Character table for the C_1v point group

C1v     E       sv         <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> 
A1        1       1        .T. T.T T.T.T T.T.T.T T.T.T.T.T T.T.T.T.T.T T.T.T.T.T.T.T
A2        1      -1        T.T .T. .T.T. .T.T.T. .T.T.T.T. .T.T.T.T.T. .T.T.T.T.T.T.


 Symmetry of Rotations and Cartesian products

A1   R+2p+3d+4f+5g+6h+7i+8j+9k+10l+11m  Ry, x, z, x2−y2, xz, z2, x(x2−3y2), z(x2−y2), xz2, z3, (x2−y2)2−4x2y2, xz(x2−3y2), z2(x2−y2), xz3, z4, x(x2−(5+2√5)y2)(x2−(5−2√5)y2), z((x2−y2)2−4x2y2), xz2(x2−3y2), z3(x2−y2), xz4, z5, x2(x2−3y2)2−y2(3x2−y2)2, xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), z2((x2−y2)2−4x2y2), xz3(x2−3y2), z4(x2−y2), xz5, z6 
A2   2R+p+2d+3f+4g+5h+6i+7j+8k+9l+10m   Rx, Rz, y, xy, yz, y(3x2−y2), xyz, yz2, xy(x2−y2), yz(3x2−y2), xyz2, yz3, y((5+2√5)x2−y2)((5−2√5)x2−y2), xyz(x2−y2), yz2(3x2−y2), xyz3, yz4, xy(x2−3y2)(3x2−y2), yz((5+2√5)x2−y2)((5−2√5)x2−y2), xyz2(x2−y2), yz3(3x2−y2), xyz4, yz5 

 Notes:

    α  The order of the C1v point group is 2, and the order of the principal axis (σv) is 2. The group has 2 irreducible representations.

    β  The C1v point group is identical to Cs in non-standard orientation (the mirror plane is xz).

    γ  The lowest nonvanishing multipole moment in C1v is 2 (dipole moment).

    δ  This is an Abelian point group (the commutative law holds between all symmetry operations).
       The C1v group is Abelian because it meets two conditions, each of one alone would have been sufficient:
       It contains only one symmetry element (σv), and there is no axis of order 3 or higher.
       In Abelian groups, all symmetry operations form a class of their own, and all irreducible representations are one-dimensional.

    ε  There are no symmetry elements of an order higher than 2 in this group.
       The symmetry-adapted Cartesian products in the table above are needlessly complicated; rather, any simple product will do.

    ζ  All characters are integers because the order of the principal axis is 1,2,3,4 or 6.
       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.