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be an odd prime. We construct a
of nilpotency class two, rank seven and exponent
, such that
on the Frattini quotient
. The constructed group
is the smallest
-group with these properties, having order
, and when
our construction gives two nonisomorphic
-groups. To show that
satisfies the specified properties, we study the action of
on the octonion algebra over
, for each power
, and explore the reducibility of the exterior square of each irreducible seven-dimensional
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