Consider the matrix \(\displaystyle P = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix}\).
Let the transpose of a matrix \( X \) be denoted by \( X^T \). Then the number of \( 3\times 3 \) invertible matrices \( Q \) with integer entries, such that \( Q^{-1}=Q^T \) and \( PQ=QP \) is:
