Character table for the C1 point group
C1 E <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————>
A 1 TTT TTT TTTTT TTTTTTT TTTTTTTTT TTTTTTTTTTT TTTTTTTTTTTTT
Symmetry of Rotations and Cartesian products
A 3R+3p+5d+7f+9g+11h+13i+15j+17k+19l+21m Rx, Ry, Rz, x, y, z, x2−y2, xy, xz, yz, z2, x(x2−3y2), y(3x2−y2), z(x2−y2), xyz, xz2, yz2, z3, (x2−y2)2−4x2y2, xy(x2−y2), xz(x2−3y2), yz(3x2−y2), z2(x2−y2), xyz2, xz3, yz3, z4, 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), xz2(x2−3y2), yz2(3x2−y2), z3(x2−y2), xyz3, xz4, yz4, z5, x2(x2−3y2)2−y2(3x2−y2)2, xy(x2−3y2)(3x2−y2), 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), xz3(x2−3y2), yz3(3x2−y2), z4(x2−y2), xyz4, xz5, yz5, z6
α The order of the C1 point group is 1, and the order of the principal axis (E) is 1. The group has 1 irreducible representations.
β The lowest nonvanishing multipole moment in C1 is 2 (dipole moment).
γ This is an Abelian point group (the commutative law holds between all symmetry operations).
The C1 group is Abelian because it contains no non-trivial symmetry operation;
therefore, its only irreducible representation is the trivial one.
δ The point group is chiral, as it does not contain any mirroring operation.
ε This is the trivial group, containing only the identity element, and thus no symmetry adaption is necessary.
Therefore, the forms of symmetry-adapted Cartesian products above are needlessly complicated; 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.