  We prove that, for every compact spaces $K_1,K_2$ and a compact group $G$, if
both $K_1$ and $K_2$ map continuously onto $G$, then the Banach space
$C(K_1\times K_2)$ contains a complemented subspace isometric to $C(G)$.
Consequently, answering a question of Alspach and Galego, we get that
$C(\beta\omega\times\beta\omega)$ contains a complemented isomorphic copy of
$C([0,1]^\kappa$) for every cardinal number $1\le\kappa\le {\mathfrak c}$ and
hence a complemented copy of $C(K)$ for every metric compact space $K$.
