  We describe the structure of the extended Clifford Group (defined to be the
group consisting of all operators, unitary and anti-unitary, which normalize
the generalized Pauli group (or Weyl-Heisenberg group as it is often called)).
We also obtain a number of results concerning the structure of the Clifford
Group proper (i.e. the group consisting just of the unitary operators which
normalize the generalized Pauli group). We then investigate the action of the
extended Clifford group operators on symmetric informationally complete POVMs
(or SIC-POVMs) covariant relative to the action of the generalized Pauli group.
We show that each of the fiducial vectors which has been constructed so far
(including all the vectors constructed numerically by Renes et al) is an
eigenvector of one of a special class of order 3 Clifford unitaries. This
suggests a strengthening of a conjuecture of Zauner's. We give a complete
characterization of the orbits and stability groups in dimensions 2-7. Finally,
we show that the problem of constructing fiducial vectors may be expected to
simplify in the infinite sequence of dimensions 7, 13, 19, 21, 31,... . We
illustrate this point by constructing exact expressions for fiducial vectors in
dimensions 7 and 19.
