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E46 2.00000 -1.87003 1.49702 -0.92945 0.24107 0.47863 -1.13613 1.64597 -1.94188 1.98542 -1.77091 1.32625 -0.70921 0.00000 0.70921 -1.32625 1.77091 -1.98542 1.94188 -1.64597 1.13613 -0.47863 -0.24107 0.92945 -1.49702 1.87003 -2.00000 1.87003 -1.49702 0.92945 -0.24107 -0.47863 1.13613 -1.64597 1.94188 -1.98542 1.77091 -1.32625 0.70921 0.00000 -0.70921 1.32625 -1.77091 1.98542 -1.94188 1.64597 -1.13613 0.47863 0.24107 -0.92945 1.49702 -1.87003 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. 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E48 2.00000 -1.94188 1.77091 -1.49702 1.13613 -0.70921 0.24107 0.24107 -0.70921 1.13613 -1.49702 1.77091 -1.94188 2.00000 -1.94188 1.77091 -1.49702 1.13613 -0.70921 0.24107 0.24107 -0.70921 1.13613 -1.49702 1.77091 -1.94188 2.00000 -1.94188 1.77091 -1.49702 1.13613 -0.70921 0.24107 0.24107 -0.70921 1.13613 -1.49702 1.77091 -1.94188 2.00000 -1.94188 1.77091 -1.49702 1.13613 -0.70921 0.24107 0.24107 -0.70921 1.13613 -1.49702 1.77091 -1.94188 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. 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E50 2.00000 -1.98542 1.94188 -1.87003 1.77091 -1.64597 1.49702 -1.32625 1.13613 -0.92945 0.70921 -0.47863 0.24107 0.00000 -0.24107 0.47863 -0.70921 0.92945 -1.13613 1.32625 -1.49702 1.64597 -1.77091 1.87003 -1.94188 1.98542 -2.00000 1.98542 -1.94188 1.87003 -1.77091 1.64597 -1.49702 1.32625 -1.13613 0.92945 -0.70921 0.47863 -0.24107 0.00000 0.24107 -0.47863 0.70921 -0.92945 1.13613 -1.32625 1.49702 -1.64597 1.77091 -1.87003 1.94188 -1.98542 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E51 2.00000 -1.99635 1.98542 -1.96724 1.94188 -1.90944 1.87003 -1.82380 1.77091 -1.71156 1.64597 -1.57437 1.49702 -1.41421 1.32625 -1.23344 1.13613 -1.03468 0.92945 -0.82083 0.70921 -0.59501 0.47863 -0.36051 0.24107 -0.12076 0.00000 0.12076 -0.24107 0.36051 -0.47863 0.59501 -0.70921 0.82083 -0.92945 1.03468 -1.13613 1.23344 -1.32625 1.41421 -1.49702 1.57437 -1.64597 1.71156 -1.77091 1.82380 -1.87003 1.90944 -1.94188 1.96724 -1.98542 1.99635 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.996351108447 = 2*cos(2*π/104) = 2*cos(π/52) 1.985417748196 = 2*cos(4*π/104) = 2*cos(π/26) 1.967239813894 = 2*cos(6*π/104) = 2*cos(3*π/52) 1.941883634852 = 2*cos(8*π/104) = 2*cos(π/13) 1.909441733017 = 2*cos(10*π/104) = 2*cos(5*π/52) 1.870032485371 = 2*cos(12*π/104) = 2*cos(3*π/26) 1.823799691984 = 2*cos(14*π/104) = 2*cos(7*π/52) 1.770912051306 = 2*cos(16*π/104) = 2*cos(2*π/13) 1.711562544603 = 2*cos(18*π/104) = 2*cos(9*π/52) 1.645967731787 = 2*cos(20*π/104) = 2*cos(5*π/26) 1.574366961218 = 2*cos(22*π/104) = 2*cos(11*π/52) 1.497021496342 = 2*cos(24*π/104) = 2*cos(3*π/13) 1.414213562373 = 2*cos(26*π/104) = 2*cos(π/4) = √2 1.326245316482 = 2*cos(28*π/104) = 2*cos(7*π/26) 1.233437745257 = 2*cos(30*π/104) = 2*cos(15*π/52) 1.136129493462 = 2*cos(32*π/104) = 2*cos(4*π/13) 1.034675628355 = 2*cos(34*π/104) = 2*cos(17*π/52) 0.929446344088 = 2*cos(36*π/104) = 2*cos(9*π/26) 0.820825610906 = 2*cos(38*π/104) = 2*cos(19*π/52) 0.709209774085 = 2*cos(40*π/104) = 2*cos(5*π/13) 0.595006107710 = 2*cos(42*π/104) = 2*cos(21*π/52) 0.478631328575 = 2*cos(44*π/104) = 2*cos(11*π/26) 0.360510075628 = 2*cos(46*π/104) = 2*cos(23*π/52) 0.241073360511 = 2*cos(48*π/104) = 2*cos(6*π/13) 0.120756994845 = 2*cos(50*π/104) = 2*cos(25*π/52) 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 D104 point group is 208, and the order of the principal axis (C104) is 104. The group has 55 irreducible representations. β The D104 point group is isomorphic to D52d and C104v. γ The D104 point group is generated by two symmetry elements, C104 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 D104 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. ι 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. While some characters of this group can be expressed using real integers and square roots alone, others need complex numbers and third roots.
D102 | ||
D103 | ||
C104 C104v C104h | D104 | D104h D104d S104 |
D105 | ||
D106 |
This Character Table for the D104 point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.