The automorphism group of the toroidal queen's graph

Denote the n x n toroidal queen's graph by Q(n)(t) We find its automorphism group Aut(Q(n)(t)) for each positive integer n, showing that for n >= 6, Aut(Q(n)(t)) is generated by the translations, the group of the square, the homotheties, and (for odd n) the automorphism (x, y) -> (y + x, y - x). For each n we find the automorphism classes of edges of Q(n)(t), in particular showing that for n > 1, Q(n)(t) is edge-transitive if and only if n is prime. We find the number of automorphism classes of regular solutions of the toroidal n-queens problem, generalizing work of Burger, Cockayne, and Mynhardt.