Invariants for critical dimension groups and permutation-Hermite equivalence

Motivated by classification, up to order isomorphism, of some dense subgroups of Euclidean space that are free of minimal rank, we obtain apparently new invariants for an equivalence relation (intermediate between Hermite and Smith) on integer matrices. These then participate in the classification of the dense subgroups. The same equivalence relation has appeared before, in the classification of lattice simplices. We discuss this equivalence relation (called {\it permutation-Hermite}), obtain fairly fine invariants for it, and have density results, and some formulas counting the numbers of equivalence classes for fixed determinant.