# An example of a group

I need an example of a finite group $G$ by the following properties:

1) Order $G$ is $336$.

2) For every prime $p$, $G$ has not any elements of $7p$.

3) the number of Sylow $7$-subgroups $G$ is $8$.

4) $G$ is not isomorphic to $PGL(2,7)$.

Can anybody help me!

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I want a question to be asked, and not commands to be given. –  amWhy Dec 28 '12 at 15:22
@amWhy Meh. ${}{}$ –  Pedro Tamaroff Dec 28 '12 at 15:35
@amWhy: It sounds like a colored menu. ;-) –  B. S. Dec 28 '12 at 15:37
Use the GAP - Groups, Algorithms, Programming - a System for Computational Discrete Algebra. See gap-system.org –  Elias Dec 28 '12 at 15:46
I am afraid that there are no groups satisfying 1) - 4). Properties 1), 2), 3) imply that the normalizer of a Sylow 7-subgroup is a Frobenius group of order 42, and hence that $G$ acts 3-transitively on its Sylow 7-subgroups. The only such group is ${\rm PGL}(2,7)$. –  Derek Holt Dec 28 '12 at 17:42