A family of quasiprimitive 2-arc transitive graphs which have non-quasiprimitive full automorphism groups (Q1266361)

A group \(G\) is said to act quasiprimitively on a graph \(\Gamma\) if every nontrivial normal subgroup of \(G\) acts transitively on the vertex set of \(\Gamma\). The action of \(G\) on \(\Gamma\) is 2-arc transitive if \(G\) acts transitively on the set of 2-arcs, that is on the set of ordered triples \((x,y,z)\) of vertices of \(\Gamma\) such that \(x\) and \(z\) are distinct vertices adjacent to \(y\). For each prime \(p\equiv 1\) or \(-1\pmod {24}\) a graph \(\Gamma\) is constructed on which \(G\cong \text{PSL}_2 (p)\) acts quasiprimitively and 2-arc transitively, such that the full automorphism group \(\Aut\Gamma\) of \(\Gamma\) is isomorphic to \(\text{PSL}_2 (p)\times \mathbb{Z}_2\) and the action of \(\Aut\Gamma\) is not quasiprimitive.