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The presence of massive gravitons in the field of massive gravity is considered as an important factor in investigating the structure of compact objects. Hence, we are encouraged to study the dark energy star structure in the Vegh's massive gravity. We consider that the equation of state governing the inner spacetime of the star is the extended Chaplygin gas, and then using this equation of state, we numerically solve the Tolman-Oppenheimer-Volkoff (TOV) equation in massive gravity. In the following, assuming different values of free parameters defined in massive gravity, we calculate the properties of dark energy star such as radial pressure, transverse pressure, anisotropy parameter, and other characteristics. Then, after obtaining the maximum mass and its corresponding radius, we compute redshift and compactness. The obtained results show that for this model of dark energy star, the maximum mass and its corresponding radius depend on the massive gravity's free parameters and anisotropy parameter. These results are consistent with the observational data, and cover the lower mass gap. We also demonstrate that all energy conditions are satisfied for this model, and in the presence of anisotropy, the dark energy star is potentially unstable.

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