We study the ground state properties, potential energy curves and potential energy surfaces of the superheavy nucleus $^{270}$Hs by using the multidimensionally-constrained relativistic mean-field model with the effective interaction PC-PK1. The binding energy, size and shape as well as single particle shell structure corresponding to the ground state of this nucleus are obtained. $^{270}$Hs is well deformed and exhibits deformed doubly magic feature in the single neutron and proton level schemes. One-dimensional potential energy curves and two-dimensional potential energy surfaces are calculated for $^{270}$Hs with various spatial symmetries imposed. We investigate in detail the effects of the reflection asymmetric and triaxial distortions on the fission barrier and fission path of $^{270}$Hs. When the axial symmetry is imposed, the reflection symmetric and reflection asymmetric fission barriers both show a double-hump structure and the former is higher. However, when triaxial shapes are allowed the reflection symmetric barrier is lowered very much and then the reflection symmetric fission path becomes favorable.
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