Character table for the C2v point group
C2v E C2 sv sd <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————>
A1 1 1 1 1 ... ..T T...T ..T...T T...T...T ..T...T...T T...T...T...T
A2 1 1 -1 -1 ..T ... .T... ...T... .T...T... ...T...T... .T...T...T...
B1 1 -1 1 -1 .T. T.. ..T.. T...T.. ..T...T.. T...T...T.. ..T...T...T..
B2 1 -1 -1 1 T.. .T. ...T. .T...T. ...T...T. .T...T...T. ...T...T...T.
Symmetry of Rotations and Cartesian products
A1 p+2d+2f+3g+3h+4i+4j+5k+5l+6m z, x2−y2, z2, z(x2−y2), z3, (x2−y2)2−4x2y2, z2(x2−y2), z4, z((x2−y2)2−4x2y2), z3(x2−y2), z5, x2(x2−3y2)2−y2(3x2−y2)2, z2((x2−y2)2−4x2y2), z4(x2−y2), z6
A2 R+d+f+2g+2h+3i+3j+4k+4l+5m Rz, xy, xyz, xy(x2−y2), xyz2, xyz(x2−y2), xyz3, xy(x2−3y2)(3x2−y2), xyz2(x2−y2), xyz4
B1 R+p+d+2f+2g+3h+3i+4j+4k+5l+5m Ry, x, xz, x(x2−3y2), xz2, xz(x2−3y2), xz3, x(x2−(5+2√5)y2)(x2−(5−2√5)y2), xz2(x2−3y2), xz4, xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), xz3(x2−3y2), xz5
B2 R+p+d+2f+2g+3h+3i+4j+4k+5l+5m Rx, y, yz, y(3x2−y2), yz2, yz(3x2−y2), yz3, y((5+2√5)x2−y2)((5−2√5)x2−y2), yz2(3x2−y2), yz4, yz((5+2√5)x2−y2)((5−2√5)x2−y2), yz3(3x2−y2), yz5
α The order of the C2v point group is 4, and the order of the principal axis (C2) is 2. The group has 4 irreducible representations.
β The C2v point group is isomorphic to C2h and D2, and also to the Klein four-group.
γ The C2v point group is generated by two two symmetry elements, C2 and σh (or, non-canonically, σd).
Another common choice is to pick the two mirror planes, σh and σd, as generators.
δ The C2v group has two nonequivalent mirror planes. By convention, the σv is the xz plane and the σd the yz plane.
ε The lowest nonvanishing multipole moment in C2v is 2 (dipole moment).
ζ This is an Abelian point group (the commutative law holds between all symmetry operations).
The C2v group is Abelian because it satisfies the sufficient condition to contain no axes of order higher than two.
In Abelian groups, all symmetry operations form a class of their own, and all irreducible representations are one-dimensional.
η 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.
θ 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.