  We study symmetries of specific left--invariant sub--Riemannian structure
with filtration $(4,7)$ and their impact on sub--Riemannian geodesics of
corresponding control problem. We show that there are two very different types
of geodesics, they either do not intersect the fixed point set of symmetries or
are contained in this set for all times. We use the symmetry reduction to study
properties of geodesics.
