Character table for the D_2d point group

D2d     E       2 S4    C2      2 C2'   2 sd       <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> 
A1        1       1       1       1       1        ... ... ....T ...T... T.......T .......T... ....T.......T
A2        1       1       1      -1      -1        ..T ... ..... ..T.... .T....... ......T.... .....T.......
B1        1      -1       1       1      -1        ... ... T.... ....... ....T.... ...T....... T.......T....
B2        1      -1       1      -1       1        ... ..T .T... ......T .....T... ..T.......T .T.......T...
E         2       0      -2       0       0        TT. TT. ..TT. TT..TT. ..TT..TT. TT..TT..TT. ..TT..TT..TT.


 Symmetry of Rotations and Cartesian products

A1   d+f+2g+h+2i+2j+3k+2l+3m        z2, xyz, (x2−y2)2−4x2y2, z4, xyz3, z2((x2−y2)2−4x2y2), z6 
A2   R+f+g+h+i+2j+2k+2l+2m          Rz, z(x2−y2), xy(x2−y2), z3(x2−y2), xyz2(x2−y2) 
B1   d+g+h+2i+j+2k+2l+3m            x2−y2, z2(x2−y2), xyz(x2−y2), x2(x2−3y2)2−y2(3x2−y2)2, z4(x2−y2) 
B2   p+d+f+g+2h+2i+2j+2k+3l+3m      z, xy, z3, xyz2, z((x2−y2)2−4x2y2), z5, xy(x2−3y2)(3x2−y2), xyz4 
E    R+p+d+2f+2g+3h+3i+4j+4k+5l+5m  {Rx, Ry}, {x, y}, {xz, yz}, {x(x2−3y2), y(3x2−y2)}, {xz2, yz2}, {xz(x2−3y2), yz(3x2−y2)}, {xz3, yz3}, {x(x2−(5+2√5)y2)(x2−(5−2√5)y2), y((5+2√5)x2−y2)((5−2√5)x2−y2)}, {xz2(x2−3y2), yz2(3x2−y2)}, {xz4, yz4}, {xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), yz((5+2√5)x2−y2)((5−2√5)x2−y2)}, {xz3(x2−3y2), yz3(3x2−y2)}, {xz5, yz5} 

 Notes:

    α  The order of the D2d point group is 8, and the order of the principal axis (S4) is 4. The group has 5 irreducible representations.

    β  The D2d point group is also known as Vd.
       The letter V derives from German ‘Vierergruppe’ (group of four) for the Klein four-group, to which D2 is isomorphic.

    γ  The D2d point group is isomorphic to C4v and D4.

    δ  The D2d point group is generated by two symmetry elements, S4 and either a perpendicular C2′ or a vertical σd.
       Also, the group may be generated from any C2′ plus any σd plane.

    ε  The group contains two C2′ symmetry axes perpendicular to the principal (z) axis, which are by convention chosen as x and y.

    ζ  The single σd set of symmetry planes contains neither the xz nor the yz planes; but it contains the median plane (x+y)z.

    η  The lowest nonvanishing multipole moment in D2d is 4 (quadrupole moment).

    θ  This point group is non-Abelian (some symmetry operations are not commutative).
       Therefore, the character table contains multi-membered classes and degenerate irreducible representations.

    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.