The Isometry Groups of $\mathbb{R}_{DH}^{3},$ $\mathbb{R}_{PD}^{3}$ and $\mathbb{R}_{TI}^{3}$

There are two aims of this paper. First one, we want to give a detailed exposition of basic properties of deltoidal hexacontahedron, pentakis dodecahedron and triakis icosahedron which are Catalan solids. Also, we construct the spaces by covering related metrics. The spheres of these spaces are deltoidal hexacontahedron, pentakis dodecahedron and triakis icosahedron. Second one is to find the isometry group of these solids. In fact, the main aim of this paper is the second one. We show that the group of isometries of the spaces covering with deltoidal hexacontahedron, pentakis dodecahedron, and triakis icosahedron metrics is the semi-direct product of the icosahedral group $I_{h}$ and $T(3)$, where $I_{h}$ is the (Euclidean) symmetry group of the icosahedron and $T(3)$ is the group of all translations of the 3-dimensional space.