Examples in which spacetime might become non-Riemannian above Planck energies occur in string theory or in the very early universe in the inflationary model. The simplest such geometry is metric-affine geometry, in which nonmetricity appears as a field strength, side by side with curvature and torsion. In this paper we obtain a solitonic solution of the metric-affine gauge theory of gravity, which is endowed with shear and dilational Weyl charges.