Nonassociative triples in involutory loops and in loops of small order

A loop of order $n$ possesses at least $3n^2-3n+1$ associative triples. However, no $n>1$ that achieves this bound seems to be known. If the is involutory, then it $3n^2-2n$ Involutory loops with triples can obtained by prolongation certain maximally nonassociative quasigroups whenever $n-1$ a prime greater than or equal $13$ $n-1=p^{2k}$, $p$ an odd prime. For orders $n\le 9$ minimum number reported for both general and involutory loops, structure corresponding described.