Centrally large subgroups permute and their product is centrally large

Statement
Suppose $$P$$ is a group of prime power order and $$A,B$$ are fact about::centrally large subgroups of $$P$$. Then, $$AB = BA$$ and $$AB$$ is also a centrally large subgroup.

Related facts

 * All minimal CL-subgroups have the same commutator subgroup
 * Centralizer-large subgroups permute and their product is centralizer-large