It seems that when $p>3$ is a prime, then each group of order $p(p^2+1)/2$ is abelian as I checked by Gap for small $p$. Is it true for each $p$?
Thanks for your answers



Groups of order $p(p^2+1)/2$
The smallest counterexample I could find is for $p=53$; then $(p^2+1)/2=5\times281$ and the cyclic group of order 281 has an automorphism of order 5, so the corresponding semidirect product will not be abelian.


