The only subgroup of $S_{10}$ with the same number of elements as $G$ which I can recall is $S_4$. Is this the subgroup?
2026-04-06 22:45:44.1775515544
Find subgroup of $S_{10}$ isomorphic to $G=\mathbb{Z}/4\mathbb{Z}\times \mathbb{Z}/6\mathbb{Z}$
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The group $G=\mathbb Z/4\mathbb Z \times \mathbb Z/6\mathbb Z$ is generated by the two commuting elements $(1,0)$ and $(0,1)$ of orders $4$ and $6$, respectively. In $S_{10}$ we can find a $4$-cycle and a $6$-cycle on disjoint sets to make them commute. Hence, consider the subgroup of $S_{10}$ generated by the cycles $(1\ 2\ 3\ 4)$ and $(5\ 6\ 7\ 8\ 9\ 10)$.