  Let $G$ be a finite non-solvable group. We prove that there exists a proper
subgroup $A$ of $G$ such that $G$ is the product of three conjugates of $A$,
thus replacing an earlier upper bound of $36$ with the smallest possible value.
The proof relies on an equivalent formulation in terms of double cosets, and
uses the following theorem which is of independent interest and wider scope:
Any group $G$ with a $BN$-pair and a finite Weyl group $W$ satisfies $G=\left(
Bn_{0}B\right) ^{2}=BB^{n_{0}}B$ where $n_{0}$ is any preimage of the longest
element of $W$. The proof of the last theorem is formulated in the dioid
consisting of all unions of double cosets of $B$ in $G$. Other results on
minimal length product covers of a group by conjugates of a proper subgroup are
given.
