  We study the recently discovered gravitational lens SDSS J1004+4112, the
first quasar lensed by a cluster of galaxies. It consists of four images with a
maximum separation of 14.62''. The system has been confirmed as a lensed quasar
at z=1.734 on the basis of deep imaging and spectroscopic follow-up
observations. We present color-magnitude relations for galaxies near the lens
plus spectroscopy of three central cluster members, which unambiguously confirm
that a cluster at z=0.68 is responsible for the large image separation. We find
a wide range of lens models consistent with the data, but they suggest four
general conclusions: (1) the brightest cluster galaxy and the center of the
cluster potential well appear to be offset by several kpc; (2) the cluster mass
distribution must be elongated in the North--South direction, which is
consistent with the observed distribution of cluster galaxies; (3) the
inference of a large tidal shear (~0.2) suggests significant substructure in
the cluster; and (4) enormous uncertainty in the predicted time delays between
the images means that measuring the delays would greatly improve constraints on
the models. We also compute the probability of such large separation lensing in
the SDSS quasar sample, on the basis of the CDM model. The lack of large
separation lenses in previous surveys and the discovery of one in SDSS together
imply a mass fluctuation normalization \sigma_8=1.0^{+0.4}_{-0.2} (95% CL), if
cluster dark matter halos have an inner slope -1.5. Shallower profiles would
require higher values of \sigma_8. Although the statistical conclusion might be
somewhat dependent on the degree of the complexity of the lens potential, the
discovery is consistent with the predictions of the abundance of cluster-scale
halos in the CDM scenario. (Abridged)
