Equivariant birational geometry of quintic del Pezzo surface

Abstract: In this paper we prove that there are exactly two $G$-minimal surfaces which are $G$-birational to the quintic del Pezzo surface, where $G \cong C_5 \rtimes C_4$. These surfaces are the quintic del Pezzo surface itself and the surface $\mathbb{P}¹ \times \mathbb{P}<sup>1$.