Character table for the D_3d point group

D3d     E       2 C3    3 C2'   i       2 S6    3 sd       <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> 
A1g       1       1       1       1       1       1        ... ... ....T ....... ...T....T ........... T......T....T
A2g       1       1      -1       1       1      -1        ..T ... ..... ....... ..T...... ........... .T....T......
Eg        2      -1       0       2      -1       0        TT. ... TTTT. ....... TT..TTTT. ........... ..TTTT..TTTT.
A1u       1       1       1      -1      -1      -1        ... ... ..... T...... ......... ....T...... .............
A2u       1       1      -1      -1      -1       1        ... ..T ..... .T....T ......... .....T....T .............
Eu        2      -1       0      -2       1       0        ... TT. ..... ..TTTT. ......... TTTT..TTTT. .............


 Symmetry of Rotations and Cartesian products

A1g  d+2g+3i+3k+4m     z2, yz(3x2−y2), z4, x2(x2−3y2)2−y2(3x2−y2)2, yz3(3x2−y2), z6 
A2g  R+g+2i+2k+3m      Rz, xz(x2−3y2), xy(x2−3y2)(3x2−y2), xz3(x2−3y2) 
Eg   R+2d+3g+4i+6k+7m  {Rx, Ry}, {x2−y2, xy}, {xz, yz}, {(x2−y2)2−4x2y2, xy(x2−y2)}, {z2(x2−y2), xyz2}, {xz3, yz3}, {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} 
A1u  f+h+2j+3l         x(x2−3y2), xz2(x2−3y2) 
A2u  p+2f+2h+3j+4l     z, y(3x2−y2), z3, yz2(3x2−y2), z5 
Eu   p+2f+4h+5j+6l     {x, y}, {z(x2−y2), xyz}, {xz2, yz2}, {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} 

 Notes:

    α  The order of the D3d point group is 12, and the order of the principal axis (S6) is 6. The group has 6 irreducible representations.

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

    γ  The D3d point group is generated by two symmetry elements, S6 and either a perpendicular C2′ or a vertical σd.
       Also, the group may be generated from any C2′ plus any σd plane.
       The canonical choice, however, is to use redundant generators: C3, C2′ and i.

    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.
       Reversely, the single set of symmetry planes denoted σd contains the yz plane but not the xz plane.

    ε  The lowest nonvanishing multipole moment in D3d 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  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.