Realizing a finite group as a subgroup of a product of two groups of permutation matrices

In this paper we prove that any finite group of order $n$ can be viewed as the group of the solutions of a certain matrix equation $XB=BY$, where the unknowns $X,Y$ are two permutation matrices of order $n$ and $(1+k)n+2$ respectively and where $k\in \mathbb{N}$ is given by Cayley’s theorem. Moreover, we show that $G$ is isomorphic to a certain subgroup formed by permutation matrices of order $(1+k)n$ obtained by permuting all the rows of the identity matrix $I_{(1+k)n}$.