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

Question

Consider the group of permutation matrices G ={I3, P1, P2, P3, P4, P5} For n = 3 the permutation matrices are I3 and the five matrices are: P1 = [1,0,0;0,0,1;0,1,0] P2 = [0,1,0;1,0,0;0,0,1] P3 = [0,1,0;0,0,1;1,0,0] P4 = [0,0,1;0,1,0;1,0,0] P5 = [0,0,1;1,0,0;0,1,0]

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->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.

Answer 1

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