  We establish the mathematical fundamentals for a unified description of
curvature, torsion, and non-metricity 2-forms in the way extending the
so-called M\"{o}bius representation of the affine group, which is the method to
convert the semi-direct product into the ordinary matrix product, to revive the
fertility of gauge theories of gravity. First of all, we illustrate the basic
concepts for constructing the metric-affine geometry. Then the curvature and
torsion 2-forms are described in a unified manner by using the Cartan
connection of the M\"{o}bius representation of the affine group. In this
unified-description, the curvature and torsion are derived by Cartan's
structure equation with respect to a common connection 1-form. After that,
extending the M\"{o}bius representation, the dilation and shear 2-forms, or
equivalently, the non-metricity 2-form, are introduced in the same unified
manner. Based on the unified-description established in this paper, introducing
a new group parametrization and applying the In\"{o}n\"{u}-Wigner group
contraction to the full theory, the relationships among symmetries, geometric
quantities, and geometries are investigated with respect to the three gauge
groups: the metric-affine group and its extension, and an extension of the
(anti)-de Sitter group in which the non-metricity exists. Finally, possible
applications to theories of gravity are briefly discussed.
