  Every topological group $G$ has up to isomorphism a unique minimal $G$-flow
that maps onto every minimal $G$-flow, the universal minimal flow $M(G).$ We
show that if $G$ has a compact normal subgroup $K$ that acts freely on $M(G)$
and there exists a uniformly continuous cross section $G/K\to G,$ then the
phase space of $M(G)$ is homeomorphic to the product of the phase space of
$M(G/K)$ with $K$. Moreover, if either the left and right uniformities on $G$
coincide or $G\cong G/K\ltimes K$, we also recover the action, in the latter
case extending a result of Kechris and Soki\'c. As an application, we show that
the phase space of $M(G)$ for any totally disconnected locally compact Polish
group $G$ with a normal open compact subgroup is homeomorphic to a finite set,
Cantor set $2^{\mathbb{N}}$, $M(\mathbb{Z})$, or $M(\mathbb{Z})\times
2^{\mathbb{N}}.$
