Let $S_n$ be the symmetric group on $n$ letters and define
$$S = \bigcup_{i=1}^{\infty} S_n.$$
That is, an element of $S$ is any permutation which permutes finitely many elements of $\mathbb{N}$. We can check it is indeed a group under the usual permutation composition operation.
Does this group have a name, or is it isomorphic to something else more standard? Any attempt to search for it gives me $\mathrm{Sym}(\mathbb{N})$ which notably is not isomorphic to $S$ since $S$ is countable (countable union of countable sets) and $\mathrm{Sym}(\mathbb{N})$ is not.