Lattices of finite abelian groups

We study certain lattices constructed from finite abelian groups. We show that such a lattice is eutactic, thereby confirming a conjecture by B"ottcher, Eisenbarth, Fukshansky, Garcia, Maharaj. Our methods also yield simpler proofs of two known results: First, such a lattice is strongly eutactic if and only if the abelian group has odd order or is elementary abelian. Second, such a lattice has a basis of minimal vectors, except for the cyclic group of order 4.