  We employ first-principles density-functional theory, to perform a
comparative investigation of the effect of the spin-orbit coupling (SOC) on the
electronic and magnetic properties of three experimentally synthesized and
characterized hexagonal perovskites Sr$_3$MIrO$_6$(M=Mg, Zn, Cd). The
electronic structure calculations show that in all the compounds, Ir is the
only magnetically active site in +4[5$d^5$] configuration, whereas M$^{+2}$
(M=Cd, Zn, Mg), remains in nonmagnetic states with Cd/Zn and Mg featuring
$d^{10}$ and $d^{0}$ electronic configurations, respectively. The insulating
gap could be opened by switching on the correlation parameter $U$ for
Sr$_3$CdrO$_6$ and Sr$_3$ZnIrO$_6$ which qualifies it to be a correlated Mott
insulator. However, in the case of Sr$_3$MgIrO$_6$ both $U$ and
antiferromagnetic ordering is not enough and the gap could only be opened by
including the SOC which classifies it to fall under the category of a typical
SOC Mott insulator. The $j_{eff}$ states are visualized from the orbital
projected band structure. The magnetism is studied from the point of view of
exchange interactions and magnetocrystalline anisotropy in the presence of the
SOC. We also present the comparative analysis of the renormalized impact of SOC
on the three compounds, which shows that all the three compounds fall under the
$intermediate$ coupling regime, where Sr$_3$MgIrO$_6$ is comparatively closer
to the atomic $j_{eff}=\frac{1}{2}$ picture from the others.
