Let $S$ be a del Pezzo surface of degree six over $\mathbb{C}$. Then $S$ is the blow-up of $\mathbb{P}^2$ in three general points $p_1,p_2,p_3$.

Is it true that its automorphism group is $((\mathbb{C}^{*})^{2}\rtimes S_2)\times S_3$?

Here $(\mathbb{C}^{*})^{2}$ are the automorphisms of $\mathbb{P}^2$ fixing $p_1,p_2,p_3$, $S_2$ is the group generated by the standard Cremona centered at $p_1,p_2,p_3$, and $S_3$ are the permutations of $p_1,p_2,p_3$.