Character table for the D18h point group

D18h    E       2 C18   2 C9    2 C6    2 C9^2  2 C18^5 2 C3    2 C18^7 2 C9^4  C2      9 C2'   9 C2"   i       2 S18   2 S9    2 S6    2 S9^2  2 S18^5 2 S3    2 S18^7 2 S9^4  sh      9 sv    9 sd       <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> 
A1g     1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000     ... ... ....T ....... ........T ........... ............T
A2g     1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000 -1.0000 -1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000 -1.0000 -1.0000     ..T ... ..... ....... ......... ........... .............
B1g     1.0000 -1.0000  1.0000 -1.0000  1.0000 -1.0000  1.0000 -1.0000  1.0000 -1.0000  1.0000 -1.0000  1.0000  1.0000 -1.0000  1.0000 -1.0000  1.0000 -1.0000  1.0000 -1.0000 -1.0000 -1.0000  1.0000     ... ... ..... ....... ......... ........... .............
B2g     1.0000 -1.0000  1.0000 -1.0000  1.0000 -1.0000  1.0000 -1.0000  1.0000 -1.0000 -1.0000  1.0000  1.0000  1.0000 -1.0000  1.0000 -1.0000  1.0000 -1.0000  1.0000 -1.0000 -1.0000  1.0000 -1.0000     ... ... ..... ....... ......... ........... .............
E1g     2.0000  1.8793  1.5320  1.0000  0.3473 -0.3473 -1.0000 -1.5320 -1.8793 -2.0000  0.0000  0.0000  2.0000 -1.8793 -1.5320 -1.0000 -0.3473  0.3473  1.0000  1.5320  1.8793 -2.0000  0.0000  0.0000     TT. ... ..TT. ....... ......TT. ........... ..........TT.
E2g     2.0000  1.5320  0.3473 -1.0000 -1.8793 -1.8793 -1.0000  0.3473  1.5320  2.0000  0.0000  0.0000  2.0000  1.5320  0.3473 -1.0000 -1.8793 -1.8793 -1.0000  0.3473  1.5320  2.0000  0.0000  0.0000     ... ... TT... ....... ....TT... ........... ........TT...
E3g     2.0000  1.0000 -1.0000 -2.0000 -1.0000  1.0000  2.0000  1.0000 -1.0000 -2.0000  0.0000  0.0000  2.0000 -1.0000  1.0000  2.0000  1.0000 -1.0000 -2.0000 -1.0000  1.0000 -2.0000  0.0000  0.0000     ... ... ..... ....... ..TT..... ........... ......TT.....
E4g     2.0000  0.3473 -1.8793 -1.0000  1.5320  1.5320 -1.0000 -1.8793  0.3473  2.0000  0.0000  0.0000  2.0000  0.3473 -1.8793 -1.0000  1.5320  1.5320 -1.0000 -1.8793  0.3473  2.0000  0.0000  0.0000     ... ... ..... ....... TT....... ........... ....TT.......
E5g     2.0000 -0.3473 -1.8793  1.0000  1.5320 -1.5320 -1.0000  1.8793  0.3473 -2.0000  0.0000  0.0000  2.0000  0.3473  1.8793 -1.0000 -1.5320  1.5320  1.0000 -1.8793 -0.3473 -2.0000  0.0000  0.0000     ... ... ..... ....... ......... ........... ..TT.........
E6g     2.0000 -1.0000 -1.0000  2.0000 -1.0000 -1.0000  2.0000 -1.0000 -1.0000  2.0000  0.0000  0.0000  2.0000 -1.0000 -1.0000  2.0000 -1.0000 -1.0000  2.0000 -1.0000 -1.0000  2.0000  0.0000  0.0000     ... ... ..... ....... ......... ........... TT...........
E7g     2.0000 -1.5320  0.3473  1.0000 -1.8793  1.8793 -1.0000 -0.3473  1.5320 -2.0000  0.0000  0.0000  2.0000  1.5320 -0.3473 -1.0000  1.8793 -1.8793  1.0000  0.3473 -1.5320 -2.0000  0.0000  0.0000     ... ... ..... ....... ......... ........... .............
E8g     2.0000 -1.8793  1.5320 -1.0000  0.3473  0.3473 -1.0000  1.5320 -1.8793  2.0000  0.0000  0.0000  2.0000 -1.8793  1.5320 -1.0000  0.3473  0.3473 -1.0000  1.5320 -1.8793  2.0000  0.0000  0.0000     ... ... ..... ....... ......... ........... .............
A1u     1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000     ... ... ..... ....... ......... ........... .............
A2u     1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000  1.0000  1.0000     ... ..T ..... ......T ......... ..........T .............
B1u     1.0000 -1.0000  1.0000 -1.0000  1.0000 -1.0000  1.0000 -1.0000  1.0000 -1.0000  1.0000 -1.0000 -1.0000 -1.0000  1.0000 -1.0000  1.0000 -1.0000  1.0000 -1.0000  1.0000  1.0000  1.0000 -1.0000     ... ... ..... ....... ......... ........... .............
B2u     1.0000 -1.0000  1.0000 -1.0000  1.0000 -1.0000  1.0000 -1.0000  1.0000 -1.0000 -1.0000  1.0000 -1.0000 -1.0000  1.0000 -1.0000  1.0000 -1.0000  1.0000 -1.0000  1.0000  1.0000 -1.0000  1.0000     ... ... ..... ....... ......... ........... .............
E1u     2.0000  1.8793  1.5320  1.0000  0.3473 -0.3473 -1.0000 -1.5320 -1.8793 -2.0000  0.0000  0.0000 -2.0000  1.8793  1.5320  1.0000  0.3473 -0.3473 -1.0000 -1.5320 -1.8793  2.0000  0.0000  0.0000     ... TT. ..... ....TT. ......... ........TT. .............
E2u     2.0000  1.5320  0.3473 -1.0000 -1.8793 -1.8793 -1.0000  0.3473  1.5320  2.0000  0.0000  0.0000 -2.0000 -1.5320 -0.3473  1.0000  1.8793  1.8793  1.0000 -0.3473 -1.5320 -2.0000  0.0000  0.0000     ... ... ..... ..TT... ......... ......TT... .............
E3u     2.0000  1.0000 -1.0000 -2.0000 -1.0000  1.0000  2.0000  1.0000 -1.0000 -2.0000  0.0000  0.0000 -2.0000  1.0000 -1.0000 -2.0000 -1.0000  1.0000  2.0000  1.0000 -1.0000  2.0000  0.0000  0.0000     ... ... ..... TT..... ......... ....TT..... .............
E4u     2.0000  0.3473 -1.8793 -1.0000  1.5320  1.5320 -1.0000 -1.8793  0.3473  2.0000  0.0000  0.0000 -2.0000 -0.3473  1.8793  1.0000 -1.5320 -1.5320  1.0000  1.8793 -0.3473 -2.0000  0.0000  0.0000     ... ... ..... ....... ......... ..TT....... .............
E5u     2.0000 -0.3473 -1.8793  1.0000  1.5320 -1.5320 -1.0000  1.8793  0.3473 -2.0000  0.0000  0.0000 -2.0000 -0.3473 -1.8793  1.0000  1.5320 -1.5320 -1.0000  1.8793  0.3473  2.0000  0.0000  0.0000     ... ... ..... ....... ......... TT......... .............
E6u     2.0000 -1.0000 -1.0000  2.0000 -1.0000 -1.0000  2.0000 -1.0000 -1.0000  2.0000  0.0000  0.0000 -2.0000  1.0000  1.0000 -2.0000  1.0000  1.0000 -2.0000  1.0000  1.0000 -2.0000  0.0000  0.0000     ... ... ..... ....... ......... ........... .............
E7u     2.0000 -1.5320  0.3473  1.0000 -1.8793  1.8793 -1.0000 -0.3473  1.5320 -2.0000  0.0000  0.0000 -2.0000 -1.5320  0.3473  1.0000 -1.8793  1.8793 -1.0000 -0.3473  1.5320  2.0000  0.0000  0.0000     ... ... ..... ....... ......... ........... .............
E8u     2.0000 -1.8793  1.5320 -1.0000  0.3473  0.3473 -1.0000  1.5320 -1.8793  2.0000  0.0000  0.0000 -2.0000  1.8793 -1.5320  1.0000 -0.3473 -0.3473  1.0000 -1.5320  1.8793 -2.0000  0.0000  0.0000     ... ... ..... ....... ......... ........... .............

 Irrational character values:  1.879385241572 = 2*cos(2*π/18) = 2*cos(π/9)
                               1.532088886238 = 2*cos(4*π/18) = 2*cos(2*π/9)
                               0.347296355334 = 2*cos(8*π/18) = 2*cos(4*π/9)



 Symmetry of Rotations and Cartesian products

A1g  d+g+i+k+m    z2, z4, z6 
A2g  R            Rz 
B1g  m 
B2g  m 
E1g  R+d+g+i+k+m  {Rx, Ry}, {xz, yz}, {xz3, yz3}, {xz5, yz5} 
E2g  d+g+i+k+m    {x2y2, xy}, {z2(x2y2), xyz2}, {z4(x2y2), xyz4} 
E3g  g+i+k+m      {xz(x2−3y2), yz(3x2y2)}, {xz3(x2−3y2), yz3(3x2y2)} 
E4g  g+i+k+m      {(x2y2)2−4x2y2, xy(x2y2)}, {z2((x2y2)2−4x2y2), xyz2(x2y2)} 
E5g  i+k+m        {xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), yz((5+2√5)x2y2)((5−2√5)x2y2)} 
E6g  i+k+m        {x2(x2−3y2)2y2(3x2y2)2, xy(x2−3y2)(3x2y2)} 
E7g  k+m 
E8g  k+2m 
A2u  p+f+h+j+l    z, z3, z5 
B1u  l 
B2u  l 
E1u  p+f+h+j+l    {x, y}, {xz2, yz2}, {xz4, yz4} 
E2u  f+h+j+l      {z(x2y2), xyz}, {z3(x2y2), xyz3} 
E3u  f+h+j+l      {x(x2−3y2), y(3x2y2)}, {xz2(x2−3y2), yz2(3x2y2)} 
E4u  h+j+l        {z((x2y2)2−4x2y2), xyz(x2y2)} 
E5u  h+j+l        {x(x2−(5+2√5)y2)(x2−(5−2√5)y2), y((5+2√5)x2y2)((5−2√5)x2y2)} 
E6u  j+l 
E7u  j+l 
E8u  l 

 Notes:

    α  The order of the D18h point group is 72, and the order of the principal axis (C18) is 18. The group has 24 irreducible representations.

    β  The D18h point group is generated by three symmetry elements that are canonically chosen C18, C2 and i.
       Other choices include σh instead of i, or any of C2, σv or σd instead of C2. Also, some ternary combinations of C2, C2, σv and σd act as generators.
       Lastly, the S18 can be chosen, together with σh or C2 and any one of C2, C2, σv or σd.

    γ  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 has the x axis as a member, while the y axis is a member of the C2 set.

    δ  There are two different sets of symmetry planes containing the principal axis (z axis in standard orientation).
       By convention, the set denoted as σv has the xz plane as a member, while the yz plane is a member of the σd set.

    ε  The lowest nonvanishing multipole moment in D18h 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 much 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.
       All characters of this group can be expressed using complex numbers, elementary arithmetic operations, square roots and third roots.

    ι  The regular nonagon or enneagon is not constructible by ruler and compass because cos(2*π/9) has an algebraic degree of 3.
       (It can be constructed by extended methods that allow angle trisection, as this corresponds to solving cubic equations).
       The value of cos(2*π/9) can be expressed using cubic roots and complex numbers, which, however, is not very useful
       for a real-valued quantity: 2*cos(2π/9) = (3−4+i*4*√3 + 3−4−i*4*√3)/2.
       Therefore, regular polygons of order 18,27,36,45,54 etc. are also inconstructible, and their cosines have no representation in real radicals.

This Character Table for the D18h point group was created by Gernot Katzer.

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