Character table for the D6d point group

D6d     E       2 S12   2 C6    2 S4    2 C3    2 S12^5 C2      6 C2'   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
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 ..... ......T ......... ..........T .T...........
E1      2.0000  1.7320  1.0000  0.0000 -1.0000 -1.7320 -2.0000  0.0000  0.0000     ... 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...
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... ....TT.......
E5      2.0000 -1.7320  1.0000  0.0000 -1.0000  1.7320 -2.0000  0.0000  0.0000     TT. ... ..TT. ....... ......TT. TT......... ..........TT.

 Irrational character values:  1.732050807569 = 2*cos(2*π/12) = 2*cos(π/6) = √3



 Symmetry of Rotations and Cartesian products

A1   d+g+i+j+k+l+m        z2, z4, z6 
A2   R+j+l                Rz 
B1   i+k+m                x2(x2−3y2)2y2(3x2y2)2 
B2   p+f+h+i+j+k+l+m      z, z3, z5, xy(x2−3y2)(3x2y2) 
E1   p+f+h+i+j+2k+l+2m    {x, y}, {xz2, yz2}, {xz4, yz4}, {xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), yz((5+2√5)x2y2)((5−2√5)x2y2)} 
E2   d+g+h+i+j+k+2l+2m    {x2y2, xy}, {z2(x2y2), xyz2}, {z((x2y2)2−4x2y2), xyz(x2y2)}, {z4(x2y2), xyz4} 
E3   f+g+h+i+j+k+2l+2m    {x(x2−3y2), y(3x2y2)}, {xz(x2−3y2), yz(3x2y2)}, {xz2(x2−3y2), yz2(3x2y2)}, {xz3(x2−3y2), yz3(3x2y2)} 
E4   f+g+h+i+j+2k+l+2m    {z(x2y2), xyz}, {(x2y2)2−4x2y2, xy(x2y2)}, {z3(x2y2), xyz3}, {z2((x2y2)2−4x2y2), xyz2(x2y2)} 
E5   R+d+g+h+i+2j+k+2l+m  {Rx, Ry}, {xz, yz}, {xz3, yz3}, {x(x2−(5+2√5)y2)(x2−(5−2√5)y2), y((5+2√5)x2y2)((5−2√5)x2y2)}, {xz5, yz5} 

 Notes:

    α  The order of the D6d point group is 24, and the order of the principal axis (S12) is 12. The group has 9 irreducible representations.

    β  The D6d point group is isomorphic to C12v and D12.

    γ  The D6d point group is generated by two symmetry elements, S12 and either a perpendicular C2 or a vertical σd.
       Also, the group may be generated from a C2 plus a σd (some pairs will yield smaller groups, though; choosing a minimum angle is safe).

    δ  The group contains one set of twofold symmetry axes (C2) perpendicular to the principal (z) axis. Both x and y axes are members of that set.

    ε  The single σd set of symmetry planes contains neither the xz nor the yz planes; but it contains the median plane (x+y)z.

    ζ  The lowest nonvanishing multipole moment in D6d is 4 (quadrupole moment).

    η  This point group is non-Abelian (some symmetry operations are not commutative).
       Therefore, the character table contains multi-membered classes and degenerate irreducible representations.

    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 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.

This Character Table for the D6d point group was created by Gernot Katzer.

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