  We present a fully analytically solvable family of models with many-body
cluster interaction and Ising interaction. This family exhibits two phases,
dubbed cluster and Ising phases, respectively. The critical point turns out to
be independent of the cluster size $n+2$ and is reached exactly when both
interactions are equally weighted. For even $n$ we prove that the cluster phase
corresponds to a nematic ordered phase and in the case of odd $n$ to a symmetry
protected topological ordered phase. Though complex, we are able to quantify
the multi-particle entanglement content of neighboring spins. We prove that
there exists no bipartite or, in more detail, no $n+1$-partite entanglement.
This is possible since the non-trivial symmetries of the Hamiltonian restrict
the state space. Indeed, only if the Ising interaction is strong enough (local)
genuine $n+2$-partite entanglement is built up. Due to their analytically
solvableness the $n$-cluster-Ising models serve as a prototype for studying non
trivial-spin orderings and due to their peculiar entanglement properties they
serve as a potential reference system for the performance of quantum
information tasks.
