 Solution (source code)

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= Solution

By transitivity and the <orbit-stabilizer theorem>,
$$
|G|=7|G_x|=7\cdot24=168=2^3\cdot3\cdot7.
$$
Inside $G_x\cong S_4$, there are three Sylow $2$-subgroups, each of order eight, and four Sylow $3$-subgroups, each of order three. These are also Sylow <subgroups> of $G$.

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