  We investigate the notion of $k$-transitivity for the quantum permutation
groups $G\subset S_N^+$, with a brief review of the known $k=1,2$ results, and
with a study of what happens at $k\geq3$. We discuss then matrix modelling
questions for the algebras $C(G)$, notably by introducing the related notions
of double and triple flat matrix model. At the level of the examples, our main
results concern the quantum groups coming from the complex Hadamard matrices,
and from the Weyl matrices.
