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