TITLE: Automorphisms of del Pezzo surfaces
QUESTION [9 upvotes]: 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 $p_1,p_2,p_3$.

REPLY [10 votes]: Not exactly. The quadratic transformation commutes with the action of $S_3$, and they both act on $(\mathbb{C}^*)^2$; so the automorphism group is $(\mathbb{C}^{*})^{2}\rtimes (S_3\times S_2)$. You'll find a detailed study of the automorphisms of del Pezzo surfaces in Chapter 8  of Dolgachev's book "Classical Algebraic Geometry: a modern view".