The \(2-(v, 13, 1)\) Designs with Block Transitive Automorphism

Abstract

This article is a contribution to the study of the automorphism groups
of \(2\)-\((v,k,1)\) designs. Let \(\mathcal{D}\) be a \(2\)-\((v,13,1)\) design and
suppose that \(G\) is a group of automorphisms of \(\mathcal{D}\) which is
block-transitive and point-primitive. Then \(\mathrm{Soc}(G)\),
the socle of \(G\), is not isomorphic to \(^2G_2(q)\) or to \(^2F_4(q^2)\)
for any prime power \(q\).