Character table for the C_6v point group

C6v     E       2 C6    2 C3    C2      3 sv    3 sd       <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> 
A1        1       1       1       1       1       1        ... ..T ....T ......T ........T ..........T T...........T
A2        1       1       1       1      -1      -1        ..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   p+d+f+g+h+2i+2j+2k+2l+2m     z, z2, z3, z4, z5, x2(x2−3y2)2−y2(3x2−y2)2, z6 
A2   R+i+j+k+l+m                  Rz, xy(x2−3y2)(3x2−y2) 
B1   f+g+h+i+j+k+2l+2m            x(x2−3y2), xz(x2−3y2), xz2(x2−3y2), xz3(x2−3y2) 
B2   f+g+h+i+j+k+2l+2m            y(3x2−y2), yz(3x2−y2), yz2(3x2−y2), yz3(3x2−y2) 
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 C6v point group is 12, and the order of the principal axis (C6) is 6. The group has 6 irreducible representations.

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

    γ  The C6v point group is generated by two symmetry elements, C6 and any σv (or, non-canonically, any σd).
       Also, the group may be generated from a σv plus a σd (some pairs will yield smaller groups, though; choosing a minimum angle is safe).

    δ  There are two different sets of symmetry planes containing the principal axis (z axis in standard orientation).
       By convention, the set denoted as σv has the xz plane as a member, while the yz plane is a member of the σd set.

    ε  The lowest nonvanishing multipole moment in C6v 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.