  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 $G \cong Aut(X)^*
\cong D_3 \times \mathbb{Z}$. We also show that there is an element $\omega$ in
$W_{14}$ is not realizable.
