Character table for the C_3v point group

C3v     E       2 C3    3 sv       <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> 
A1        1       1       1        ... ..T ....T T.....T ..T.....T ....T.....T T.....T.....T
A2        1       1      -1        ..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   p+d+2f+2g+2h+3i+3j+3k+4l+4m     z, z2, x(x2−3y2), z3, xz(x2−3y2), z4, xz2(x2−3y2), z5, x2(x2−3y2)2−y2(3x2−y2)2, xz3(x2−3y2), z6 
A2   R+f+g+h+2i+2j+2k+3l+3m          Rz, y(3x2−y2), yz(3x2−y2), yz2(3x2−y2), xy(x2−3y2)(3x2−y2), yz3(3x2−y2) 
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 C3v point group is 6, and the order of the principal axis (C3) is 3. The group has 3 irreducible representations.

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

    γ  The C3v point group is generated by two symmetry elements, C3 and any σv.
       Also, the group may be generated from any two σv planes.

    δ  The group contains one set of symmetry planes σv intersecting in the principal (z) axis. The xz plane (but not the yz plane) is a member of that set.

    ε  The lowest nonvanishing multipole moment in C3v is 2 (dipole 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.