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The virtual braid group $VB_n$, the virtual twin group $VT_n$ and the virtualtriplet group $VL_n$ are extensions of the symmetric group $S_n$, which aremotivated by the Alexander-Markov correspondence for virtual knot theories. Thekernels of natural epimorphisms of these groups onto the symmetric group $S_n$are the pure virtual braid group $VP_n$, the pure virtual twin group $PVT_n$and the pure virtual triplet group $PVL_n$, respectively. In this paper, weinvestigate commutator subgroups, pure subgroups and crystallographic quotientsof these groups. We derive explicit finite presentations of the pure virtualtriplet group $PVL_n$, the commutator subgroup $VT_n^{'}$ of $VT_n$ and thecommutator subgroup $VL_n^{'}$ of $VL_n$. Our results complete theunderstanding of these groups, except that of $VB_n^{'}$, for which theexistence of a finite presentations is not known for $n \ge 4$. We also provethat $VL_n/PVL_n^{'}$ is a crystallographic group and give an explicitconstruction of infinitely many torsion elements in it.

Commutator subgroups and crystallographic quotients of virtual extensions of symmetric groups