  In this paper we study a sub-Finsler geometric problem on the free-nilpotent
group of rank 2 and step 3. Such a group is also called Cartan group and has a
natural structure of Carnot group, which we metrize considering the
$\ell_\infty$ norm on its first layer. We adopt the point of view of
time-optimal control theory. We characterize extremal curves via Pontryagin
maximum principle. We describe abnormal and singular arcs, and construct the
bang-bang flow.
