Flag-Transitive, Point-Imprimitive 2-Designs and Direct Products of Symmetric Groups

Consider the direct product of symmetric groups Sc× Sₙ and its natural action on P=C× N, where $|C|=c$ and $|N|=n$. We characterize the structure of 2-designs with point set P admitting flag-transitive, point-imprimitive automorphism groups H≤ Sc× Sₙ. As an example of its applications, we show that H cannot be any subgroup of D2c× Sₙ or Sc× D₂ₙ. Besides, some families of 2-designs admitting flag-transitive automorphism groups Sc× Sₙ are constructed by using complete bipartite graphs and cycles. Two families of these also admit flag-transitive, point-primitive automorphism groups Sc S₂, a family of which attain the Cameron-Praeger upper bound v=(k-2)².