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The semisimple bismash product Hopf algebra $J_n=k^{S_{n-1}}\#kC_n$ for analgebraically closed field $k$ is constructed using the matched pair actions of$C_n$ and $S_{n-1}$ on each other. In this work, we reinterpret these actionsand use an understanding of the involutions of $S_{n-1}$ to derive a newFroebnius-Schur indicator formula for irreps of $J_n$ and show that for $n$odd, all indicators of $J_n$ are nonnegative. We also derive a variety ofcounting formulas including Theorem 6.2.2 which fully describes the indicatorsof all $2$-dimensional irreps of $J_n$ and Theorem 6.1.2 which fully describesthe indicators of all odd-dimensional irreps of $J_n$ and use these formulas toshow that nonzero indicators become rare for large $n$.

An Indicator Formula for the Hopf Algebra $k^{S_{n-1}}\#kC_n$