A quotient of the Artin braid groups related to crystallographic groups

Let n≥3. In this paper, we study the quotient group Bn/[Pn,Pn] of the Artin braid group Bn by the commutator subgroup of its pure Artin braid group Pn. We show that Bn/[Pn,Pn] is a crystallographic group, and in the case n=3, we analyse explicitly some of its subgroups. We also prove that Bn/[Pn,Pn] possesses torsion, and we show that there is a one-to-one correspondence between the conjugacy classes of the finite-order elements of Bn/[Pn,Pn] with the conjugacy classes of the elements of odd order of the symmetric group Sn, and that the isomorphism class of any Abelian subgroup of odd order of Sn is realised by a subgroup of Bn/[Pn,Pn]. Finally, we discuss the realisation of non-Abelian subgroups of Sn of odd order as subgroups of Bn/[Pn,Pn], and we show that the Frobenius group of order 21, which is the smallest non-Abelian group of odd order, embeds in Bn/[Pn,Pn] for all n≥7.