Character table for the S₄ point group

S4      E       2 S4    C2         <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> 
A         1       1       1        ..T ... ....T ..TT... TT......T ......TT... ....TT......T
B         1      -1       1        ... ..T TT... ......T ....TT... ..TT......T TT......TT...
E   *     2       0      -2        TT. TT. ..TT. TT..TT. ..TT..TT. TT..TT..TT. ..TT..TT..TT.


 Symmetry of Rotations and Cartesian products

A    R+d+2f+3g+2h+3i+4j+5k+4l+5m    Rz, z2, z(x2−y2), xyz, (x2−y2)2−4x2y2, xy(x2−y2), z4, z3(x2−y2), xyz3, z2((x2−y2)2−4x2y2), xyz2(x2−y2), z6 
B    p+2d+f+2g+3h+4i+3j+4k+5l+6m    z, x2−y2, xy, z3, z2(x2−y2), xyz2, z((x2−y2)2−4x2y2), xyz(x2−y2), z5, x2(x2−3y2)2−y2(3x2−y2)2, xy(x2−3y2)(3x2−y2), z4(x2−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 S4 point group is 4, and the order of the principal axis (S4) is 4. The group has 3 irreducible representations.

    β  The S4 point group could also be named C4i, because the S4 axis is identical to a roto-inversion axis of order 4.

    γ  The S4 point group is isomorphic to C4.

    δ  The S4 point group is generated by one single symmetry element, S4. Therefore, it is a cyclic group.

    ε  The lowest nonvanishing multipole moment in S4 is 4 (quadrupole moment).

    ζ  This is an Abelian point group (the commutative law holds between all symmetry operations).
       The S4 group is Abelian because it contains only one symmetry element, all the powers of which necessarily commute (sufficient condition).
       In Abelian groups, all symmetry operations form a class of their own, and all irreducible representations are one-dimensional.

    η  Because the group is Abelian and the maximum order of rotation is >2, some irreducible representations have complex characters.
       These two cases have been combined into one two-dimensional representation that is no longer irreducible but has real-valued characters.
       Accordingly, the left and the right S4 rotation have been combined into one two-membered pseudo-class.

    θ  The single “E” representation is reducible but almost behaves like a true irreducible representation.
       Its norm, however, is twice the group order. Therefore, is has 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.

    ι  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.