The band-touching points of stable, three-dimensional, Kramers-degenerate, Dirac semimetals are singularities of a five-component, unit vector field and non-Abelian, $SO(5)$-Berrys connections, whose topological classification is an important, open problem. We solve this problem by performing second homotopy classification of Berrys 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 Berrys connections.
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