Smallest $n$ such that $G$ is a subgroup of the symmetric group $S_n$ [duplicate]

The following question from MathOverflow should answer your questions and more: Smallest permutation representation of a finite group?

The answer to 3) is yes, and more generally if $G$ is a finite abelian group such that

$$G \cong \mathbb{Z}_{p_1^{a_1}} \times \cdots \times \mathbb{Z}_{p_t^{a_t}}$$

where $p_i$ are prime and $a_i \geq 1$, then the minimal $n$ is $p_1^{a_1} + \cdots + p_t^{a_t}$. A proof can be found in the following paper that Jack Schmidt mentions in his MO answer.

Johnson, D. L. "Minimal permutation representations of finite groups." Amer. J. Math. 93 (1971), 857-866. MR 316540 DOI: 10.2307/2373739.