The electromagnetic form factors of the spin-3/2 $Omega$ baryons, namely $Omega$, $Omega_c^ast$, $Omega_{cc}^ast$ and $Omega_{ccc}$, are calculated in full QCD on $32^3times 64$ PACS-CS lattices with a pion mass of 156(9) MeV. The electric charge radii and magnetic moments from the $E0$ and $M1$ multipole form factors are extracted. Results for the electric quadrupole form factors, $E2$, are also given. Quark sector contributions are computed individually for each observable and then combined to obtain the baryon properties. We find that the charm quark contributions are systematically smaller than the strange-quark contributions in the case of the charge radii and magnetic moments. $E2$ moments of the $Omega_{cc}^ast$ and $Omega_{ccc}$ provide a statistically significant data to conclude that their electric charge distributions are deformed to an oblate shape. Properties of the spin-1/2 $Omega_c$ and $Omega_{cc}$ baryons are also computed and a thorough comparison is given. This complete study gives valuable hints about the heavy-quark dynamics in charmed hadrons.
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