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The 6H-perovskites Ba$_{3}$(R/M)Ru$_{2}$O$_{9}$ (R = rare earth, M = transition metal) exhibit complex magnetism and have been extensively studied recently for their magnetodielectric (MD) properties. Here, we present a detailed study of structural, magnetic, thermodynamic and MD properties of a 6H-perovskite Ba$_{3}$DyRu$_{2}$O$_{9}$. This compound is found to undergo long range antiferromagnetic (AFM) ordering below $\sim$ 5.8 K ({\it{${T_N}$}}), along with the presence of metamagnetic transition at low temperatures. The heat capacity shows two additional anomalies at $\sim$ 28 K ({\it{${T_1}$}}) and $\sim$ 33 K ({\it{${T_2}$}}), besides the anomaly at {\it{${T_N}$}}. Signature of these anomalies is also visible in the derivative of magnetization curve. The dielectric response also shows weak anomalies around {\it{${T_1}$}} and {\it{${T_2}$}} at zero field whereas anomaly at {\it{${T_2}$}} gets suppressed at 80 kOe. The observed MD coupling of $\sim$ 2- 4 {\%} at 80 kOe field below $\sim$ 30 K temperature range is among the highest values observed for the compounds of this family. Low temperature crystal structures of the compound show the sharp distortion of {Ru$_{2}$O$_{9}$} octahedra near {\it{${T_2}$}}. Our study points towards the emergence of structurally driven spin correlations of Ru moments resulting in the observed MD coupling in this compound.
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