TITLE: Properties about order of a general group
QUESTION [5 upvotes]: For a group, $G$, is it true that $o(Z(G))\cdot o([G,G]) \leq o(G)$ where $Z(G)$ denotes the centre of $G$ and $[G,G]$ denotes the commutator subgroup of $G$?

REPLY [6 votes]: Any perfect group $G$ (ie. $[G,G]=G$) with a nontrivial center is a counterexample; take for example $G= SL(2,5)$ (mentionned here).