Finite derived subgroup not implies FZ

Statement
It is possible to have a group with finite derived subgroup (i.e., a group whose derived subgroup is a finite group) that is not a FZ-group -- in other words, the index of the center is finite.

Related facts

 * FZ implies finite derived subgroup (this result is also called the Schur-Baer theorem).

Proof
The idea is to construct an infinite extraspecial group, for instance, by taking a central product of infinitely many copies of unitriangular matrix group:UT(3,p) with all the centers identified with each other. More generally, start with any group of nilpotency class two that is not abelian, and take a restricted external central product of a countably infinite number of copies, identifying the centers of all copies (in fact, identifying any subgroup intermediate between the center and the derived subgroup is good enough).