On the order of vertex-stabilisers in vertex-transitive graphs with local group Cp×Cp or CpwrC2

Let p be a prime and let L be either the intransitive permutation group Cp×Cp of degree 2p or the transitive permutation group CpwrC2 of degree 2p. Let Γ be a connected G-vertex-transitive and G-edge-transitive graph and let v be a vertex of Γ. We show that if the permutation group induced by the vertex-stabiliser Gv on the neighbourhood Γ(v) is isomorphic to L then either |V(Γ)|≥p|Gv|logp(|Gv|/2), or |V(Γ)| is bounded by a constant depending only on p, or Γ is a very-well understood graph. This generalises a few recent results.