Groups having minimal covering number 2 of the diagonal type

Garonzi and Lucchini explored finite groups (Formula presented.) possessing a normal 2-covering, where no proper quotient of (Formula presented.) 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 (Formula presented.) has such a covering. With deep appreciation to Martino Garonzi and Andrea Lucchini, for keeping us entertained.