Character table for the C_2h point group

C2h     E       C2      i       sh         <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> 
Ag        1       1       1       1        ..T ... TT..T ....... TT..TT..T ........... TT..TT..TT..T
Bg        1      -1       1      -1        TT. ... ..TT. ....... ..TT..TT. ........... ..TT..TT..TT.
Au        1       1      -1      -1        ... ..T ..... ..TT..T ......... ..TT..TT..T .............
Bu        1      -1      -1       1        ... TT. ..... TT..TT. ......... TT..TT..TT. .............


 Symmetry of Rotations and Cartesian products

Ag   R+3d+5g+7i+9k+11m   Rz, x2−y2, xy, z2, (x2−y2)2−4x2y2, xy(x2−y2), z2(x2−y2), xyz2, z4, x2(x2−3y2)2−y2(3x2−y2)2, xy(x2−3y2)(3x2−y2), z2((x2−y2)2−4x2y2), xyz2(x2−y2), z4(x2−y2), xyz4, z6 
Bg   2R+2d+4g+6i+8k+10m  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+3f+5h+7j+9l       z, z(x2−y2), xyz, z3, z((x2−y2)2−4x2y2), xyz(x2−y2), z3(x2−y2), xyz3, z5 
Bu   2p+4f+6h+8j+10l     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 C2h point group is 4, and the order of the principal axis (C2) is 2. The group has 4 irreducible representations.

    β  The C2h point group is isomorphic to C2v and D2, and also to the Klein four-group.

    γ  The C2h point group is generated by two two symmetry elements, C2 and i; non-canonically, by C2 and σh or by i and σh.

    δ  The lowest nonvanishing multipole moment in C2h is 4 (quadrupole moment).

    ε  This is an Abelian point group (the commutative law holds between all symmetry operations).
       The C2h group is Abelian because it meets two sufficient conditions: Its symmetry elements are coaxial, and none is of of order 3 or higher.
       In Abelian groups, all symmetry operations form a class of their own, and all irreducible representations are one-dimensional.

    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.