# Consider group of permutation matrices and write out elements isomorphic to the group and exhibit it

Consider the group of permutation matrices $G =\{I_3, P_1, P_2, P_3, P_4, P_5\}$ For $n=3$ the permutation matrices are $I_3$ and the five matrices are:

\begin{equation*} P_1 = [1,0,0;0,0,1;0,1,0] \\ P_2 = [0,1,0;1,0,0;0,0,1] \\ P_3 = [0,1,0;0,0,1;1,0,0] \\ P_4 = [0,0,1;0,1,0;1,0,0] \\ P_5 = [0,0,1;1,0,0;0,1,0] \end{equation*}

Write out the elements of a group of permutations that is isomorphic to $G$, and exhibit an isomorphism from $G$ to this group!

I think it has to do with Cayley's Theorem. With $f_a:G\to G$ defined by $f_a(x) = ax$ for each $a$ that exists in $G$...

I thought about making a table, but realize I don't know how to since I am dealing with matrices.

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HINT: Look at the result of multiplying each of those matrices by the vector $(1,2,3)^T$.