Let $U$ denote a set of $N$ points $x_1, \ldots, x_N$ that live on the $(d-1)$-dimensional sphere. The set $U$ admits the following group actions:
- The permutation group $S_N$ acts on $U$ by changing the labels of the $x_i$
- The orthogonal group $\mathcal O_d$ acts on $U$ by rotating each $x_i$ individually.
My question is: how can I construct a group $G$ that has a copy of $S_N$ and $\mathcal O_d$ as subgroups, such that the action of $G$ on $U$ is consistent with the individual actions of $S_N$ and $\mathcal O_d$ above?
I assume that $G$ should be some kind of product of $S_N$ and $\mathcal O_d$, but I am not sure which kind is appropriate here (semi-product? wreath product?). Any help is very much appreciated!
If you mean that an element of $\mathcal{O}_d$ acts by simultaneously rotating every $x_i$, then you just want the direct product $S_N\times\mathcal{O}_d$. This works because the actions of $S_N$ and $\mathcal{O}_d$ commute with each other: permuting the points and then rotating all of them is the same as rotating all of them and then permuting them.
Alternatively, if you want $\mathcal{O}_d$ to be able to act by rotating just one point while leaving the others fixed (so you really have $N$ different actions, once for each point), this is exactly what the wreath product is for. Specifically, you would want the wreath product of $\mathcal{O}_d$ by $S_N$ using the permutation action of $S_N$ on the set $\{1,\dots,N\}$. Explicitly, this group can be described as a semidirect product $\mathcal{O}_d^N \rtimes S_N$ where $S_N$ acts on $\mathcal{O}_d^N$ by permuting the coordinates.