(1,2)-Groups with p3-regulator quotient

For a prime p and a poset (1,2)=(τ1,τ2<τ3) of types, p-reduced almost completely decomposable groups with critical typeset (1,2) and a p-power as regulating index are called (1,2)-groups. The number of near-isomorphism types of indecomposable (1,2)-groups depends on the exponent pk of the regulator quotient. It is shown that indecomposable (1,2)-groups with a regulator quotient of exponent p3 have rank 4, and if the types τi and the prime p are fixed, then there are precisely four near-isomorphism types of indecomposable groups. It is unknown for which exponent pk0 of the regulator quotient exist infinitely many near-isomorphism types of indecomposable (1,2)-groups.