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Describe the structure of the Sylow $2$-subgroups of the symmetric group of degree $22$.

The only thing I've managed to deduce about the structure of $P\in \operatorname{Syl}_p(G)$ is that $|P| = 2^{12}$.

Help please :)

edit: I obviously can't count. Silly, I (for some reason) only counted $8$ as one $2$ and $16$ as one $2$. But as far as I can tell now, $2^{17}$ is the highest power of $2$ dividing $22$.

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    See another [thread](http://math.stackexchange.com/q/155785/11619) for a description of the Sylow subgroups of symmetric groups. In some sense this question is a duplicate of that one, but that assumes that you are familiar with a construction called wreath product. May be we should close this question as an abstract duplicate? OTOH small specific cases can be described explicitly without the language of wreath products, and some users may appreciate that?2012-06-10
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    But isn't $2^{19}$ the highest power of two that is a factor of $22!$?2012-06-10
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    $2^1\mid 2,6,10,14,18,22$; $2^2\mid 4,12,20$; $2^3\mid 8$; $2^4\mid 16$. $$6\cdot1+3\cdot2+1\cdot3+1\cdot4=19,$$ so $2^{19}\mid 22!$2012-06-13
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    OK agreed :) $2^{19}$ is the highest power dividing $22!$. thanks heaps :)2012-06-20

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