  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.
