  The band-touching points of stable, three-dimensional, Kramers-degenerate,
Dirac semimetals are singularities of a five-component, unit vector field and
non-Abelian, $SO(5)$-Berry's connections, whose topological classification is
an important, open problem. We solve this problem by performing second homotopy
classification of Berry's connections. Using Abelian projected connections, the
generic planes, orthogonal to the direction of nodal separation, and lying
between two Dirac points are shown to be higher-order topological insulators,
which support quantized, chromo-magnetic flux or relative Chern number, and
gapped, edge states. The Dirac points are identified as a pair of
unit-strength, $SO(5)$- monopole and anti-monopole, where the relative Chern
number jumps by $\pm 1$. Using these bulk invariants, we determine the
topological universality class of different types of Dirac semimetals. We also
describe a universal recipe for computing quantized, non-Abelian flux for Dirac
materials from the windings of spectra of planar Wilson loops, displaying
$SO(5)$-gauge invariance. With non-perturbative, analytical solutions of
surface-states, we show the absence of helical Fermi arcs, and predict the
fermiology and the spin-orbital textures. We also discuss the similarities and
important topological distinction between the surface-states Hamiltonian and
the generator of Polyakov loop of Berry's connections.
