Character table for the D1h point group
D1h E C2' sh sv <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
A1" 1 1 -1 -1 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. .T...T...T...
A2" 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 x, x2−y2, z2, x(x2−3y2), xz2, (x2−y2)2−4x2y2, z2(x2−y2), z4, x(x2−(5+2√5)y2)(x2−(5−2√5)y2), xz2(x2−3y2), xz4, x2(x2−3y2)2−y2(3x2−y2)2, z2((x2−y2)2−4x2y2), z4(x2−y2), z6
A1" R+d+f+2g+2h+3i+3j+4k+4l+5m Rx, yz, xyz, yz(3x2−y2), yz3, xyz(x2−y2), xyz3, yz((5+2√5)x2−y2)((5−2√5)x2−y2), yz3(3x2−y2), yz5
A2' R+p+d+2f+2g+3h+3i+4j+4k+5l+5m Rz, y, xy, y(3x2−y2), yz2, xy(x2−y2), xyz2, y((5+2√5)x2−y2)((5−2√5)x2−y2), yz2(3x2−y2), yz4, xy(x2−3y2)(3x2−y2), xyz2(x2−y2), xyz4
A2" R+p+d+2f+2g+3h+3i+4j+4k+5l+5m Ry, z, xz, z(x2−y2), z3, xz(x2−3y2), xz3, z((x2−y2)2−4x2y2), z3(x2−y2), z5, xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), xz3(x2−3y2), xz5
α The order of the D1h point group is 4, and the order of the principal axis (C2′) is 2. The group has 4 irreducible representations.
β The D1h point group is identical to C2v in non-standard orientation (the C2′ axis is x, the σh plane is xy and the σv plane is xz).
γ The lowest nonvanishing multipole moment in D1h is 2 (dipole moment).
δ This is an Abelian point group (the commutative law holds between all symmetry operations).
The D1h 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.
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.