C18h E 2 C18 2 C9 2 C6 2 C9^2 2 C18^5 2 C3 2 C18^7 2 C9^4 C2 i 2 S18 2 S9 2 S6 2 S9^2 2 S18^5 2 S3 2 S18^7 2 S9^4 sh <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> Ag 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 ........... ............T Bg 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 2.0000 -1.8793 -1.5320 -1.0000 -0.3473 0.3473 1.0000 1.5320 1.8793 -2.0000 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 2.0000 1.5320 0.3473 -1.0000 -1.8793 -1.8793 -1.0000 0.3473 1.5320 2.0000 ... ... TT... ....... ....TT... ........... ........TT... E3g * 2.0000 1.0000 -1.0000 -2.0000 -1.0000 1.0000 2.0000 1.0000 -1.0000 -2.0000 2.0000 -1.0000 1.0000 2.0000 1.0000 -1.0000 -2.0000 -1.0000 1.0000 -2.0000 ... ... ..... ....... ..TT..... ........... ......TT..... E4g * 2.0000 0.3473 -1.8793 -1.0000 1.5320 1.5320 -1.0000 -1.8793 0.3473 2.0000 2.0000 0.3473 -1.8793 -1.0000 1.5320 1.5320 -1.0000 -1.8793 0.3473 2.0000 ... ... ..... ....... TT....... ........... ....TT....... E5g * 2.0000 -0.3473 -1.8793 1.0000 1.5320 -1.5320 -1.0000 1.8793 0.3473 -2.0000 2.0000 0.3473 1.8793 -1.0000 -1.5320 1.5320 1.0000 -1.8793 -0.3473 -2.0000 ... ... ..... ....... ......... ........... ..TT......... E6g * 2.0000 -1.0000 -1.0000 2.0000 -1.0000 -1.0000 2.0000 -1.0000 -1.0000 2.0000 2.0000 -1.0000 -1.0000 2.0000 -1.0000 -1.0000 2.0000 -1.0000 -1.0000 2.0000 ... ... ..... ....... ......... ........... TT........... E7g * 2.0000 -1.5320 0.3473 1.0000 -1.8793 1.8793 -1.0000 -0.3473 1.5320 -2.0000 2.0000 1.5320 -0.3473 -1.0000 1.8793 -1.8793 1.0000 0.3473 -1.5320 -2.0000 ... ... ..... ....... ......... ........... ............. E8g * 2.0000 -1.8793 1.5320 -1.0000 0.3473 0.3473 -1.0000 1.5320 -1.8793 2.0000 2.0000 -1.8793 1.5320 -1.0000 0.3473 0.3473 -1.0000 1.5320 -1.8793 2.0000 ... ... ..... ....... ......... ........... ............. Au 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 ............. Bu 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 -2.0000 1.8793 1.5320 1.0000 0.3473 -0.3473 -1.0000 -1.5320 -1.8793 2.0000 ... TT. ..... ....TT. ......... ........TT. ............. E2u * 2.0000 1.5320 0.3473 -1.0000 -1.8793 -1.8793 -1.0000 0.3473 1.5320 2.0000 -2.0000 -1.5320 -0.3473 1.0000 1.8793 1.8793 1.0000 -0.3473 -1.5320 -2.0000 ... ... ..... ..TT... ......... ......TT... ............. E3u * 2.0000 1.0000 -1.0000 -2.0000 -1.0000 1.0000 2.0000 1.0000 -1.0000 -2.0000 -2.0000 1.0000 -1.0000 -2.0000 -1.0000 1.0000 2.0000 1.0000 -1.0000 2.0000 ... ... ..... TT..... ......... ....TT..... ............. E4u * 2.0000 0.3473 -1.8793 -1.0000 1.5320 1.5320 -1.0000 -1.8793 0.3473 2.0000 -2.0000 -0.3473 1.8793 1.0000 -1.5320 -1.5320 1.0000 1.8793 -0.3473 -2.0000 ... ... ..... ....... ......... ..TT....... ............. E5u * 2.0000 -0.3473 -1.8793 1.0000 1.5320 -1.5320 -1.0000 1.8793 0.3473 -2.0000 -2.0000 -0.3473 -1.8793 1.0000 1.5320 -1.5320 -1.0000 1.8793 0.3473 2.0000 ... ... ..... ....... ......... TT......... ............. E6u * 2.0000 -1.0000 -1.0000 2.0000 -1.0000 -1.0000 2.0000 -1.0000 -1.0000 2.0000 -2.0000 1.0000 1.0000 -2.0000 1.0000 1.0000 -2.0000 1.0000 1.0000 -2.0000 ... ... ..... ....... ......... ........... ............. E7u * 2.0000 -1.5320 0.3473 1.0000 -1.8793 1.8793 -1.0000 -0.3473 1.5320 -2.0000 -2.0000 -1.5320 0.3473 1.0000 -1.8793 1.8793 -1.0000 -0.3473 1.5320 2.0000 ... ... ..... ....... ......... ........... ............. E8u * 2.0000 -1.8793 1.5320 -1.0000 0.3473 0.3473 -1.0000 1.5320 -1.8793 2.0000 -2.0000 1.8793 -1.5320 1.0000 -0.3473 -0.3473 1.0000 -1.5320 1.8793 -2.0000 ... ... ..... ....... ......... ........... ............. 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 Ag R+d+g+i+k+m Rz, z2, z4, z6 Bg 2m E1g R+d+g+i+k+m {Rx, Ry}, {xz, yz}, {xz3, yz3}, {xz5, yz5} E2g d+g+i+k+m {x2−y2, xy}, {z2(x2−y2), xyz2}, {z4(x2−y2), xyz4} E3g g+i+k+m {xz(x2−3y2), yz(3x2−y2)}, {xz3(x2−3y2), yz3(3x2−y2)} E4g g+i+k+m {(x2−y2)2−4x2y2, xy(x2−y2)}, {z2((x2−y2)2−4x2y2), xyz2(x2−y2)} E5g i+k+m {xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), yz((5+2√5)x2−y2)((5−2√5)x2−y2)} E6g i+k+m {x2(x2−3y2)2−y2(3x2−y2)2, xy(x2−3y2)(3x2−y2)} E7g k+m E8g k+2m Au p+f+h+j+l z, z3, z5 Bu 2l E1u p+f+h+j+l {x, y}, {xz2, yz2}, {xz4, yz4} E2u f+h+j+l {z(x2−y2), xyz}, {z3(x2−y2), xyz3} E3u f+h+j+l {x(x2−3y2), y(3x2−y2)}, {xz2(x2−3y2), yz2(3x2−y2)} E4u h+j+l {z((x2−y2)2−4x2y2), xyz(x2−y2)} E5u h+j+l {x(x2−(5+2√5)y2)(x2−(5−2√5)y2), y((5+2√5)x2−y2)((5−2√5)x2−y2)} E6u j+l E7u j+l E8u l Notes: α The order of the C18h point group is 36, and the order of the principal axis (C18) is 18. The group has 20 irreducible representations. β The C18h point group is generated by two symmetry elements, which are canonically chosen as C18 and i. Other possible choices are C18 and σh, or less commonly S18 with either C2 or σh. γ The lowest nonvanishing multipole moment in C18h is 4 (quadrupole moment). δ This is an Abelian point group (the commutative law holds between all symmetry operations). The C18h group is Abelian because all its symmetry operations are coaxial. This is a sufficient condition. In Abelian groups, all symmetry operations form a class of their own, and all irreducible representations are one-dimensional. ε Because the group is Abelian and the maximum order of rotation is >2, some irreducible representations have complex characters. These 32 cases have been combined into 16 two-dimensional representations that are no longer irreducible but have real-valued characters. Accordingly, 16 pairs of left and right rotations have been combined into one two-membered pseudo-class each. ζ The 16 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. 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.
| C16h | ||
| C17h | ||
| C18 C18v | C18h | D18 D18h D18d S18 |
| C19h | ||
| C20h |
This Character Table for the C18h point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.