  We consider both facial reduction, \FRp, and symmetry reduction, \SRp,
techniques for semidefinite programming, \SDPp. We show that the two together
fit surprisingly well in an alternating direction method of multipliers,
\ADMMp, approach. In fact, this approach allows for simply adding on
nonnegativity constraints, and solving the doubly nonnegative, \DNN, relaxation
of many classes of hard combinatorial problems. We also show that the
singularity degree remains the same after \SRp, and that the \DNN relaxations
considered here have singularity degree one, that is reduced to zero after
\FRp. The combination of \FR and \SR leads to a significant improvement in both
numerical stability and running time for both the \ADMM and interior point
approaches.
  We test our method on various \DNN relaxations of hard combinatorial problems
including quadratic assignment problems with sizes of more than $n=500$. This
translates to a semidefinite constraint of order $250,000$ and $625\times 10^8$
nonnegative constrained variables, before applying the reduction techniques.
