  We study models in which the Higgs sector is extended by a single scalar
electroweak multiplet Z with isospin T=5/2 (sextet) or 7/2 (octet) and the same
hypercharge as the Standard Model Higgs doublet, in which Z is odd under a
global Z2 symmetry. This Z2 symmetry keeps the lightest (neutral) member of Z
stable and has interesting implications for phenomenology. We determine the
constraints on these models from precision electroweak measurements and Higgs
boson decays to two photons. We compute the thermal relic density of the stable
member of Z and show that, for masses below 1 TeV, it can make up at most 1% of
the dark matter in the universe. We also show that current dark matter direct
detection experiments do not constrain the models, but future ton-scale
experiments will probe their parameter space.
