Here's the problem:
Let $G$ be a finite group with a normal subgroup $N$ such that $C_{G}(N) \leq N$. Show that $|G| \leq |N|!$.
Also, show that this upper bound is achieved when $|N|=4$. Identify $G$ in this case.
Now, I've already proved the first part of the problem. What I'm having trouble with is the second part, I don't believe it.
Let $G = N = K_{4}$. Then all assumptions of the statement are met, and $|N| = 4$, but clearly the upper bound is not achieved. What am I missing?