  By exploiting the permutation symmetry of Dick states, we derive closed
analytical expressions of Schmidt decompositions for {\it all} possible
bipartitions of a system described by this kind of state. This allows us to
exhaustively compute the entropy of entanglement of the bipartitions and, thus,
compare the their entanglement extent. We also address the multipartite
character of Dicke states by calculating the purity of balanced bipartitions to
determine the potential of multipartite entanglement (the average purity). In
particular, we found that the entanglement of $W$ states remains constant as
the number of qubits is increased. As a final application we define a family of
multipartite entanglement witnesses and compute their resistance against random
and systematic imperfections. It is shown that in some circumstances, for a
fixed white noise fraction, the entanglement becomes detectable only if one
{\it increases} the amount of systematic imperfection in the state.
