Character table for the C13v point group

C13v    E       2 C13   2 C13^2 2 C13^3 2 C13^4 2 C13^5 2 C13^6 13 sv      <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> 
A1      1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000     ... ..T ....T ......T ........T ..........T ............T
A2      1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000 -1.0000     ..T ... ..... ....... ......... ........... .............
E1      2.0000  1.7709  1.1361  0.2410 -0.7092 -1.4970 -1.9418  0.0000     TT. TT. ..TT. ....TT. ......TT. ........TT. ..........TT.
E2      2.0000  1.1361 -0.7092 -1.9418 -1.4970  0.2410  1.7709  0.0000     ... ... TT... ..TT... ....TT... ......TT... ........TT...
E3      2.0000  0.2410 -1.9418 -0.7092  1.7709  1.1361 -1.4970  0.0000     ... ... ..... TT..... ..TT..... ....TT..... ......TT.....
E4      2.0000 -0.7092 -1.4970  1.7709  0.2410 -1.9418  1.1361  0.0000     ... ... ..... ....... TT....... ..TT....... ....TT.......
E5      2.0000 -1.4970  0.2410  1.1361 -1.9418  1.7709 -0.7092  0.0000     ... ... ..... ....... ......... TT......... ..TT.........
E6      2.0000 -1.9418  1.7709 -1.4970  1.1361 -0.7092  0.2410  0.0000     ... ... ..... ....... ......... ........... TT...........

 Irrational character values:  1.770912051306 = 2*cos(2*π/13)
                               1.136129493462 = 2*cos(4*π/13)
                               0.241073360511 = 2*cos(6*π/13)
                              -0.709209774085 = 2*cos(8*π/13)
                              -1.497021496342 = 2*cos(10*π/13)
                              -1.941883634852 = 2*cos(12*π/13)



 Symmetry of Rotations and Cartesian products

A1   p+d+f+g+h+i+j+k+l+m    z, z2, z3, z4, z5, z6 
A2   R                      Rz 
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      {x2y2, xy}, {z(x2y2), xyz}, {z2(x2y2), xyz2}, {z3(x2y2), xyz3}, {z4(x2y2), xyz4} 
E3   f+g+h+i+j+k+l+2m       {x(x2−3y2), y(3x2y2)}, {xz(x2−3y2), yz(3x2y2)}, {xz2(x2−3y2), yz2(3x2y2)}, {xz3(x2−3y2), yz3(3x2y2)} 
E4   g+h+i+j+k+2l+2m        {(x2y2)2−4x2y2, xy(x2y2)}, {z((x2y2)2−4x2y2), xyz(x2y2)}, {z2((x2y2)2−4x2y2), xyz2(x2y2)} 
E5   h+i+j+2k+2l+2m         {x(x2−(5+2√5)y2)(x2−(5−2√5)y2), y((5+2√5)x2y2)((5−2√5)x2y2)}, {xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), yz((5+2√5)x2y2)((5−2√5)x2y2)} 
E6   i+2j+2k+2l+2m          {x2(x2−3y2)2y2(3x2y2)2, xy(x2−3y2)(3x2y2)} 

 Notes:

    α  The order of the C13v point group is 26, and the order of the principal axis (C13) is 13. The group has 8 irreducible representations.

    β  The C13v point group is isomorphic to D13.

    γ  The C13v point group is generated by two symmetry elements, C13 and any σv.
       Also, the group may be generated from any two σv planes.

    δ  The group contains one set of symmetry planes σv intersecting in the principal (z) axis. The xz plane (but not the yz plane) is a member of that set.

    ε  The lowest nonvanishing multipole moment in C13v is 2 (dipole 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) = ( 3104−20*√13+12*√39*i + 3104−20*√13−12*√39*i + √13−1 )/6

This Character Table for the C13v point group was created by Gernot Katzer.

For other groups and some explanations, see the Main Page.