ON p-GROUPS with AUTOMORPHISM GROUPS RELATED to the CHEVALLEY GROUP G₂(p)

Let p be an odd prime. We construct a p -group P of nilpotency class two, rank seven and exponent p , such that Aut(P) induces NGL(7,p)(G2(p))=Z(GL(7,p))G2(p) on the Frattini quotient P/Φ(P) . The constructed group P is the smallest p -group with these properties, having order p14 , and when p=3 our construction gives two nonisomorphic p -groups. To show that P satisfies the specified properties, we study the action of G2(q) on the octonion algebra over Fq , for each power q of p , and explore the reducibility of the exterior square of each irreducible seven-dimensional Fq[G2(q)] -module.