# Is there a way to classify all finite groups with trivial centre?

Is there a way to classify all finite groups with trivial centre?

As far as I know, all finite symmetric groups, all non-abelian finite simple groups, $A_4$, the trivial group, and all direct products of aforementioned groups satisfy that condition.

But are there any other ones?

Any help will be appreciated.

• direct products of any of these. – Lord Shark the Unknown Dec 27 '17 at 12:44
• $A_4$ is not in your list. – Mark Bennet Dec 27 '17 at 12:53
• All (nontrivial) semidirect products with faithful actions. – tomasz Dec 27 '17 at 13:08