Integer group determinants of order 16 (Q6631623)

Let \(G\) be a finite group, let \(|G|\) be its order, and let \(S(G)\) be the set of the values of its integer group determinants. If \(|G|\le 15\), then \(S(G)\) is known. If \(|G|=16\), then \(S(G)\) is known in all but two cases, and the authors resolve them. Let \(C_n\) and \(Q_n\) be the cyclic group and the generalized quaternion group of order \(n\), respectively. The authors find \(S(G)\) for\N\begin{align*}\NG&=C_8{\times|_3}C_2:=\{g_1,g_2:g_1^8=g_2^2=e,\,g_2g_1=g_1^3g_2\} \\\N&=\{g_1^rg_2^s:0\le r\le 7,\,0\le s\le 1\}\N\end{align*}\Nand for\N\begin{align*}\NG&=Q_8{\times|}C_2 \\\N&:=\{g_1,g_2,g_3:g_1^4=g_3^2=e,\,g_1^2=g_2^2,\,g_2g_1=g_1g_2,\, g_3g_2=g_2g_3,\,g_3g_1=g_1^3g_3\} \\\N&=\{g_1^rg_2^sg_3^t:0\le r\le 3,\,0\le s,t\le 1\}.\N\end{align*}\NHere \(e\) denotes the unit element. \textit{B. Serrano} et al. [Comb. Number Theory 13, 59--65 (2024; Zbl 07856610)] have independently resolved the latter case. The present authors also describe the set inclusions in the family \(\{S(G): |G|=16\}\).