How many automorphisms does $S_3\times S_3$ have?

Question

I've shown that $|\text{Aut}(S_3\times S_3)|\ge 72$, how can I show that $|\text{Aut}(S_3\times S_3)|\le 72$ ?

Answer 1

You need to show that the direct factors are the only normal subgroups of $G = S_3 \times S_3$ that are isomorphic to $S_3$. That implies that any automorphism must either fix both direct factors or interchange them. Since ${\rm Aut}(S_3) \cong S_3$, there are $36$ that fix them both, and another $36$ that interchange them.

You could show first that there are only two normal subgroups of order $3$, and then show that there is only one way to extend each of them to a normal subgroup of order $6$.