Character table for the D2 point group
D2 E C2 C2' C2" <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————>
A 1 1 1 1 ... ... T...T ...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..
B3 1 -1 1 -1 T.. T.. ...T. T...T.. ...T...T. T...T...T.. ...T...T...T.
Symmetry of Rotations and Cartesian products
A 2d+f+3g+2h+4i+3j+5k+4l+6m x2−y2, z2, xyz, (x2−y2)2−4x2y2, z2(x2−y2), z4, xyz(x2−y2), xyz3, x2(x2−3y2)2−y2(3x2−y2)2, z2((x2−y2)2−4x2y2), z4(x2−y2), z6
B1 R+p+d+2f+2g+3h+3i+4j+4k+5l+5m Rz, z, xy, z(x2−y2), z3, xy(x2−y2), xyz2, z((x2−y2)2−4x2y2), z3(x2−y2), z5, xy(x2−3y2)(3x2−y2), xyz2(x2−y2), xyz4
B2 R+p+d+2f+2g+3h+3i+4j+4k+5l+5m Ry, y, xz, y(3x2−y2), yz2, xz(x2−3y2), xz3, y((5+2√5)x2−y2)((5−2√5)x2−y2), yz2(3x2−y2), yz4, xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), xz3(x2−3y2), xz5
B3 R+p+d+2f+2g+3h+3i+4j+4k+5l+5m Rx, x, yz, x(x2−3y2), xz2, yz(3x2−y2), yz3, x(x2−(5+2√5)y2)(x2−(5−2√5)y2), xz2(x2−3y2), xz4, yz((5+2√5)x2−y2)((5−2√5)x2−y2), yz3(3x2−y2), yz5
α The order of the D2 point group is 4, and the order of the principal axis (C2) is 2. The group has 4 irreducible representations.
β The D2 point group is also known as V. The letter V derives from German ‘Vierergruppe’ (group of four) for the Klein four-group.
γ The D2 point group is isomorphic to C2v and C2h, and also to the Klein four-group.
δ The D2 point group is generated by any two perpendicular C2 axes, canonically C2 and C2′.
ε The D2 group has three nonequivalent C2 axes. The one labelled “C2” is the principal axis (z in canonical orientation).
The other two are perpendicular to the principal axis. By convention, the C2′ is the x axis and C2″ the y axis.
ζ The lowest nonvanishing multipole moment in D2 is 4 (quadrupole moment).
η This is an Abelian point group (the commutative law holds between all symmetry operations).
The D2 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.
θ The point group is chiral, as it does not contain any mirroring operation.
ι The naming of irreducible representations in this group as B1,B2,B3 is purely conventional.
κ 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.
λ 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.