On Flag-Transitive $2$-$(k^2,k,\lambda )$ Designs with $\lambda \mid k$

Abstract

It is shown that, apart from the smallest Ree group, a flag-transitive automorphism group $G$ of a $2$-$(k^2,k,\lambda )$ design D, with $\lambda \mid k$, is either an affine group or an almost simple classical group. Moreover, when $G$ is the smallest Ree group, $\mathcal{D}$ is isomorphic either to the $2$-$(62,6,2)$ design or to one of the three $2$- $(62,6,6)$ designs constructed in this paper. All the four $2$-designs have the $36$ secants of a nondegenerate conic $\mathcal{C}$ of $P{G}_2(8)$ as a point set and 6-sets of secants in a remarkable configuration as a block set.