  A class of $AdS_2\times \Sigma_2$, with $\Sigma_2$ being a two-sphere or a
hyperbolic space, solutions within four-dimensional $N=4$ gauged supergravity
coupled to three-vector multiplets with dyonic gauging is identified. The
gauged supergravity has non-semisimple $SO(3)\ltimes
(\mathbf{T}^3,\hat{\mathbf{T}}^3)$ gauge group and can be obtained from a
consistent truncation of eleven-dimensional supergravity on a tri-sasakian
manifold. The maximally symmetric vacua contain $AdS_4$ geometries with $N=1,3$
supersymmetry corresponding to $N=1$ and $N=3$ superconformal field theories
(SCFTs) in three dimensions. We find supersymmetric solutions of the form
$AdS_2\times \Sigma_2$ preserving two supercharges. These solutions describe
twisted compactifications of the dual $N=1$ and $N=3$ SCFTs and should arise as
near horizon geometries of dyonic black holes in asymptotically $AdS_4$
space-time. Most solutions have hyperbolic horizons although some of them
exhibit spherical horizons. These provide a new class of $AdS_2\times \Sigma_2$
geometries with known M-theory origin.
