Cubic edge-transitive graphs of order $\ell p^2$

A graph is edge-transitive if its automorphism group acts transitively on its edge set. This paper is devoted to a systematic study of cubic edge-transitive graphs of order $\ell p^2$, where $p$ is a prime and $\ell$ is an even integer. We first show that either $Γ$ is the unique cubic edge-transitive graph of order $18$, or its full automorphism group $\mathrm{Aut}(Γ)$ contains a normal Sylow $p$-subgroup $P$ which is isomorphic to either $Z_{p^2}$ or $Z_{p}^2$. For the case $P\cong Z_{p^2}$, we prove that $Γ$ is either arc-regular or edge-regular, and provide a complete classification of such graphs. For the case $P\cong Z_{p}^2$, we prove that $Γ$ is a normal cover of certain known graphs. This work generalizes previous classifications by Zhou and Feng (2024), Lu et al.~(2004), and others. As an application, we obtain a classification of cubic edge-transitive graphs of order $\ell p^2$ in the case where $\ell$ is square-free.