Groups having minimal covering number 2 of the diagonal type

Garonzi and Lucchini explored finite groups $G$ possessing a normal 2-covering, where no proper quotient of $G$ exhibits such a covering. Their investigation offered a comprehensive overview of these groups, delineating that such groups fall into distinct categories: almost simple, affine, product action, or diagonal.

In this paper, we focus on the family falling under the diagonal type. Specifically, we present a thorough classification of finite diagonal groups possessing a normal 2-covering, with the attribute that no proper quotient of $G$ has such a covering.

With deep appreciation to Martino Garonzi and Andrea Lucchini, for keeping us entertained.