Order Equal to 288

Suppose that $\mid G \mid = 288 = 2^{5} \times 3^{2}$. By Sylow's Theorem:

$$\begin{tabular}{\vert c\vert c\vert l\vert} \hline prime & Sy... ...e 2 & 32 & 1, 3 \\ \hline 3 & 9 & 1, 4 \\ \hline \end{tabular}$$

If G is not simple, then n₃ = 4. But then $\mid G:N_{G}(Syl_{3}) \mid = 4$. But $\mid G \mid$ does not divide 4!. Therefore, by the Index Theorem G is not simple!