Can someone show me the proof for :
Any nonabelian group of order 12 is isomorphic to $D_6$, $A_4$, or $\mathbb{Z}_3 \times \mathbb{Z}_4$
I've seen a few proofs where this is included in also proving $A_4$ and $\mathbb{Z}_3 \times \mathbb{Z}_4$ but $D_6$ always seems to be left as an exercise.
There is a nice note on the classification of all groups of order $12$ by Keith Conrad here, see Theorem $1$. There are exactly three different non-abelian groups of order $12$, namely $A_4, D_6$ and $\mathbb{Z}/3 \rtimes \mathbb{Z}/4$. The proof uses, of course, Sylow theorems and an isomorphism property of semidirect products. Special attention also is paid to the semidirect product $\mathbb{Z}/3 \rtimes \mathbb{Z}/4$, which is the group you should take instead of the abelian direct product.