Let $G= \Bbb{Z}_3$ then determine the holomorph of $G$ which is
$$\Bbb{Z}_3 \rtimes_\phi \text{Aut}(\Bbb{Z}_3)$$
where $\phi :{\rm Aut}(\Bbb{Z}_3) \rightarrow {\rm Aut}(\Bbb{Z}_3$) is the identity.
My gut intuition tells me this is either $\Bbb{Z}_6$ or $S_3$ as the order is $3$ times $\phi(3)=2$ which is 6. How do I show what the holomorph is of this given group?
I know the binary operation is defined via
$$(h_1,k_1)(h_2,k_2)=(h_1 \phi(k_1)(h_2),k_1k_2)$$
where $h_i \in \Bbb{Z}_3$ and $k_i \in{\rm Aut}(\Bbb{Z}_3)$. So is it enough to determine whether or not the semi direct product is abelian or not? If so how do you show commutativity of the semi direct product?