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 Ω 1 , … , Ω k and Ω 1 ′ , … , Ω k ′ ′ . One can construct a transitive 1–design with | Ω i | , i = 1 , … , k , points and | Ω j ′ | , j = 1 , … , k ′ , blocks (or | Ω i | , i = 1 , … , k , blocks and | Ω j ′ | , j = 1 , … , k ′ , points) whose base block is ∪ i = 1 s G α ⋅ β i , α ∈ Ω j , β 1 , … , β s ∈ Ω i . Combining incidence matrices of those transitive 1–designs we construct an incidence matrix of a non-transitive design.