Character table for the Cs point group
C1h E sh <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————>
A' 1 1 ..T TT. TT..T TT..TT. TT..TT..T TT..TT..TT. TT..TT..TT..T
A" 1 -1 TT. ..T ..TT. ..TT..T ..TT..TT. ..TT..TT..T ..TT..TT..TT.
Symmetry of Rotations and Cartesian products
A' R+2p+3d+4f+5g+6h+7i+8j+9k+10l+11m Rz, x, y, x2−y2, xy, z2, x(x2−3y2), y(3x2−y2), xz2, yz2, (x2−y2)2−4x2y2, xy(x2−y2), z2(x2−y2), xyz2, z4, x(x2−(5+2√5)y2)(x2−(5−2√5)y2), y((5+2√5)x2−y2)((5−2√5)x2−y2), xz2(x2−3y2), yz2(3x2−y2), xz4, yz4, x2(x2−3y2)2−y2(3x2−y2)2, xy(x2−3y2)(3x2−y2), z2((x2−y2)2−4x2y2), xyz2(x2−y2), z4(x2−y2), xyz4, z6
A" 2R+p+2d+3f+4g+5h+6i+7j+8k+9l+10m Rx, Ry, z, xz, yz, z(x2−y2), xyz, z3, xz(x2−3y2), yz(3x2−y2), xz3, yz3, z((x2−y2)2−4x2y2), xyz(x2−y2), z3(x2−y2), xyz3, z5, xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), yz((5+2√5)x2−y2)((5−2√5)x2−y2), xz3(x2−3y2), yz3(3x2−y2), xz5, yz5
α The order of the C1h point group is 2, and the order of the principal axis (σh) is 2. The group has 2 irreducible representations.
β The C1h point group is usually referred to as Cs. It could also be named S1 or C2i.
γ The Cs point group is isomorphic to C2 and Ci, and also to the Symmetric Group Sym(2).
δ The Cs point group is generated by one single symmetry element, σh. Therefore, it is a cyclic group.
ε The lowest nonvanishing multipole moment in C1h is 2 (dipole moment).
ζ This is an Abelian point group (the commutative law holds between all symmetry operations).
The Cs group is Abelian because it meets two conditions, each of one alone would have been sufficient:
It contains only one symmetry element (σh), 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.
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 Cs is often termed a nonaxial group, although the mirror plane does have a preferred direction.
This term is also used for C1 and Ci, and occasionally for C2.
ι 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.