$S$-matrix in permutation orbifolds

For a fixed positive integer $k$, any element $g$ of the permutation group $S_{k}$ acts on the tensor product vertex operator algebra $V^{\otimes k}$ in the obvious way. In this paper, we determine the $S$-matrix of $\left(V^{\otimes k}\right)^{G}$ if $G=\left\langle g\right\rangle$ is the cyclic group generated by $g=\left(1,\ 2,\cdots,k\right).$