  We consider quantum two-block group algebra (2BGA) codes, a previously
unstudied family of smallest lifted-product (LP) codes. These codes are related
to generalized-bicycle (GB) codes, except a cyclic group is replaced with an
arbitrary finite group, generally non-abelian. As special cases, 2BGA codes
include a subset of square-matrix LP codes over abelian groups, including
quasi-cyclic codes, and all square-matrix hypergraph-product codes constructed
from a pair of classical group codes. We establish criteria for permutation
equivalence of 2BGA codes and give bounds for their parameters, both explicit
and in relation to other quantum and classical codes. We also enumerate the
optimal parameters of all inequivalent connected 2BGA codes with stabilizer
generator weights $W \le 8$, of length $n \le 100$ for abelian groups, and $n
\le 200$ for non-abelian groups.
