Character table for the Ci point group
S2 E i <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————>
Ag 1 1 TTT ... TTTTT ....... TTTTTTTTT ........... TTTTTTTTTTTTT
Au 1 -1 ... TTT ..... TTTTTTT ......... TTTTTTTTTTT .............
Symmetry of Rotations and Cartesian products
Ag 3R+5d+9g+13i+17k+21m Rx, Ry, Rz, x2−y2, xy, xz, yz, z2, (x2−y2)2−4x2y2, xy(x2−y2), xz(x2−3y2), yz(3x2−y2), z2(x2−y2), xyz2, xz3, yz3, z4, 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
Au 3p+7f+11h+15j+19l x, y, z, x(x2−3y2), y(3x2−y2), z(x2−y2), xyz, xz2, yz2, z3, 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
α The order of the S2 point group is 2, and the order of the principal axis (i) is 2. The group has 2 irreducible representations.
β The S2 point group is usually referred to as Ci.
γ The Ci point group is isomorphic to C2 and Cs, and also to the Symmetric Group Sym(2).
δ The Ci point group is generated by one single symmetry element, i. Therefore, it is a cyclic group.
ε The lowest nonvanishing multipole moment in S2 is 4 (quadrupole moment).
ζ This is an Abelian point group (the commutative law holds between all symmetry operations).
The Ci group is Abelian because it meets two conditions, each of one alone would have been sufficient:
It contains only one symmetry element (i), 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 Ci is a nonaxial group, because the center of inversion has no preferred direction. Therefore, x, y and z fall into the same irreducible representation.
This term is also used for C1, and with less justification for Cs. Also, C2 has in some way “nonaxial” properties, as it is isomorphic to Ci.
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.