Block-transitive, point-imprimitive designs with \(\lambda \Relbar 1\) (Q1801702)

Let \(D\) be a 2-\((v,k,1)\) design with a block-transitive group \(G\) of automorphisms transitive and imprimitive on points. Suppose that each block of \(D\) contains exactly one pair of points of the same class. If the subgroup of elements of \(G\) fixing (setwise) each imprimitivity class is not trivial it is elementary abelian of odd order (and transitive on each imprimitivity class) or isomorphic to \(S_ 3\) (and then \(D\) is PG(2,4)). If \(v\) is maximal (i.e. \(v=(k(k-1)/2-1)^ 2\), according to a recent result) then \(v=729\) and \(k=8\), and \(G\) is a product of a point- regular group \(N\) and a group \(H\) of order 13 or 39. There exist 467 pairwise nonisomorphic designs of this type, according to some remarks and a machine enumeration described in a previous paper.