Character table for the D₁ point group

D1      E       C2'        <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> 
A1        1       1        T.. T.. T..TT T..TT.. T..TT..TT T..TT..TT.. T..TT..TT..TT
A2        1      -1        .TT .TT .TT.. .TT..TT .TT..TT.. .TT..TT..TT .TT..TT..TT..


 Symmetry of Rotations and Cartesian products

A1   R+p+3d+3f+5g+5h+7i+7j+9k+9l+11m     Rx, x, x2−y2, yz, z2, x(x2−3y2), xyz, xz2, (x2−y2)2−4x2y2, yz(3x2−y2), z2(x2−y2), yz3, z4, x(x2−(5+2√5)y2)(x2−(5−2√5)y2), xyz(x2−y2), xz2(x2−3y2), xyz3, xz4, x2(x2−3y2)2−y2(3x2−y2)2, yz((5+2√5)x2−y2)((5−2√5)x2−y2), z2((x2−y2)2−4x2y2), yz3(3x2−y2), z4(x2−y2), yz5, z6 
A2   2R+2p+2d+4f+4g+6h+6i+8j+8k+10l+10m  Ry, Rz, y, z, xy, xz, y(3x2−y2), z(x2−y2), yz2, z3, xy(x2−y2), xz(x2−3y2), xyz2, xz3, y((5+2√5)x2−y2)((5−2√5)x2−y2), z((x2−y2)2−4x2y2), yz2(3x2−y2), z3(x2−y2), yz4, z5, xy(x2−3y2)(3x2−y2), xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), xyz2(x2−y2), xz3(x2−3y2), xyz4, xz5 

 Notes:

    α  The order of the D1 point group is 2, and the order of the principal axis (C2′) is 2. The group has 2 irreducible representations.

    β  The D1 point group is identical to C2 in non-standard orientation (the C2′ axis is x).

    γ  The lowest nonvanishing multipole moment in D1 is 2 (dipole moment).

    δ  This is an Abelian point group (the commutative law holds between all symmetry operations).
       The D1 group is Abelian because it meets two conditions, each of one alone would have been sufficient:
       It contains only one symmetry element (C2′), 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.

    ε  The point group is chiral, as it does not contain any mirroring operation.

    ζ  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.

    η  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.