We study the subgroups of the full linear group GL(n, R) over a Dedekind ring R that contain the group of quasidiagonal matrices of fixed type with diagonal blocks of at least third order, each of which is generated by elementary matrices. For any such subgroup H there exists a unique D-net σ of ideals of R such that[Figure not available: see fulltext.], where E(σ) is the subgroup generated by all transvections of the net subgroup G(σ). and[Figure not available: see fulltext.] is the normalizer of G(σ). The subgroup E(σ) is normal in[Figure not available: see fulltext.]. To study the factor group[Figure not available: see fulltext.] we introduce an intermediate subgroup F(σ), E(σ) ≤ F(σ) ≤ G(σ). The group[Figure not available: see fulltext.] is finite and is connected with permutations in the symmetric group. The factor group G(σ)/F(σ) is Abelian - these are the values of a certain "determinant." In the calculation of F(σ)/E(σ) appears the SK₁-functor. Results are stated without proof.