C12v E 2 C12 2 C6 2 C4 2 C3 2 C12^5 C2 6 sv 6 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 ... ..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 ..T ... ..... ....... ......... ........... ............. B1 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 1.0000 -1.0000 ... ... ..... ....... ......... ........... T............ B2 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 ... ... ..... ....... ......... ........... .T........... E1 2.0000 1.7320 1.0000 0.0000 -1.0000 -1.7320 -2.0000 0.0000 0.0000 TT. TT. ..TT. ....TT. ......TT. ........TT. ..........TT. E2 2.0000 1.0000 -1.0000 -2.0000 -1.0000 1.0000 2.0000 0.0000 0.0000 ... ... TT... ..TT... ....TT... ......TT... ........TT... E3 2.0000 0.0000 -2.0000 0.0000 2.0000 0.0000 -2.0000 0.0000 0.0000 ... ... ..... TT..... ..TT..... ....TT..... ......TT..... E4 2.0000 -1.0000 -1.0000 2.0000 -1.0000 -1.0000 2.0000 0.0000 0.0000 ... ... ..... ....... TT....... ..TT....... ....TT....... E5 2.0000 -1.7320 1.0000 0.0000 -1.0000 1.7320 -2.0000 0.0000 0.0000 ... ... ..... ....... ......... TT......... ..TT......... Irrational character values: 1.732050807569 = 2*cos(2*π/12) = 2*cos(π/6) = √3 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 i+j+k+l+m x2(x2−3y2)2−y2(3x2−y2)2 B2 i+j+k+l+m xy(x2−3y2)(3x2−y2) 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+2m {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+2l+2m {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+2k+2l+2m {(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+2j+2k+2l+2m {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)} Notes: α The order of the C12v point group is 24, and the order of the principal axis (C12) is 12. The group has 9 irreducible representations. β The C12v point group is isomorphic to D6d and D12. γ The C12v point group is generated by two symmetry elements, C12 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 contains both the xz and the yz planes. ε The lowest nonvanishing multipole moment in C12v 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 constructible polygon, as the order of the principal axis is a product of any number of different Fermat primes (3,5,17,257,65537) times an arbitrary power of two. Therefore, all characters have an algebraic degree which is a power of two and can be expressed as radicals involving only square roots and integer numbers.
| C10v | ||
| C11v | ||
| C12 | C12v | C12h D12 D12h D12d S12 |
| C13v | ||
| C14v |
This Character Table for the C12v point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.