G2 and hypergeometric sheaves

We determine, in every finite characteristic p, those hypergeometric sheaves of type (7,m) with 7 ≥ m whose geometric monodromy group Ggeom lies in G2, cf. Theorem 3.1 and Theorem 6.1. For each of these we determine Ggeom exactly, cf. Theorem 9.1. Each of the five primitive irreducible finite subgroups of G2, namely L2(8), U3(3), U3(3).2 = G2(2), L2(7).2 = PGL2(7), L2(13) turns out to occur as Ggeom in a single characteristic p, namely p = 2, 3, 7, 7, 13 for the groups as listed, and for essentially just one hypergeometric sheaf in that characteristic. It would be interesting to find conceptual, rather than classificational/computational, proofs of these results.