Let $Q\cong\mathbb{P^{1}_{1}}\times\mathbb{P^{1}_{2}}\subset\mathbb{P}^{3}$ be a smooth quadric surface. We have the following two actions on $Q$: $$S_2\times Q\rightarrow Q,\; (\sigma,(x,y))\mapsto\sigma(x,y)$$ where $S_2$ is the symmetric group of order $2$, and $$Aut(\mathbb{P^{1}_{1}})\times Aut(\mathbb{P^{1}_{2}})\times Q\rightarrow Q,\;((f,g),(x,y))\mapsto (f(x),g(y)).$$ These two actions do not commute. Furthermore $h^{0}(Q,T_{Q}) = 6$, that is $dim(Aut(Q)) = 6$.

Is it true that $Aut(Q)$ is the semi-direct product $$Aut(Q)\cong(Aut(\mathbb{P^{1}_{1}})\times Aut(\mathbb{P^{1}_{2}}))\rtimes S_2 ?$$