I have two matrices which are all permutations $C = \begin{pmatrix} 0 & 1 & 0 \\ 1& 0& 0\\0 & 0 &1\end{pmatrix}$and $ A= \begin{pmatrix} 0 & 0 & 1 \\ 0& 1& 0\\1 & 0 &0\end{pmatrix}$of the identity matrix. The question I want to answer is : Is it possible to create the remaining four matrices in this $S_3$ group. Also, can other permutations be generated from matrix products $\otimes$ of these two?
I have computed $C^2$ and I got a (321) permutation $C^3$ led to the identity matrix. So I concluded that no, it is impossible. $A^2 \rightarrow I_3$
On the matrix product part, I simply considered the fact that the product will be a 9 x 9 matrix and I said no.