<s>[INST] Disjoint Cycle Question I'm working on this question:

List the permutations of $\{1,2,3\}$ in disjoint cycle form.

I already know what a disjoint cycle is. It's basically means that every cycle contains numbers that are not in any other cycle. So with that in mind, do I write all the possible permutations of
\begin{pmatrix} 
1 & 2 & 3\\
? & ? & ?
\end{pmatrix}
such that I could write down all of it's permutations in disjoint cycle form? A push to get me started is all I'm asking for here.
EDIT
Please see finalized answer below.
 [/INST] Here is my attempted:
\begin{pmatrix} 1 & 2 & 3 \\ 3 & 2 & 1  \end{pmatrix} is given by $(1 3)(2)$.
Then \begin{pmatrix} 1 & 2 & 3\\3 & 1 & 2 \end{pmatrix}is given by $(1 3 2)$.
Then \begin{pmatrix} 1 & 2 & 3\\ 2& 3 & 1\end{pmatrix} is given by $(1 2 3)$.
Then \begin{pmatrix} 1 & 2 & 3 \\2 & 1 & 3 \end{pmatrix} is given by $(1 2)(3)$.
Then \begin{pmatrix} 1 & 2 & 3 \\1 & 3 & 2 \end{pmatrix} is given by $(1)(2 3)$.
Lastly, \begin{pmatrix} 1 & 2 & 3\\1 & 2 & 3 \end{pmatrix} is given by $(1)(2)(3)$, but these are just the identity matrices, so we can eliminate $(2)$ and $(3)$ and are only left with $(1)$.
So therefore, the final list of permutations is given by $$(1),(12),(13),(23),(123),(132).$$ 
</s>