Let the symmetric group $S_n$ on $n$ letters act on $S_n^d=S_n\times\cdots\times S_n$ by simultaneous conjugation, i.e. $\pi\in S_n$ acts on $(\sigma_1,\ldots,\sigma_d)\in S_n^d$ by $\pi.(\sigma_1,\ldots,\sigma_d)=(\pi\sigma_1\pi^{-1},\ldots,\pi\sigma_d\pi^{-1})$.

I would like to know if there is a classification (a combinatorial description) of the orbits of this action, at least in the case $d=2$. For $d=1$, a combinatorial description is given by the cycle structure of the permutation.

There is a related question that deals with *deciding* whether two elements of $S_n^d$ are in the same orbit and the author of that question also defines a canonical representative for each orbit, but it is quite indirect. It seems to me that a classification for arbitrary $d$ is probably not known.

In another related question, the *number* of orbits is discussed, but I am really interested in a combinatorial description that classifies them.

**Edit.** To clarify what I mean by *"combinatorial description"*: At the very least, I am looking for an algorithm that enumerates exactly one representative from each simultaneous conjugacy class in $S_n^d$ with polynomial delay (polynomial in $n$ and $d$ will be fine).