  The Triplet extension of the MSSM (TMSSM) alleviates the little hierarchy
problem and provides a significant enhancement of the loop-induced diphoton
rate of the lightest CP-even Higgs h. In this paper we pursue the analysis of
the TMSSM Higgs phenomenology by computing for the first time the h into Z +
gamma decay. Interestingly we find that the rates of loop-induced decays are
correlated and their signal strengths can rise up to 40% - 60% depending on the
channel. We furthermore study the dark matter phenomenology of the TMSSM. The
lightest neutralino is a good dark matter candidate in two regions. The first
one is related to the Higgs and Z resonances and the LSP is mostly Bino. The
second one is achieved for a mass larger than 90 GeV and the LSP behaves as the
well-tempered neutralino. An advantage of the triplet contribution is that the
well-tempered neutralino can be a Bino-Triplino mixture, relieving the problem
of achieving M_2 ~ M_1 in unified scenarios. The dark matter constraints
strongly affect the Higgs phenomenology, reducing the potential enhancements of
the diphoton and of the Z + photon channels by 20% at most. These enhancements
are however larger than the MSSM ones. In the near future, complementarity of
dark matter direct searches and collider experiments will be crucial to probe
most of the parameter space where the neutralino is the dark matter candidate.
