  We study $P(n,3)$, the size of the largest subset of the set of all
permutations $S_n$ with minimum Kendall $\tau$-distance $3$. Using a
combination of group theory and integer programming, we reduced the upper bound
of $P(p,3)$ from $(p-1)!-1$ to $(p-1)!-\lceil\frac{p}{3}\rceil+2\leq (p-1)!-2$
for all primes $p\geq 11$. In special cases where $n$ is equal to
$6,7,11,13,14,15$ and $17$ we reduced the upper bound of $P(n,3)$ by
$3,3,9,11,1,1$ and $4$, respectively.
