  We study supersymmetric $AdS_4\times \Sigma_2$ and $AdS_3\times \Sigma_3$
solutions in half-maximal gauged supergravity in six dimensions with
$SU(2)_R\times SU(2)$ gauge group. The gauged supergravity is obtained by
coupling three vector multiplets to the pure $F(4)$ gauged supergravity. The
$SU(2)_R$ R-symmetry together with the $SO(3)\sim SU(2)$ symmetry of the vector
multiplets are gauged. The resulting gauged supergravity admits supersymmetric
$AdS_6$ critical points with $SO(4)\sim SU(2)\times SU(2)$ and $SO(3)\sim
SU(2)_{\textrm{diag}}$ symmetries. The former corresponds to five-dimensional
$N=2$ superconformal field theories (SCFTs) with $E_1\sim SU(2)$ symmetry. We
find new classes of supersymmetric $AdS_4\times \Sigma_2$ and $AdS_3\times
\Sigma_3$ solutions with $\Sigma_{2,3}$ being $S^{2,3}$ and $H^{2,3}$. These
solutions describe SCFTs in three and two dimensions obtained from twisted
compactifications of the aforementioned five-dimensional SCFTs with different
numbers of unbroken supersymmetry and various types of global symmetries.
