Construction of 1-designs from group action

Let G be a finite group acting on sets Ω and Ω′. Denote the orbits of those actions by Ω1,…,Ωk and . One can construct a transitive 1–design with |Ωi|,i=1,…,k, points and |, blocks (or |Ωi|,i=1,…,k, blocks and |, points) whose base block is . Combining incidence matrices of those transitive 1–designs we construct an incidence matrix of a non-transitive design.