Powerful 2-Engel groups II

We conclude our classification of powerful 2-Engel groups of class three that are minimal in the sense that every proper powerful section is nilpotent of class at most two. In the predecessor to this paper we obtained three families of minimal groups. Here we get a fourth family of minimal examples that is described in terms of irreducible polynomials over the field of three elements. We also get one isolated minimal example of rank 5 and exponent 27. The last one has a related algebraic structure that we call a “symplectic alternating algebra.” To each symplectic alternating algebra over the field of three elements there corresponds a unique 2-Engel group of exponent 27.