Frobenius partitions in extraspecial groups

Let K be a field and G be the group of the upper unitriangular (n + 2) × (n + 2) K-matrices with nonzero entries only in the first row and in the last column. Then G has a normal subgroup N with a complement H which are K-vector spaces respectively of dimensions n + 1 and n. In the present paper we show that the orbit of H under a group of automorphisms of G together with N, forms a partition of G, provided that there exists a commutative (possibly nonassociative) division algebra of dimension n + 1 over K. This algebra exists when K is a finite field.