  In this paper, we classify a type of abstract groups by the central products
of dihedral groups and quaternion groups. We recognize them as abstract error
groups which are often not isomorphic to the Pauli groups in the literature. We
show the corresponding nice error bases equivalent to the Pauli error bases
modulo phase factors. The extension of these abstract groups by the symmetric
group are finite images of the Jones unitary representations (or modulo a phase
factor) of the braid group at q=i or r=4. We hope this work can finally lead to
new families of quantum error correction codes via the representation theory of
the braid group.
