  We find a non-trivial representation of the symmetric group $S_n$ on the
$n$-fold Deligne product $\mathcal{C}^{\boxtimes n}$ of a modular tensor
category $\mathcal{C}$ for any $n \geq 2$. This is accomplished by checking
that a particular family of $\mathcal{C}^{\boxtimes n}$-bimodule categories
representing adjacent transpositions satisfies the symmetric group relations
with respect to the relative Deligne product. The bimodule categories are based
on a permutation action of $S_2$ on $\mathcal{C}\boxtimes\mathcal{C}$ discussed
by Fuchs and Schweigert in hep-th/1310.1329 , for which we show that it is, in
a certain sense, unique. In the context of condensed matter physics, the
$S_n$-representation corresponds to the specification of permutation twist
surface defects in a (2+1)-dimensional topological multilayer phase, which are
relevant to topological quantum computation and could promote the explicit
construction of the data of an $S_n$-gauged phase.
