Consider $O(n)$, the Lie group of $n{\times}n$ orthogonal matrices. Let $\pi$ be a given permutation of $(1,2,\ldots,n)$; the elements of the corresponding permutation matrix are \begin{eqnarray} P_{i,j} &\equiv& \left\{ \begin{array} {ccl} 1 & \quad & j = \pi(i)\\ 0 & & \text{else} \end{array} \right. \end{eqnarray} with $i,j = 1,\ldots,n$. The set of matrices $\{I,P,P^2,\ldots,P^{k-1}\}$ with $k$ denoting the order of $\pi$, form a subgroup of $O(n)$, since $P^m \cdot (P^m)^T = I$ with unit matrix $I$ and $0\,{\le}\,m\,{\le}\,{k-1}$. Now let's introduce an equivalence relation on the matrices $R, S \in O(n)$ \begin{eqnarray} R \sim S \iff S = P^m \cdot R \cdot (P^m)^T \quad\text{with}\quad 0\,{\le}\,m\,{\le}\,{k-1} \end{eqnarray} If I'm not mistaken, this equivalence relation partitions the Lie group $O(n)$ into an infinite set $X$ of equivalence classes with each class containing $k$ matrices.
What kind of object is $X$? Can it be expressed as a quotient group? Naïve as I am, I would expect $X$ to be a group, too. But I'm stuck at defining a meaningful way to 'multiply' two elements of $X$. Any advice would be greatly appeciated.