  In this paper we characterize the geodesic dimension $N_{GEO}$ and give a new
lower bound to the curvature exponent $N_{CE}$ on Sard-regular Carnot groups.
As an application, we give an example of step-two Carnot group on which $N_{CE}
> N_{GEO}$: this answers a question posed by Rizzi in arXiv:1510.05960v4
[math.MG].
