arXiv2026
Let $n, m \ge 4$. We study the Zappa--Szép products $G = HK$ with $H = \langle x\rangle \rtimes \langle y\rangle \cong SD_{2^n}$ and $K = \langle z\rangle \rtimes \langle w\rangle \cong SD_{2^m}$, together with the fixed generators $x, y, z, w$, and we describe the parameter tuples which encode the mixed commutators of these generators, according to the cores of $\langle x\rangle$ and $\langle z\rangle$ in~$G$. First, when both $\langle x\rangle$ and $\langle z\rangle$ are normal in~$G$, we show that the admissible parameter tuples are exactly the solutions of an explicit system of six polynomial congruences. Second, when the cores $\langle x\rangle_G$ and $\langle z\rangle_G$ are arbitrary subgroups of $\langle x\rangle$ and $\langle z\rangle$, under the simplifying assumption $[x, z] = 1$, and for generating systems for which all mixed commutators lie in $\langle x\rangle\langle z\rangle$, we obtain an analogous classification by twelve congruences together with two order conditions; this is the semidihedral counterpart of the Hu--Yu classification~\cite{HuYu2025} for dihedral groups $D_{2n}$ and $D_{2m}$ with $n, m$ odd. In contrast with that case, we further construct an explicit exact product with both cores non-trivial and $[x, z] \ne 1$, so that the semidihedral setting contains additional families of exact products.