Division polynomials with galois group su₃(3):2 ≌ g₂(2)

We use a rigidity argument to prove the existence of two related degree 28 covers of the projective plane with Galois group SU₃(3):2 ≌ G₂(2). Constructing corresponding two-parameter polynomials directly from the defining group-theoretic data seems beyond feasibility. Instead we provide two independent constructions of these polynomials, one from 3-division points on covers of the projective line studied by Deligne and Mostow, and one from 2-division points of genus three curves studied by Shioda. We explain how one of the covers also arises as a 2-division polynomial for a family of G₂ motives in the classification of Dettweiler and Reiter. We conclude by specializing our two covers to get interesting three-point covers and number fields which would be hard to construct directly.