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This is quite a general question. The case for the previous $S_p$ is trivial and it seems the case with $S_{p^2}$ is different. For all $2\le n\le p-1$, those $p$-Sylows will have $p^n$ order.

So to make that happen, my idea is generally, for each case, I use $n$ different cyclic's generator (like $n$ different $p$-cycles) to generate a new group, which should contain $p^n$ elements.

Am I right?

Shaun
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qwertymask
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    Sylow subgroups of symmetric groups are well understood and are direct products of iterated wreath products of cyclic groups of order $p$. For $S_{p^k}$ you get $C_p \wr C_p \wr \cdots k\hbox{ times } \cdots \wr C_p$. – Derek Holt Nov 11 '19 at 08:27
  • ... Sorry, I don't understand the direct products of iterated wreath products... What is that? By the way, is there a way to produce $S_{p^k}$ with the semi-direct product? – qwertymask Nov 11 '19 at 16:49
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    What is the question? – verret Nov 11 '19 at 17:29
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    You cannot possibly understand the Sylow subgroups of $S_n$ without knowing what a wreath product is. – Derek Holt Nov 11 '19 at 17:48

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