Does the semidirect product $(H⋊K)$ is the smallest group contained the groups $H$ and $K$?

Question

I recently learn from the semidirect product lesson, and I occur a problem that if given the $2$ arbitrary groups $H$ and $K$, and we also construct the semidirect product group $(G=H⋊K)$ from permutation representation, I wonder that whether the $G$ is the smallest order contain the two groups $H$ and $K$ ? Since I "guess" there are lots of approaches to create the group from $H$ and $K$, thus the smallest must be important to us.

Thanks a lot.

Answer 1

No. What if $G=H{}{}{}{}{}{}{}{}$? If the orders are coprime then the direct product is minimal.