The automorphism group of the toroidal queen's graph

Denote the n × n toroidal queen’s graph by Qn. We find its automorphism group Aut(Qn) for each positive integer n, showing that for n ≥ 6, Aut(Qn) 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 Qn, in particular showing that for n > 1, Qn 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.