Constructing flag-transitive,point-imprimitive designs

We give a construction of a family of designs with a specified point-partition and determine the subgroup of automorphisms leaving invariant the point-partition. We give necessary and sufficient conditions for a design in the family to possess a flag-transitive group of automorphisms preserving the specified point-partition. We give examples of flag-transitive designs in the family, including a new symmetric 2-(1408,336,80) design with automorphism group (2^{12}:((3cdot mathrm {M}_{22}):2)) and a construction of one of the families of the symplectic designs (the designs (S^-(n))) exhibiting a flag-transitive, point-imprimitive automorphism group.