Starting with only three of the six relations defining the standard (Manin) GLq(2), we try to construct a quantum group. The antipode condition requires some new relations, but the process stops at a Hopf algebra with a Birkhoff–Witt basis of irreducible monomials. The quantum determinant is group-like but not central, even when q = 1. So, the two Hopf algebras constructed in this way are not isomorphic to the Manin GLq(2), all of whose group-like elements are central. Analogous constructions can be made starting with the Dipper–Donkin version of GLq(2), but these turn out to be included in the two classes of Hopf algebras described above.