  We study four-dimensional $N=4$ gauged supergravity coupled to six vector
multiplets with semisimple gauge groups $SO(4)\times SO(4)$, $SO(3,1)\times
SO(3,1)$ and $SO(4)\times SO(3,1)$. All of these gauge groups are dyonically
embedded in the global symmetry group $SO(6,6)$ via its maximal subgroup
$SO(3,3)\times SO(3,3)$. For $SO(4)\times SO(4)$ gauge group, there are four
$N=4$ supersymmetric $AdS_4$ vacua with $SO(4)\times SO(4)$, $SO(4)\times
SO(3)$, $SO(3)\times SO(4)$ and $SO(3)\times SO(3)$ symmetries, respectively.
These $AdS_4$ vacua correspond to $N=4$ SCFTs in three dimensions with $SO(4)$
R-symmetry and different flavor symmetries. We explicitly compute the full
scalar mass spectra at all these vacua. Holographic RG flows interpolating
between these conformal fixed points are also given. The solutions describe
supersymmetric deformations of $N=4$ SCFTs by relevant operators of dimensions
$\Delta=1,2$. A number of these solutions can be found analytically although
some of them can only be obtained numerically. These results provide a rich and
novel class of $N=4$ fixed points in three-dimensional Chern-Simons-Matter
theories and possible RG flows between them in the framework of $N=4$ gauged
supergravity in four dimensions. Similar studies are carried out for
non-compact gauge groups, but the $SO(4)\times SO(4)$ gauge group exhibits a
much richer structure.

High Energy Physics - Theory