A Family of Quasiprimitive 2-arc Transitive Graphs which Have Non-quasiprimitive Full Automorphism Groups

For each primep ≡ 1 or −1(mod 24), we construct a (PSL(2,p), 2)-arc transitive graph Γ of valency 4 such that Aut Γ = PSL(2,p) × Z2. Thus there exists an infinite family of quasiprimitive 2-arc transitive graphs which have non-quasiprimitive full automorphism groups.