x2–y2), xyz3} E47 R+d+g+i {Rx, Ry}, {xz, yz}, {xz3, yz3}, {xz5, yz5} Notes: α The order of the S96 point group is 96, and the order of the principal axis is 96. β The S96 point group could also be named C96i, because the S96 axis is identical to a roto-inversion axis of order 96. γ This is an Abelian point group (the commutative law holds between all symmetry operations). The S96 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 94 cases have been combined into 47 two-dimensional representations that are no longer irreducible but have real-valued characters. Accordingly, 47 pairs of left and right rotations have been combined into one two-membered pseudo-class each. ε The 47 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). η The point group corresponds to a constructible polygon, as the order of the principal axis is a product of any number of different Fermat primes (3,5,17,257,65537) times an arbitrary power of two. Therefore, all characters have an algebraic degree which is a power of two and can be expressed as radicals involving only square roots and integer numbers.
| S94 | ||
| C96 C96v C96h D96 D96h D96d | S96 | |
| S98 |
This Character Table for the S96 point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.