An automorphism group of a rational surface: not too big not too small

We study the realization problem of subgroups of Coxeter groups. We construct a subgroup G of the Coxeter group \(W_{15}\) $W_{15}$ such that G is realized as automorphism groups of a rational surface X. The automorphism group \(\textrm{Aut}\hspace{0.55542pt}(X)\) $\text{Aut}(X)$ has six generators with positive entropy and \(G \cong \textrm{Aut}\hspace{0.55542pt}(X)^*\! \cong D_3 \hspace{0.55542pt}{\rtimes }\hspace{1.111pt}{\mathbb {Z}}\) $G\cong \text{Aut}{(X)}^{\ast }\cong D_3⋊\mathbb{Z}$ . We also show that there is an element \(\omega \) $\omega$ in \(W_{14}\) $W_{14}$ that is not realizable.