On Two Group Functors Extending Schur Multipliers

Liedtke has introduced group functors K and (K) over tilde, which are used in the context of describing certain invariants for complex algebraic surfaces. He proved that these functors are connected to the theory of central extensions and Schur multipliers. In this work, we relate K and (K) over tilde to a group functor tau arising in the construction of the non-abelian exterior square of a group. In contrast to (K) over tilde, there exist efficient algorithms for constructing tau, especially for polycyclic groups. Supported by computations with the computer algebra system GAP, we investigate when K(G, 3) is a quotient of tau(G), and when tau(G) and (K) over tilde (G, 3) are isomorphic.