a nonabelian group $G$ such that it is the internal direct product of $H,K$ with $N$ a normal subgroup in the center trivially intersecting $H,K$

Question

In sequal to my previous question Does it really matter that $G$ must be a nonabelian group in one exercise from Hungerford’s algebra book?, can one find a nonabelian group $G$ that is the internal direct product of $H,K$ such that $N$ is a normal subgroup in the center of $G$ but not nontrivially intersecting $H$ and $K$?

Answer 1

Take your favorite nonabelian group $G$ with nontrivial center $Z=Z(G)$. Consider the direct product $G\times G$; the center is $Z\times Z$ and $\{(x,x):x\in Z\}$ satisfies the requirement.

(Internal or external product doesn't matter, of course.)