Designs derived from permutation groups

Let G be a transitive permutation group on a set Ω of v points {1, 2, …, v}. Let H be an intransitive subgroup of G and let Δ a set of k points where Δ consists of complete orbits of H. Then the images Δ^x of Δ under permutations x of Δ have been shown by the first author to be a partially balanced block design D with G as a group of automorphisms. Under certain circumstances D is a balanced incomplete block design. Here a representation of the simple group PSL₃(4) of order 20,160 on 56 letters leads to a new symmetric block design with parameters v=56, k-11, λ=2. A representation of the simple group of order 25,920 as U₄(4) on 45 isotropic points gives a symmetric design with v=45, k=12, λ=3. One representation of U₄(4) on 40 points, gives the design of planes in PG(3, 3) and exhibits the isomorphism of this group to the symplectic group S₄(3).