Let $p> 3$ be a prime number and $G$ be a finite group of order $p(p^2-1) / 4 $. If $ 4 \mid (p+1)$ then easily we can see that the $ p$-Sylow subgroup of $G$ is a normal subgroup of $G$. As I checked it seems that for each $p$ this result holds i.e. the $ p$-Sylow subgroup of $G$ is a normal subgroup of $G$. Any references or hints would be highly appreciated. Thanks in advance.