Confused with Cayley's Theorem in group theory. 
Cayley's Theorem: Every group is isomorphic to a group of permutations.

$\mathbb Z_6$ is a group and $S_3$ is a permutation, but $\mathbb Z_6$ is not isomorphic to $S_3$.
$\mathbb Z_6$ is abelian while $S_3$ is not, hence they are not isomorphic.
So, what is going on here? I thought every group is isomorphic to a group of permutations.
 A: A more precise statement of Cayley's theorem states that if $|G| = n$, then $G$ is a subgroup of $S_n$.
In this case, $|\mathbb{Z}_6| = 6$, so $\mathbb{Z}_6 \leq S_6$.  In particular, it will be the subgroup generated by the $6$-cycle $\sigma = (1,2,3,4,5,6)$.  
A: It can be a subgroup of a group of permutations, not the whole group! For example,
$$C_6\cong \langle\; (1,2,3,4,5,6)\;\rangle\le S_6\;$$
A: The group $S_n$ has order $n!$ (in group theory it's customary calling order the number of elements in the group).
If your interpretation were correct, there wouldn't exist groups with order $3$, $4$, $5$ and so on. But for every natural number $m>0$, there is a group of order $m$, for instance the cyclic group $\mathbb{Z}/m\mathbb{Z}$. Moreover, $S_n$ is not abelian as soon as $n>2$, so general abelian groups would not be covered.
What Cayley's theorem says is that for every group $G$ there are $n>0$ and a subgroup $H$ of $S_n$ such that $G$ is isomorphic to $H$. With different terminology, there are $n>0$ and an injective homomorphism (or embedding) $G\to S_n$.
For instance, $\mathbb{Z}/6\mathbb{Z}$ is isomorphic to the subgroup of $S_6$
$$
\{\mathit{id},(123456),(135)(246),(14)(25)(36),(153)(264),(165432)\}
$$
that is
$$
\{(123456)^0,(123456)^1,(123456)^2,(123456)^3,(123456)^4,(123456)^5\}
$$
It is also isomorphic to the subgroup of $S_5$
$$
\{\mathit{id},(123)(45),(132),(45),(123),(132)(45)\}
$$
that is
$$
\{(123)(45)^0,(123)(45)^1,(123)(45)^2,(123)(45)^3,(123)(45)^4,(123)(45)^5\}.
$$
(here id stands for the identity permutation).
Note also that the usual proof of Cayley's theorem will embed $S_3$ in $S_6$, because it proves the existence of an injective homomorphism $G\to S_n$ where $n=|G|$.
