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Let $G$ be a finite group multiplicatively written. The small Davenportconstant of $G$ is the maximum positive integer ${\sf d}(G)$ such that thereexists a sequence $S$ of length ${\sf d}(G)$ for which every subsequence of $S$is product-one free. Let $s^2 \equiv 1 \pmod n$, where $s \not\equiv \pm1 \pmodn$. It has been proven that ${\sf d}(C_n \rtimes_s C_2) = n$ (see Lemma 6 of[Zhuang, Gao; Europ. J. Combin. 26 (2005), 1053-1059]). In this paper, wedetermine all sequences over $C_n \rtimes_s C_2$ of length $n$ which areproduct-one free. It completes the classification of all product-one freesequences over every group of the form $C_n \rtimes_s C_2$, including thequasidihedral groups and the modular maximal-cyclic groups.

Extremal product-one free sequences over $C_n \rtimes_s C_2$