  The analogies between symplectic and orthogonal groups, regarded as
symmetries of real bilinear forms, are manifest in their (metaplectic and spin)
projective representations. In finite dimensions, those are true
representations of doubly covering groups; but one may also use group
extensions by a circle. Here we lay out a parallel treatment of of the
Mp$^\mathrm{c}$ and Spin$^\mathrm{c}$ covering groups, acting on the respective
Fock spaces by permuting certain Gaussian vectors. The cocycles of these
extensions exhibit interesting similarities.
