Character table for the D_1d point group

D1d     E       C2'     i       sd         <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> 
A1g       1       1       1       1        T.. ... T..TT ....... T..TT..TT ........... T..TT..TT..TT
A2g       1      -1       1      -1        .TT ... .TT.. ....... .TT..TT.. ........... .TT..TT..TT..
A1u       1       1      -1      -1        ... T.. ..... T..TT.. ......... T..TT..TT.. .............
A2u       1      -1      -1       1        ... .TT ..... .TT..TT ......... .TT..TT..TT .............


 Symmetry of Rotations and Cartesian products

A1g  R+3d+5g+7i+9k+11m   Rx, x2−y2, yz, z2, (x2−y2)2−4x2y2, yz(3x2−y2), z2(x2−y2), yz3, z4, 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 
A2g  2R+2d+4g+6i+8k+10m  Ry, Rz, xy, xz, xy(x2−y2), xz(x2−3y2), xyz2, xz3, xy(x2−3y2)(3x2−y2), xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), xyz2(x2−y2), xz3(x2−3y2), xyz4, xz5 
A1u  p+3f+5h+7j+9l       x, x(x2−3y2), xyz, xz2, x(x2−(5+2√5)y2)(x2−(5−2√5)y2), xyz(x2−y2), xz2(x2−3y2), xyz3, xz4 
A2u  2p+4f+6h+8j+10l     y, z, y(3x2−y2), z(x2−y2), yz2, z3, y((5+2√5)x2−y2)((5−2√5)x2−y2), z((x2−y2)2−4x2y2), yz2(3x2−y2), z3(x2−y2), yz4, z5 

 Notes:

    α  The order of the D1d point group is 4, and the order of the principal axis (C2′) is 2. The group has 4 irreducible representations.

    β  The D1d point group is identical to C2h in non-standard orientation (the C2′ axis is x, and the σd plane is yz).

    γ  The lowest nonvanishing multipole moment in D1d is 4 (quadrupole moment).

    δ  This is an Abelian point group (the commutative law holds between all symmetry operations).
       The D1d 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.