On Flag-Transitive $2$-$(k^{2}, k, λ)$ Designs with $λ\mid k$

It is shown that, apart from the smallest Ree group, a flag-transitive automorphism group $G$ of a $2$-$(k^{2}, k, λ)$ design D, with $λ\mid k$, 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 $PG_{2}(8)$ as a point set and 6-sets of secants in a remarkable configuration as a block set.

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