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

TL;DR

This paper classifies all flag-transitive, point-imprimitive symmetric 2-designs with fewer than 100 points, discovering two new designs constructed via theoretical methods involving group representations.

Contribution

It provides a complete classification of such designs with v<100, including the construction and proof of automorphism groups of two new designs.

Findings

Identified two new non-isomorphic 2-(64,28,12) designs.

Proved their automorphism groups are flag-transitive and point-imprimitive.

Completed the classification of flag-transitive 2-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 $\lambda \leq 10$, the complementary design of $PG_{5}(2)$, 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 $PSL_{2}(7)$. Our result, together with that about the flag-transitive point-primitive symmetric $2$-designs with $v<2500$ by Brai\'{c}-Golemac-Mandi\'{c}-Vu\v{c}i\v{c}i\'{c} \cite{BGMV}, provides a complete classification of the flag-transitive $2$-designs with $v<100$.