A classification of flag-transitive block designs

In this article, we investigate 2- $$(v,k,\lambda )$$ designs with $$\gcd (r,\lambda )=1$$ admitting flag-transitive automorphism groups G. We prove that if G is an almost simple group, then such a design belongs to one of the eight infinite families 2-designs or it eleven well-known examples. describe all these examples designs. We, in particular, $${\mathcal {D}}$$ symmetric (k,\lambda group G, either $$G\leqslant \mathrm {A}\Gamma {L}_{1}(q)$$ for some odd prime power q, point-hyperplane unique Hadamard parameters (11, 5, 2).