Commutator subgroups of irreducible $\mathrm C$-group

A classification up to isomorphism is given of groups that are irreducible orientable $\mathrm C$-groups in the sense of Kulikov and have commutator subgroups that are either free of rank 2 or the Heisenberg group $\mathscr H_3$. In addition, it is shown that the commutator subgroup of every Coxeter group generated by a single conjugacy class of elements is the commutator subgroup of some irreducible orientable $\mathrm C$-group.