C18v E 2 C18 2 C9 2 C6 2 C9^2 2 C18^5 2 C3 2 C18^7 2 C9^4 C2 9 sv 9 sd <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> A1 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 ........T ..........T ............T A2 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 ... ..... ....... ......... ........... ............. B1 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 ... ... ..... ....... ......... ........... ............. B2 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 -1.0000 1.0000 ... ... ..... ....... ......... ........... ............. E1 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. ......TT. ........TT. ..........TT. E2 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... ......TT... ........TT... E3 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..... ....TT..... ......TT..... E4 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....... ....TT....... E5 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......... E6 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........... E7 2.0000 -1.5320 0.3473 1.0000 -1.8793 1.8793 -1.0000 -0.3473 1.5320 -2.0000 0.0000 0.0000 ... ... ..... ....... ......... ........... ............. E8 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 A1 p+d+f+g+h+i+j+k+l+m z, z2, z3, z4, z5, z6 A2 R Rz B1 l+m B2 l+m 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+2m Notes: α The order of the C18v point group is 36, and the order of the principal axis (C18) is 18. The group has 12 irreducible representations. β The C18v point group is isomorphic to D9d, D9h and D18. γ The C18v point group is generated by two symmetry elements, C18 and any σv (or, non-canonically, any σd). Also, the group may be generated from a σv plus a σd (some pairs will yield smaller groups, though; choosing a minimum angle is safe). 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 C18v 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 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.
| C16v | ||
| C17v | ||
| C18 | C18v | C18h D18 D18h D18d S18 |
| C19v | ||
| C20v |
This Character Table for the C18v point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.