Thanks very much once again. It is very clear and very helpful :-)

Hi Geoff, I may ask again for your help as I am really stuck on this one: Now $G$ has cyclic Sylow $7$-subgroups for otherwise, $G$ contains a element of order $7$ with eigenvalues $1,\omega, \omega^{-1}$, where $\omega = e^{\frac{2 \pi i}{7}}$, I could show that the $7-$Sylow is abelian but that's all.

Thanks very much Geoff for the explanation. I would not have found this one by myself :-) This glimpse at modular representation theory shows it is very powerful !

@ Geoff. Do you have a preferred reference for some examples of reduction? Also, how do you prove that a Sylow $-$normalizer must have order 21. All I have is $C_G(P<N)G(P)$ by Burnside and $[N_G(P):C_G(P)]\in \{2,3,6\}$ by the N/C lemma.

Thanks Derek. I found Mitchell's article. To answer your question, I only have access to articles freely available on the internet.

Thanks vm Geoff. It will take me some time to go through your proof but it is extremely precious ! May I just ask: when you mention reduction, do you refer to the use of modular representations or something else ?