An Automorphism group of a rational surface: Not too big not too small

Abstract

This article concerns the realization problem of subgroups of Coxeter groups. We construct a subgroup $G$ of the Coxeter group $W_{15}$ such that $G$ is realized as automorphism groups of a rational surface $X$ and $G\cong Aut(X{)}^{\ast }\cong D_3\times \mathbb{Z}$. We also show that there is an element $\omega$ in $W_{14}$ is not realizable.