Character table for the D₆ point group

D6      E       2 C6    2 C3    C2      3 C2'   3 C2"      <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> 
A1        1       1       1       1       1       1        ... ... ....T ....... ........T ........... T...........T
A2        1       1       1       1      -1      -1        ..T ..T ..... ......T ......... ..........T .T...........
B1        1      -1       1      -1       1      -1        ... ... ..... T...... ...T..... ....T...... .......T.....
B2        1      -1       1      -1      -1       1        ... ... ..... .T..... ..T...... .....T..... ......T......
E1        2       1      -1      -2       0       0        TT. TT. ..TT. ....TT. ......TT. TT......TT. ..TT......TT.
E2        2      -1      -1       2       0       0        ... ... TT... ..TT... TT..TT... ..TT..TT... ....TT..TT...


 Symmetry of Rotations and Cartesian products

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

 Notes:

    α  The order of the D6 point group is 12, and the order of the principal axis (C6) is 6. The group has 6 irreducible representations.

    β  The D6 point group is isomorphic to D3d, D3h and C6v.

    γ  The D6 point group is generated by two symmetry elements, C6 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 D6 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  All characters are integers because the order of the principal axis is 1,2,3,4 or 6.
       This implies that the point group corresponds to a constructible polygon which can be used for tiling the plane.
       Such point groups are also referred to as “crystallographic point groups”, as they are compatible with periodic lattice symmetry.
       There are exactly 32 such groups: C1,Cs,Ci,C2,C2h,C2v,C3,C3h,C3v,C4,C4h,C4v,C6,C6h,C6v,D2,D2d,D2h,D3,D3d,D3h,D4,D4h,D6,D6h,S4,S6,T,Td,Th,O,Oh.