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Density functional theory (DFT) approaches have been ubiquitously used to predict topological order and non-trivial band crossings in real materials, like Dirac, Weyl semimetals and so on. However, use of less accurate exchange-correlation functional often yields false prediction of non-trivial band order leading to misguide the experimental judgment about such materials. Using relatively more accurate hybrid functional exchange-correlation, we explore a set of (already) experimentally synthesized materials (crystallizing in space group P6_3/mmc) Our calculations based on more accurate functional helps to correct various previous predictions for this material class. Based on point group symmetry analysis and ab-initio calculations, we systematically show how lattice symmetry breaking via alloy engineering manifests different fermionic behavior, namely Dirac, triple point and Weyl in a single material. Out of various compounds, XAgBi (X=Ba,Sr) turn out to be two ideal candidates, in which the topological nodal point lie very close to the Fermi level, within minimal/no extra Fermi pocket. We further studied the surface states and Fermi arc topology on the surface of Dirac, triple point and Weyl semimetallic phases of BaAgBi. We firmly believe that, while the crystal symmetry is essential to protect the band crossings, the use of accurate exchange correlation functional in any DFT calculation is an important necessity for correct prediction of band order which can be trusted and explored in future experiments.