Constructing representations of the finite symplectic group Sp(4,q)

Let G be a finite group and χ be an irreducible character. We say that a subgroup H is a χ-subgroup if the restriction χH of χ to H has at least one linear constituent of multiplicity 1. Not every pair (G,χ) has a χ-subgroup, but χ-subgroups can be found in many cases. The existence of such subgroups is of interest for several reasons, one being that knowledge of a χ-subgroup enables us to give a simple construction of a matrix representation of G affording χ. In this paper we show that, when G=Sp(4,q) where q is a power of an odd prime p and H is a Sylow p-subgroup of G, then H is a χ-subgroup for every irreducible character χ (with one exception). We also find a p-subgroup which is a χ-subgroup for the exceptional character.