On difference sets in groups of order 4p2

If p is a prime greater than or equal to 5, and G is a group of order 4p2 containing a (Menon type) difference set, then either G has an irreducible complex representation of degree 4, or G ≌ 〈x, y, z | xp = yp = z4 = 1, xy = yx, xz = zx, zyz−1 = y−1〉 (mod4). The proof involves representation theory, algebraic number theory, and a generalization of Fourierʹs inversion formula. The six remaining isomorphism classes are considered in part II.