We present comprehensive characterization of the Galactic open cluster M 36. Some two hundred member candidates, with an estimated contamination rate of $sim$8%, have been identified on the basis of proper motion and parallax measured by the $Gaia$ DR2. The cluster has a proper motion grouping around ($mu_{alpha} cosdelta = -$0.15 $pm$ 0.01 mas yr$^{-1}$, and $mu_{delta} = -$3.35 $pm$ 0.02 mas yr$^{-1}$), distinctly separated from the field population. Most member candidates have parallax values 0.7$-$0.9 mas, with a median value of 0.82 $pm$ 0.07 mas (distance $sim$1.20 $pm$ 0.13 kpc). The angular diameter of 27$$ $pm$ $0farcm4$ determined from the radial density profile then corresponds to a linear extent of 9.42 $pm$ 0.14 pc. With an estimated age of $sim$15 Myr, M 36 is free of nebulosity. To the south-west of the cluster, we discover a highly obscured ($A_{V}$ up to $sim$23 mag), compact ($sim$ $1farcm9 times 1farcm2$) dense cloud, within which three young stellar objects in their infancy (ages $lesssim$ 0.2 Myr) are identified. The molecular gas, 3.6 pc in extent, contains a total mass of (2$-$3)$times$10$^{2}$ M$_{odot}$, and has a uniform velocity continuity across the cloud, with a velocity range of $-$20 to $-$22 km s$^{-1}$, consistent with the radial velocities of known star members. In addition, the cloud has a derived kinematic distance marginally in agreement with that of the star cluster. If physical association between M 36 and the young stellar population can be unambiguously established, this manifests a convincing example of prolonged star formation activity spanning up to tens of Myrs in molecular clouds.
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