$G=\mathbb{Z}_2\times\mathbb{Z}_3$ is isomorphic to
$S_3$
A subgroup of $S_4$.
A proper subgroup of $S_5$
$G$ is not isomorphic to a subgroup of $S_n$ for all $n\ge 3$
What I know is Any finite group is isomorphic to a subgroup of $S_n$ for some suitable $n$(Caley's Theorem), 1 is not true as $G$ is abelian but $S_3$ is not,$4$ violates Caleys Theorem, for 3 I saw that there is an order $6$ element in $G$ namely $(1,1)$ but No element of the subgroup of $S_4$ have order $6$ right? Not sure about 3, Thank you for the help.