On base sizes for primitive groups of product type

Let G ⩽ Sym ( Ω ) be a finite permutation group and recall that the base size of is minimal subset with trivial pointwise stabiliser. There an extensive literature on sizes for primitive groups, but there are very few results groups product type. In this paper, we initiate systematic study bases in setting. Our first main result determines every type form L ≀ P soluble point stabilisers, where = Γ k , S transitive. This extends recent work Burness almost simple groups. We also obtain expression number regular suborbits any classify unique suborbit under assumption primitive, which involves extending earlier due to Seress Dolfi. present applications Saxl graphs base-two conclude by establishing several new general