Block-transitive 3-$(v,4,\lambda )$ designs with sporadic or alternating socle

Abstract

This paper is a contribution to the classification of block-transitive 3-designs. Let ${{\mathcal {D}}=({\mathcal {P}},{\mathcal {B}})}$ be a nontrivial 3-$(v,4,\lambda )$ design and $G \le Aut({\mathcal {D}})$ acts block-transitively on ${\mathcal {D}}$ with sporadic or alternating socle, then there are exactly 6 incomplete designs as follows:

• (i) ${\mathcal {D}}$ is isomorphic to a 3-$(12,4,\lambda )$ design with $\lambda \in \{3,6\}$, and $G \cong M_{11}$;

• (ii) ${\mathcal {D}}$ is isomorphic to a 3-$(22,4,\lambda )$ design with $\lambda \in \{3,16\}$, and $Soc(G)=M_{22}$;

• (iii) ${\mathcal {D}}$ is isomorphic to a 3-(10, 4, 1) design, and $G \cong M_{10}$, $PGL_2(9)$ or $P\Gamma L_2(9)$;

• (iv) ${\mathcal {D}}$ is isomorphic to a 3-(10, 4, 6) design, and $Soc(G)=A_6$.