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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!

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    @DerekHolt: $T$hank you so much.2012-12-28

1 Answers 1

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This question had already been answered in comments by Derek Holt who gave purely theoretical explanation: a comment of him says "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 $PGL(2,7)$".

As another comment by @Elias suggests to use GAP, I would also show how to check this in GAP, so that the question may be removed from the unanswered queue:

Step 1: List all groups of order 336:

gap> l:=AllSmallGroups(336);; gap> Length(l); 228 

Step 2: Filter those from condition (2):

gap> l1:=Filtered(l, g -> ForAll(ConjugacyClasses(g),  > c -> not IsPrimeInt(Order(Representative(c))/7))); [ Group([ (1,4,6,8,5,2,7,3), (1,3,8,6,5,4,7) ]) ] 

Observe that this condition already lefts us with only one group.

Step 3: Filter those from condition (3):

gap> l2:=Filtered(l1, g -> Size(First(ConjugacyClassesSubgroups(g),  > c -> Order(Representative(c))=7))=8); [ Group([ (1,4,6,8,5,2,7,3), (1,3,8,6,5,4,7) ]) ] 

Step 4: Check the isomorphism type of the group: it is $PGL(2,7)$ since it thas the same IdGroup:

gap> IdGroup(l2[1]); [ 336, 208 ] gap> IdGroup(PGL(2,7)); [ 336, 208 ] 

One could also see this from here:

gap> StructureDescription(l2[1]); "PSL(3,2) : C2" gap> StructureDescription(PGL(2,7)); "PSL(3,2) : C2"