We illustrate the relationship between geometric properties of mappings between stratified groups and a system of first order nonlinear PDEs, namely, the “contact equations”. We show how these equations characterize continuously Pansu differentiable mappings. Then we present an implicit function theorem and a rank theorem for this class of mappings. This allows of defining intrinsically regular subsets modeled on groups. When contact equations are a.e. satisfied they can be used to characterize existence of Lipschitz extensions for mappings of stratified groups. As an application, we show how Allcock’s quadratic isoperimetric inequality in higher dimensional Heisenberg groups leads to a Lipschitz extension result for mappings defined on subsets of the Euclidean plane that take values in higher dimensional Heisenberg groups. In the three dimensional Heisenberg group, the same equations lead to nonexistence of Legendrian submanifolds with suitably “low regularity”.