On flag-transitive 2- ( k 2 , k , λ ) $({k}^{2},k,\lambda )$ designs with λ ∣ k $\lambda | k$

Abstract

It is shown that, apart from the smallest Ree group, a flag-transitive automorphism group $G$ of a 2-$\left(k^2,k,\lambda \right)$ 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, $D$ is isomorphic either to the 2-$\left(6^2,6,2\right)$ design or to one of the three 2-$\left(6^2,6,6\right)$ designs constructed in this paper. All the four 2-designs have the 36 secants of a non-degenerate conic $C$ of $P{G}_2\left(8\right)$ as a point set and 6-sets of secants in a remarkable configuration as a block set.