Hurwitz Equivalence in Tuples of Dihedral Groups, Dicyclic Groups, and Semidihedral Groups

Abstract

Let $D_{2N}$ be the dihedral group of order $2N$, ${\it Dic}_{4M}$ the dicyclic group of order $4M$, $SD_{2^m}$ the semidihedral group of order $2^m$, and $M_{2^m}$ the group of order $2^m$ with presentation $$M_{2^m} = \langle \alpha, \beta \mid \alpha^{2^{m-1}} = \beta^2 = 1,\ \beta\alpha\beta^{-1} = \alpha^{2^{m-2}+1} \rangle.$$ We classify the orbits in $D_{2N}^n$, ${\it Dic}_{4M}^n$, $SD_{2^m}^n$, and $M_{2^m}^n$ under the Hurwitz action.