# Consider team of permutation matrices and also draw up components isomorphic to the team and also show it

Consider the team 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 also the 5 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 components of a team of permutations that is isomorphic to $G$, and also show an isomorphism from $G$ to this team!

I assume it concerns Cayley is Theorem. With $f_a:G\to G$ specified by $f_a(x) = ax$ for each and every $a$ that exists in $G$ ...

I thought of making a table, yet understand I do not recognize just how to given that I am managing matrices.

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

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