Character table for the D_3h point group

D3h     E       2 C3    3 C2'   sh      2 S3    3 sv       <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> 
A1'       1       1       1       1       1       1        ... ... ....T T...... ........T ....T...... T...........T
A1"       1       1       1      -1      -1      -1        ... ... ..... ....... ...T..... ........... .......T.....
A2'       1       1      -1       1       1      -1        ..T ... ..... .T..... ......... .....T..... .T...........
A2"       1       1      -1      -1      -1       1        ... ..T ..... ......T ..T...... ..........T ......T......
E'        2      -1       0       2      -1       0        ... TT. TT... ....TT. TT..TT... TT......TT. ....TT..TT...
E"        2      -1       0      -2       1       0        TT. ... ..TT. ..TT... ......TT. ..TT..TT... ..TT......TT.


 Symmetry of Rotations and Cartesian products

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

 Notes:

    α  The order of the D3h point group is 12, and the order of the principal axis (S3) is 6. The group has 6 irreducible representations.

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

    γ  The D3h point group is generated by two symmetry elements, S3 and either a perpendicular C2′ or a vertical σv.
       Also, the group may be generated from any two σv planes, or any σv and a non-coplanar C2′.
       The canonical choice, however, is to use redundant generators: C3, C2′ and σh.

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

    ε  The lowest nonvanishing multipole moment in D3h 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.