D13h E 2 C13 2 C13^2 2 C13^3 2 C13^4 2 C13^5 2 C13^6 13 C2' sh 2 S13 2 S13^3 2 S13^5 2 S13^7 2 S13^9 2 S13^11 13 sv <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> A1' 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 A1" 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 ... ... ..... ....... ......... ........... ............. A2' 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 ... ..... ....... ......... ........... ............. A2" 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 1.77091 1.13613 0.24107 -0.70921 -1.49702 -1.94188 0.00000 2.00000 1.77091 0.24107 -1.49702 -1.94188 -0.70921 1.13613 0.00000 ... TT. ..... ....TT. ......... ........TT. ............. E1" 2.00000 1.77091 1.13613 0.24107 -0.70921 -1.49702 -1.94188 0.00000 -2.00000 -1.77091 -0.24107 1.49702 1.94188 0.70921 -1.13613 0.00000 TT. ... ..TT. ....... ......TT. ........... ..........TT. E2' 2.00000 1.13613 -0.70921 -1.94188 -1.49702 0.24107 1.77091 0.00000 2.00000 1.13613 -1.94188 0.24107 1.77091 -1.49702 -0.70921 0.00000 ... ... TT... ....... ....TT... ........... ........TT... E2" 2.00000 1.13613 -0.70921 -1.94188 -1.49702 0.24107 1.77091 0.00000 -2.00000 -1.13613 1.94188 -0.24107 -1.77091 1.49702 0.70921 0.00000 ... ... ..... ..TT... ......... ......TT... ............. E3' 2.00000 0.24107 -1.94188 -0.70921 1.77091 1.13613 -1.49702 0.00000 2.00000 0.24107 -0.70921 1.13613 -1.49702 1.77091 -1.94188 0.00000 ... ... ..... TT..... ......... ....TT..... ............. E3" 2.00000 0.24107 -1.94188 -0.70921 1.77091 1.13613 -1.49702 0.00000 -2.00000 -0.24107 0.70921 -1.13613 1.49702 -1.77091 1.94188 0.00000 ... ... ..... ....... ..TT..... ........... ......TT..... E4' 2.00000 -0.70921 -1.49702 1.77091 0.24107 -1.94188 1.13613 0.00000 2.00000 -0.70921 1.77091 -1.94188 1.13613 0.24107 -1.49702 0.00000 ... ... ..... ....... TT....... ........... ....TT....... E4" 2.00000 -0.70921 -1.49702 1.77091 0.24107 -1.94188 1.13613 0.00000 -2.00000 0.70921 -1.77091 1.94188 -1.13613 -0.24107 1.49702 0.00000 ... ... ..... ....... ......... ..TT....... ............. E5' 2.00000 -1.49702 0.24107 1.13613 -1.94188 1.77091 -0.70921 0.00000 2.00000 -1.49702 1.13613 1.77091 -0.70921 -1.94188 0.24107 0.00000 ... ... ..... ....... ......... TT......... ............. E5" 2.00000 -1.49702 0.24107 1.13613 -1.94188 1.77091 -0.70921 0.00000 -2.00000 1.49702 -1.13613 -1.77091 0.70921 1.94188 -0.24107 0.00000 ... ... ..... ....... ......... ........... ..TT......... E6' 2.00000 -1.94188 1.77091 -1.49702 1.13613 -0.70921 0.24107 0.00000 2.00000 -1.94188 -1.49702 -0.70921 0.24107 1.13613 1.77091 0.00000 ... ... ..... ....... ......... ........... TT........... E6" 2.00000 -1.94188 1.77091 -1.49702 1.13613 -0.70921 0.24107 0.00000 -2.00000 1.94188 1.49702 0.70921 -0.24107 -1.13613 -1.77091 0.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.941883634852 = 2*cos(2*π/26) = 2*cos(π/13) 1.770912051306 = 2*cos(4*π/26) = 2*cos(2*π/13) 1.497021496342 = 2*cos(6*π/26) = 2*cos(3*π/13) 1.136129493462 = 2*cos(8*π/26) = 2*cos(4*π/13) 0.709209774085 = 2*cos(10*π/26) = 2*cos(5*π/13) 0.241073360511 = 2*cos(12*π/26) = 2*cos(6*π/13) Symmetry of Rotations and Cartesian products A1' d+g+i+k+m z2, z4, z6 A2' R Rz A2" p+f+h+j+l z, z3, z5 E1' p+f+h+j+l {x, y}, {xz2, yz2}, {xz4, yz4} E1" R+d+g+i+k+m {Rx, Ry}, {xz, yz}, {xz3, yz3}, {xz5, yz5} E2' d+g+i+k+m {x2−y2, xy}, {z2(x2−y2), xyz2}, {z4(x2−y2), xyz4} E2" f+h+j+l {z(x2−y2), xyz}, {z3(x2−y2), xyz3} E3' f+h+j+l+m {x(x2−3y2), y(3x2−y2)}, {xz2(x2−3y2), yz2(3x2−y2)} E3" g+i+k+m {xz(x2−3y2), yz(3x2−y2)}, {xz3(x2−3y2), yz3(3x2−y2)} E4' g+i+k+l+m {(x2−y2)2−4x2y2, xy(x2−y2)}, {z2((x2−y2)2−4x2y2), xyz2(x2−y2)} E4" h+j+l+m {z((x2−y2)2−4x2y2), xyz(x2−y2)} E5' h+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)} E5" i+k+l+m {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)} E6" j+k+l+m Notes: α The order of the D13h point group is 52, and the order of the principal axis (S13) is 26. The group has 16 irreducible representations. β The D13h point group is isomorphic to D13d, C26v and D26. γ The D13h point group is generated by two symmetry elements, S13 and either a perpendicular C2′ or a vertical σv. Also, the group may be generated from any two σv planes, or any σv and a non-coplanar C2′. The canonical choice, however, is to use redundant generators: C13, C2′ and σh. δ The group contains one set of C2′ symmetry axes perpendicular to the principal (z) axis. The x axis (but not the y axis) is a member of that set. Similarly, the single set of symmetry planes denoted σd contains the xz plane but not the yz plane. ε The lowest nonvanishing multipole moment in D13h 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 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 just less than half the order of the principal axis. θ 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. ι The regular tridecagon or triskaidecagon is not constructible by ruler and compass because cos(2*π/13) has an algebraic degree of 6. (It can be constructed by extended methods that allow angle trisection, as this corresponds to solving cubic equations). The value of cos(2*π/13) can be expressed using cubic roots and complex numbers, which is hardly useful for a real quantity. 2*cos(2π/13) = ( 3√104−20*√13+12*√39*i + 3√104−20*√13−12*√39*i + √13−1 )/6
| D11h | ||
| D12h | ||
| C13 C13v C13h D13 | D13h | D13d |
| D14h | ||
| D15h |
This Character Table for the D13h point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.