How to convert a permutation group into linear transformation matrix? is there any example about apply isomorphism to permutation group
and how to convert linear transformation matrix to permutation group and convert back to linear transformation matrix

It seems that for each natural number $n$ there is an isomorphic embedding $i$ of the permutation group $S_n$ into the group of all non-degerated matrices of order $n$ over $\mathbb R$, defined as $i(\sigma)=A_\sigma=\|a_{ij}\|$ for each $\sigma\in S_n$, where $a_{ij}=1$ provided $\sigma(i)=j$, and $a_{ij}=0$ in the opposite case.
From the other side, if $G$ is any group (and, in particular, a matrix group of linear transormations) then any element $g\in G$ induces a permutation $j(g)$ of the set $G$ such that $j(g)h=gh$ for each $h\in G$. Then the map $j:G\to S(G)$ should be an isomorphic embedding of the group $G$ into the group $S(G)$ of all permutations of the set $G$. You can read more details about such embedding here.
