The smallest vertex-primitive $2$-arc-transitive digraph

TL;DR

This paper proves that a specific large vertex-primitive 2-arc-transitive digraph constructed in 2017 is the smallest known of its kind with valency at least 2, establishing a minimal example in this class.

Contribution

It demonstrates that the previously constructed digraph is the smallest vertex-primitive 2-arc-transitive digraph with valency at least 2.

Findings

The digraph has 30,758,154,560 vertices.

It is the smallest such digraph with these properties.

The automorphism group is PSL_3(49) with vertex-stabilizer A_6.

Abstract

In 2017, Giudici, Li and the third author constructed the first known family of vertex-primitive $2$-arc-transitive digraphs of valency at least $2$. The smallest digraph in this family admits $\mathrm{PSL}_3(49)$ acting $2$-arc-transitively with vertex-stabilizer $\mathrm{A}_6$ and hence has $30758154560$ vertices. In this paper, we prove that this digraph is the vertex-primitive $2$-arc-transitive digraph of valency at least $2$ with fewest vertices.