Character table for the C13h point group

C13h    E        2 C13    2 C13^2  2 C13^3  2 C13^4  2 C13^5  2 C13^6  sh       2 S13    2 S13^3  2 S13^5  2 S13^7  2 S13^9  2 S13^11    <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> 
A'      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 ........... ............T
A"      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  2.00000  1.77091  0.24107 -1.49702 -1.94188 -0.70921  1.13613     ... TT. ..... ....TT. ......... ........TT. .............
E1" *   2.00000  1.77091  1.13613  0.24107 -0.70921 -1.49702 -1.94188 -2.00000 -1.77091 -0.24107  1.49702  1.94188  0.70921 -1.13613     TT. ... ..TT. ....... ......TT. ........... ..........TT.
E2' *   2.00000  1.13613 -0.70921 -1.94188 -1.49702  0.24107  1.77091  2.00000  1.13613 -1.94188  0.24107  1.77091 -1.49702 -0.70921     ... ... TT... ....... ....TT... ........... ........TT...
E2" *   2.00000  1.13613 -0.70921 -1.94188 -1.49702  0.24107  1.77091 -2.00000 -1.13613  1.94188 -0.24107 -1.77091  1.49702  0.70921     ... ... ..... ..TT... ......... ......TT... .............
E3' *   2.00000  0.24107 -1.94188 -0.70921  1.77091  1.13613 -1.49702  2.00000  0.24107 -0.70921  1.13613 -1.49702  1.77091 -1.94188     ... ... ..... TT..... ......... ....TT..... .............
E3" *   2.00000  0.24107 -1.94188 -0.70921  1.77091  1.13613 -1.49702 -2.00000 -0.24107  0.70921 -1.13613  1.49702 -1.77091  1.94188     ... ... ..... ....... ..TT..... ........... ......TT.....
E4' *   2.00000 -0.70921 -1.49702  1.77091  0.24107 -1.94188  1.13613  2.00000 -0.70921  1.77091 -1.94188  1.13613  0.24107 -1.49702     ... ... ..... ....... TT....... ........... ....TT.......
E4" *   2.00000 -0.70921 -1.49702  1.77091  0.24107 -1.94188  1.13613 -2.00000  0.70921 -1.77091  1.94188 -1.13613 -0.24107  1.49702     ... ... ..... ....... ......... ..TT....... .............
E5' *   2.00000 -1.49702  0.24107  1.13613 -1.94188  1.77091 -0.70921  2.00000 -1.49702  1.13613  1.77091 -0.70921 -1.94188  0.24107     ... ... ..... ....... ......... TT......... .............
E5" *   2.00000 -1.49702  0.24107  1.13613 -1.94188  1.77091 -0.70921 -2.00000  1.49702 -1.13613 -1.77091  0.70921  1.94188 -0.24107     ... ... ..... ....... ......... ........... ..TT.........
E6' *   2.00000 -1.94188  1.77091 -1.49702  1.13613 -0.70921  0.24107  2.00000 -1.94188 -1.49702 -0.70921  0.24107  1.13613  1.77091     ... ... ..... ....... ......... ........... TT...........
E6" *   2.00000 -1.94188  1.77091 -1.49702  1.13613 -0.70921  0.24107 -2.00000  1.94188  1.49702  0.70921 -0.24107 -1.13613 -1.77091     ... ... ..... ....... ......... ........... .............

 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

A'   R+d+g+i+k+m  Rz, z2, z4, z6 
A"   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    {x2y2, xy}, {z2(x2y2), xyz2}, {z4(x2y2), xyz4} 
E2"  f+h+j+l      {z(x2y2), xyz}, {z3(x2y2), xyz3} 
E3'  f+h+j+l+m    {x(x2−3y2), y(3x2y2)}, {xz2(x2−3y2), yz2(3x2y2)} 
E3"  g+i+k+m      {xz(x2−3y2), yz(3x2y2)}, {xz3(x2−3y2), yz3(3x2y2)} 
E4'  g+i+k+l+m    {(x2y2)2−4x2y2, xy(x2y2)}, {z2((x2y2)2−4x2y2), xyz2(x2y2)} 
E4"  h+j+l+m      {z((x2y2)2−4x2y2), xyz(x2y2)} 
E5'  h+j+k+l+m    {x(x2−(5+2√5)y2)(x2−(5−2√5)y2), y((5+2√5)x2y2)((5−2√5)x2y2)} 
E5"  i+k+l+m      {xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), yz((5+2√5)x2y2)((5−2√5)x2y2)} 
E6'  i+j+k+l+m    {x2(x2−3y2)2y2(3x2y2)2, xy(x2−3y2)(3x2y2)} 
E6"  j+k+l+m 

 Notes:

    α  The order of the C13h point group is 26, and the order of the principal axis (S13) is 26. The group has 14 irreducible representations.

    β  The C13h point group could also be named S13, as it contains the S13 axis as its only symmetry element.
       Another rare designation is C26i because the S13 axis is identical to a roto-inversion axis of order 26.

    γ  The C13h point group is isomorphic to C26 and S26.

    δ  The C13h point group is generated by one single symmetry element, S13. Therefore, it is a cyclic group.
       The canonical choice, however, is to use redundant generators: C13 and σh.

    ε  The lowest nonvanishing multipole moment in C13h is 4 (quadrupole moment).

    ζ  This is an Abelian point group (the commutative law holds between all symmetry operations).
       The C13h 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.

    η  Because the group is Abelian and the maximum order of rotation is >2, some irreducible representations have complex characters.
       These 24 cases have been combined into 12 two-dimensional representations that are no longer irreducible but have real-valued characters.
       Accordingly, 12 pairs of left and right rotations have been combined into one two-membered pseudo-class each.

    θ  The 12 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.

    ι  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.

    κ  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.

    λ  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 C13h point group was created by Gernot Katzer.

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