  We consider 3XOR games with perfect commuting operator strategies. Given any
3XOR game, we show existence of a perfect commuting operator strategy for the
game can be decided in polynomial time. Previously this problem was not known
to be decidable. Our proof leads to a construction, showing a 3XOR game has a
perfect commuting operator strategy iff it has a perfect tensor product
strategy using a 3 qubit (8 dimensional) GHZ state. This shows that for perfect
3XOR games the advantage of a quantum strategy over a classical strategy
(defined by the quantum-classical bias ratio) is bounded. This is in contrast
to the general 3XOR case where the optimal quantum strategies can require high
dimensional states and there is no bound on the quantum advantage.
  To prove these results, we first show equivalence between deciding the value
of an XOR game and solving an instance of the subgroup membership problem on a
class of right angled Coxeter groups. We then show, in a proof that consumes
most of this paper, that the instances of this problem corresponding to 3XOR
games can be solved in polynomial time.
