Let $G$ be a finite group such that $G$ has a normal subgroup $N$ of order $p(p^2+1)/2$, where $p>13$ is an odd prime and $p\ne 239$. Also $G/N\cong PSL(2,p)$. Can we say that there exists a prime divisor $t$ of $|N|$ such that $2t\not\mid \chi(1)$ for every $\chi\in \text{Irr}(G)$?

Of course we know that $N$ is isomorphic to ${\Bbb Z_p}\times H$, where $H$ is a solvable group of order $(p^2+1)/2$.

Thanks for your helps.

Let $G$ be a finite group such that $G$ has a normal subgroup $N$ of order $p(p^2+1)/2$, where $p>13$ is an odd prime and $p\ne 239$. Also $G/N\cong \text{PSL}(2,p)$. Can we say that there exists a prime divisor $t$ of $|N|$ such that $2t\not\mid \chi(1)$ for every $\chi\in \text{Irr}(G)$?

Of course we know that $N$ is isomorphic to ${\Bbb Z_p}\times H$, where $H$ is a solvable group of order $(p^2+1)/2$.

Thanks for your helps.