Flag-transitive block designs with automorphism group SnwrS2

In this paper we consider the possibility that groups SnwrS2,4≤n≤63, act flag-transitively on block designs with n2 points. We rule out n∈{7,11,15,19,23,31,35,43,47,59} and confirm the required action for other n in the given range through performing the construction of base blocks of the desired designs. Our construction makes use of flag-transitive incidence structures and weakly flag-transitive incidence structures.