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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!

  • 6
    I want a question to be asked, and not commands to be given.2012-12-28
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    @amWhy Meh. ${}{}$2012-12-28
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    @amWhy: It sounds like a colored menu. ;-)2012-12-28
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    Use the GAP - Groups, Algorithms, Programming - a System for Computational Discrete Algebra. See http://www.gap-system.org/2012-12-28
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    What do you need this group for? (I am sorry to have to tell you that you are out of luck.)2012-12-28
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    @Derek Holt: I am working on the group $PGL(2,q)$. I am looking for a counterexample when that properties is valid, but $G$ is not isomorphic to $PGL(2,q)$.2012-12-28
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    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 ${\rm PGL}(2,7)$.2012-12-28
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    @DerekHolt: Thank you so much.2012-12-28

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