The Flag-Transitive and Point-Imprimitive Symmetric $(v,k,\lambda )$ Designs with $v<100$

Abstract

A complete classification of the flag-transitive point-imprimitive symmetric $2$-$(v,k,\lambda )$ designs with $v<100$ is provided. Apart from the known examples with $\lambda \le 10$, the complementary design of $P{G}_5(2)$, and the $2$-design ${\mathcal{S}}^{-}(3)$ constructed by Kantor in \cite{Ka75}, we found two non isomorphic $2$-$(64,28,12)$ designs. They were constructed via computer as developments of $(64,28,12)$-difference sets by AbuGhneim in \cite{OAG}. In the present paper, independently from \cite{OAG}, we construct the aforementioned two $2$-designs and we prove that their full automorhpism group is flag-transitive and point-imprimitive. The construction is theoretical and relies on the the absolutely irreducible $8$-dimensional ${\mathbb{F}}_2$-representation of $PS{L}_2(7)$. Our result, together with that about the flag-transitive point-primitive symmetric $2$-designs with $v<2500$ by Braić-Golemac-Mandić-Vučičić \cite{BGMV}, provides a complete classification of the flag-transitive $2$-designs with $v<100$.