D18 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" <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 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 ..T ..... ......T ......... ..........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 d+g+i+k+m z2, z4, z6 A2 R+p+f+h+j+l Rz, z, z3, z5 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 D18 point group is 36, and the order of the principal axis (C18) is 18. The group has 12 irreducible representations. β The D18 point group is isomorphic to D9d, D9h and C18v. γ The D18 point group is generated by two symmetry elements, C18 and a perpendicular C2′ (or, non-canonically, C2″). Also, the group may be generated from a C2′ plus a C2″ (some pairs will yield smaller groups, though; choosing a minimum angle is safe). δ 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. ε The lowest nonvanishing multipole moment in D18 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 The point group is chiral, as it does not contain any mirroring operation. θ 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. ι 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. κ 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.
| D16 | ||
| D17 | ||
| C18 C18v C18h | D18 | D18h D18d S18 |
| D19 | ||
| D20 |
This Character Table for the D18 point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.