TITLE: Sylow 3-subgroups of symmetric group
QUESTION [5 upvotes]: Is there any routine technique to find a set of permutations which generate a Sylow $3$-subgroup of the symmetric group $S_{n}$?

REPLY [11 votes]: Answer by example:
Let's take $n=16=9+3+3+1$.
The 3-Sylow of $S_n$ is a subgroup of the symmety group of this graph:
$$
\bullet\!\!\!\stackrel{\textstyle/}{\phantom{\bullet}}\stackrel{\textstyle\bullet}{\stackrel{\textstyle |}\bullet}\stackrel{\textstyle\backslash}{\phantom{\bullet}}
\!\!\!\!\bullet
\!\!\!
\begin{matrix}\diagup\\\phantom{\Big|}\\\phantom{.}\\\phantom{.}\\\end{matrix}
\!\!
\bullet\!\!\!\stackrel{\textstyle/}{\phantom{\bullet}}
\stackrel{\textstyle\bullet}{\stackrel{\textstyle |}
{\stackrel{\textstyle\bullet}{\stackrel{\textstyle |}\bullet}}}\stackrel{\textstyle\backslash}{\phantom{\bullet}}
\!\!\!\!\bullet
\!\!
\begin{matrix}
\diagdown\\\phantom{.}\\\phantom{\Big|}\\\phantom{.}\\
\end{matrix}
\!\!\!
\bullet\!\!\!\stackrel{\textstyle/}{\phantom{\bullet}}\stackrel{\textstyle\bullet}{\stackrel{\textstyle |}\bullet}\stackrel{\textstyle\backslash}{\phantom{\bullet}}
\!\!\!\!\bullet
\quad
\bullet\!\!\!\stackrel{\textstyle/}{\phantom{\bullet}}\stackrel{\textstyle\bullet}{\stackrel{\textstyle |}\bullet}\stackrel{\textstyle\backslash}{\phantom{\bullet}}
\!\!\!\!\bullet\,\,\,\,\,
\bullet\!\!\!\stackrel{\textstyle/}{\phantom{\bullet}}\stackrel{\textstyle\bullet}{\stackrel{\textstyle |}\bullet}\stackrel{\textstyle\backslash}{\phantom{\bullet}}
\!\!\!\!\bullet\quad
\bullet
$$
It consists of those symmetries
that (1) do not permule the connected components, and (2) preserve the cyclic order among the children of each vertex.
A minimal set of generators is given by:
$$
(1,2,3),\quad (1,4,7)(2,5,8)(3,6,9),\quad (10,11,12),\quad (13,14,15).
$$