Every topological group $G$ admits a universal minimal flow, $M(G),$ that is, a minimal flow that factors onto any minimal flow, that is unique up to isomorphism. Thus understanding $M(G)$ sheds light on how complicated minimal dynamics of $G$ can be. In the case of infinitely countable discrete groups, the underlying space of $M(G)$ is always the Gleason space of the Cantor cube of weight continuum, $\text{Gl}(2^{\mathfrak{c}})$, that can also be seen as the Stone space of the free completion of the free Boolean algebra on continuum many generators. Totally disconnected locally compact (TDLC) Polish non-discrete groups, in other words locally compact automorphism groups of countable structures, are topologically homeomorphic to the product of a countable discrete set and the Cantor space. In case a group $G$ is also algebraically isomorphic to a product of an infinitely countable discrete group and the Cantor group, $D\times 2^{\omega}$, then the underlying space of $M(G)$ is homeomorphic to the product $\text{Gl}(2^{\mathfrak{c}})\times 2^{\omega}$. The question is whether that is always the case. We show that the answer is positive in various scenarios, covering for instance the example of automorphism groups of finitely-branching regular countable tree. This is a joint work in progress with Andy Zucker.