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

Abstract

Let D[subscript 2N] be the dihedral group of order 2N, Dic[subscript 4M] the dicyclic group of order 4M, SD[subscript 2m] the semidihedral group of order 2[superscript m], and M[subscript 2m] the group of order 2[superscript m] with presentation M[subscript 2m] = ⟨α,β∣α[superscript 2m−1] = β[superscript 2] = 1, βαβ[superscript −1] = α[superscript 2m−2+1]⟩. We classify the orbits in D[n over 2N], Dic[n over 4M], SD[n over 2m], and M[n over 2m] under the Hurwitz action.