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 from
$G/K$ 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 is isomorphic to a semidirect product
$G/K\ltimes K$, we also recover the action, in the latter case extending a result of Kechris and Sokić. 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, the Cantor set
$2^{\mathbb {N}}$,
$M(\mathbb {Z})$, or
$M(\mathbb {Z})\times 2^{\mathbb {N}}.$