Classification of subgroups of symplectic groups over finite fields   containing a transvection
  In this note we give a self-contained proof of the following classification (up to conjugation) of subgroups of the general symplectic group of dimension n over a finite field of characteristic l, for l at least 5, which can be derived from work of Kantor: G is either reducible, symplectically imprimitive or it contains Sp(n, l). This result is for instance useful for proving "big image" results for symplectic Galois representations. 
Sara Arias-de-Reyna, Luis Dieulefait, Gabor Wiese
math.NT math.GR