Character table for the C_4h point group

C4h     E       2 C4    C2      i       2 S4    sh         <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> 
Ag        1       1       1       1       1       1        ..T ... ....T ....... TT......T ........... ....TT......T
Bg        1      -1       1       1      -1       1        ... ... TT... ....... ....TT... ........... TT......TT...
Eg  *     2       0      -2       2       0      -2        TT. ... ..TT. ....... ..TT..TT. ........... ..TT..TT..TT.
Au        1       1       1      -1      -1      -1        ... ..T ..... ......T ......... ..TT......T .............
Bu        1      -1       1      -1       1      -1        ... ... ..... ..TT... ......... ......TT... .............
Eu  *     2       0      -2      -2       0       2        ... TT. ..... TT..TT. ......... TT..TT..TT. .............


 Symmetry of Rotations and Cartesian products

Ag   R+d+3g+3i+5k+5m  Rz, z2, (x2−y2)2−4x2y2, xy(x2−y2), z4, z2((x2−y2)2−4x2y2), xyz2(x2−y2), z6 
Bg   2d+2g+4i+4k+6m   x2−y2, xy, z2(x2−y2), xyz2, x2(x2−3y2)2−y2(3x2−y2)2, xy(x2−3y2)(3x2−y2), z4(x2−y2), xyz4 
Eg   R+d+2g+3i+4k+5m  {Rx, Ry}, {xz, yz}, {xz(x2−3y2), yz(3x2−y2)}, {xz3, yz3}, {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} 
Au   p+f+3h+3j+5l     z, z3, z((x2−y2)2−4x2y2), xyz(x2−y2), z5 
Bu   2f+2h+4j+4l      z(x2−y2), xyz, z3(x2−y2), xyz3 
Eu   p+2f+3h+4j+5l    {x, y}, {x(x2−3y2), y(3x2−y2)}, {xz2, yz2}, {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} 

 Notes:

    α  The order of the C4h point group is 8, and the order of the principal axis (C4) is 4. The group has 6 irreducible representations.

    β  The C4h point group is generated by two symmetry elements, which are canonically chosen as C4 and i.
       Other possible choices are C4 and σh, or less commonly S4 with either i or σh.

    γ  The lowest nonvanishing multipole moment in C4h is 4 (quadrupole moment).

    δ  This is an Abelian point group (the commutative law holds between all symmetry operations).
       The C4h group is Abelian because all its symmetry operations are coaxial. This is a sufficient condition.
       In Abelian groups, all symmetry operations form a class of their own, and all irreducible representations are one-dimensional.

    ε  Because the group is Abelian and the maximum order of rotation is >2, some irreducible representations have complex characters.
       These 4 cases have been combined into 2 two-dimensional representations that are no longer irreducible but have real-valued characters.
       Accordingly, 2 pairs of left and right rotations have been combined into one two-membered pseudo-class each.

    ζ  The 2 reducible “E” representations almost behave like true irreducible representations.
       Their norm, however, is twice the group order. Therefore, they have been marked with an asterisk in the table.
       This is essential when trying to decompose a reducible representation into “irreducible” ones using the familiar projection formula.

    η  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.