Let G=PSp₄(q), q=p^k odd. We show that the geometry of root subgroups of G is the tangent envelope of a system of conics that comprise the (q,q)generalized quadrangle associated with G. The flags of this geometry form a rank 3 chamber complex in the sense of Tits [9], as one would expect from the theory of symmetric spaces for Lie groups. By way of application, we give an intrinsic interpretation of symplectic 2-transvections. We then show that the subgroup generated by a pair of shortroot subgroups not contained in a pSylow is determined by the geometry. In particular, we describe the incidence conditions under which such pairs are contained in the maximal subgroups of G corresponding to the pluspoint and minuspoint stabilizers in the orthogonal construction of G ([3], xii).