Character table for the D₃ point group

D3      E       2 C3    3 C2'      <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> 
A1        1       1       1        ... ... ....T T...... ...T....T ....T...... T......T....T
A2        1       1      -1        ..T ..T ..... .T....T ..T...... .....T....T .T....T......
E         2      -1       0        TT. TT. TTTT. ..TTTT. TT..TTTT. TTTT..TTTT. ..TTTT..TTTT.


 Symmetry of Rotations and Cartesian products

A1   d+f+2g+h+3i+2j+3k+3l+4m         z2, x(x2−3y2), yz(3x2−y2), z4, xz2(x2−3y2), x2(x2−3y2)2−y2(3x2−y2)2, yz3(3x2−y2), z6 
A2   R+p+2f+g+2h+2i+3j+2k+4l+3m      Rz, z, y(3x2−y2), z3, xz(x2−3y2), yz2(3x2−y2), z5, xy(x2−3y2)(3x2−y2), xz3(x2−3y2) 
E    R+p+2d+2f+3g+4h+4i+5j+6k+6l+7m  {Rx, Ry}, {x, y}, {x2−y2, xy}, {xz, yz}, {z(x2−y2), xyz}, {xz2, yz2}, {(x2−y2)2−4x2y2, xy(x2−y2)}, {z2(x2−y2), xyz2}, {xz3, yz3}, {x(x2−(5+2√5)y2)(x2−(5−2√5)y2), y((5+2√5)x2−y2)((5−2√5)x2−y2)}, {z((x2−y2)2−4x2y2), xyz(x2−y2)}, {z3(x2−y2), xyz3}, {xz4, yz4}, {xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), yz((5+2√5)x2−y2)((5−2√5)x2−y2)}, {z2((x2−y2)2−4x2y2), xyz2(x2−y2)}, {z4(x2−y2), xyz4}, {xz5, yz5} 

 Notes:

    α  The order of the D3 point group is 6, and the order of the principal axis (C3) is 3. The group has 3 irreducible representations.

    β  The D3 point group is isomorphic to C3v.
       It is also isomorphic to the Symmetric Group Sym(3).

    γ  The D3 point group is generated by two symmetry elements, C3 and a perpendicular C2′.
       Also, the group may be generated from any two C2′ axes.

    δ  The group contains one set of C2′ symmetry axes perpendicular to the principal (z) axis. The x axis (but not the y axis) is a member of that set.

    ε  The lowest nonvanishing multipole moment in D3 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.