Character table for the D4 point group

D4      E       2 C4    C2      2 C2'   2 C2"      <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> 
A1        1       1       1       1       1        ... ... ....T ....... T.......T ...T....... ....T.......T
A2        1       1       1      -1      -1        ..T ..T ..... ......T .T....... ..T.......T .....T.......
B1        1      -1       1       1      -1        ... ... T.... ...T... ....T.... .......T... T.......T....
B2        1      -1       1      -1       1        ... ... .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+2g+h+2i+j+3k+2l+3m           z2, (x2y2)2−4x2y2, z4, xyz(x2y2), z2((x2y2)2−4x2y2), z6 
A2   R+p+f+g+2h+i+2j+2k+3l+2m       Rz, z, z3, xy(x2y2), z((x2y2)2−4x2y2), z5, xyz2(x2y2) 
B1   d+f+g+h+2i+2j+2k+2l+3m         x2y2, xyz, z2(x2y2), xyz3, x2(x2−3y2)2y2(3x2y2)2, z4(x2y2) 
B2   d+f+g+h+2i+2j+2k+2l+3m         xy, z(x2y2), xyz2, z3(x2y2), xy(x2−3y2)(3x2y2), xyz4 
E    R+p+d+2f+2g+3h+3i+4j+4k+5l+5m  {Rx, Ry}, {x, y}, {xz, yz}, {x(x2−3y2), y(3x2y2)}, {xz2, yz2}, {xz(x2−3y2), yz(3x2y2)}, {xz3, yz3}, {x(x2−(5+2√5)y2)(x2−(5−2√5)y2), y((5+2√5)x2y2)((5−2√5)x2y2)}, {xz2(x2−3y2), yz2(3x2y2)}, {xz4, yz4}, {xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), yz((5+2√5)x2y2)((5−2√5)x2y2)}, {xz3(x2−3y2), yz3(3x2y2)}, {xz5, yz5} 

 Notes:

    α  The order of the D4 point group is 8, and the order of the principal axis (C4) is 4. The group has 5 irreducible representations.

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

    γ  The D4 point group is generated by two symmetry elements, C4 and a perpendicular C2 (or, non-canonically, C2).
       Also, the group may be generated from any C2 plus any C2 axes.

    δ  There are two different sets of twofold symmetry axes perpendicular to the principal axis (z axis in standard orientation).
       By convention, the set denoted as C2 contains both the x and y axes.

    ε  The lowest nonvanishing multipole moment in D4 is 4 (quadrupole moment).

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

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

    θ  All characters are integers because the order of the principal axis is 1,2,3,4 or 6.
       This implies that the point group corresponds to a constructible polygon which can be used for tiling the plane.
       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.

This Character Table for the D4 point group was created by Gernot Katzer.

For other groups and some explanations, see the Main Page.