  The reduction of ten-dimensional heterotic supergravity with Yang-Mills
symmetry group K is performed on an arbitrary n-dimensional group manifold G.
The reduction involves a nonvanishing 3-form flux, and the Lie algebra of G
must have traceless structure constants to ensure the consistency of the
reduction at the level of the action. A large class of gauged supergravities in
d=10-n with (non)compact gaugings is obtained. The resulting models describe
half-maximal gauged supergravities coupled to $ (n + {\rm dim} K)$ vector
multiplets. We uncover their hidden $SO(n,n+{\rm dim} K)$ duality symmetry, and
the $SO(n,n+{\rm dim} K) / SO(n)\times SO(n+{\rm dim} K)$ coset structure that
governs the couplings of the scalar fields. We find that the local gauge
symmetry of the d-dimensional theory is $K\times G \ltimes R^n$. Differences
from the existing gauged supergravities are highlighted. The consistent
truncation to pure half-maximal gauged supergravity in any dimension is shown,
and the obstacle to performing a chiral truncation of the theory in d=6
dimensions is found. Among the results obtained are the complete
diagonalisation of the fermionic kinetic terms, and other reduction formulae
that are applicable to group reductions of supergravities in arbitrary
dimensions.
