Vertices of the 4-dimensional semi-regular polytope, \textit{ snub 24-cell} and its symmetry group $W(D_{ 4}):C_{ 3} $ of order 576 are represented in terms of quaternions with unit norm. It follows from the icosian representation of \textbf{ $E_{ 8} $} root system. The quaternionic root system of $H_{ 4} $ splits as the vertices of 24-cell and the \textit{ snub 24-cell} under the symmetry group of the \textit{ snub 24-cell} which is one of the maximal subgroups of the group \textbf{ $W(H_{ 4})$} as well as $W(F_{ 4})$. It is noted that the group is isomorphic to the\textbf{ }semi-direct product of the Weyl group of $D_{ 4}$ with the cyclic group of order 3 denoted by $W(D_{ 4}):C_{ 3} $, the Coxeter notation for which is $[3,4,3^{ +}]$. We analyze the vertex structure of the \textit{ snub 24-cell} and decompose the orbits of \textbf{ $W(H_{ 4})$} under the orbits of $W(D_{ 4}):C_{ 3} $. The cell structure of the snub 24-cell has been explicitly analyzed with quaternions by using the subgroups of the group $W(D_{ 4}):C_{ 3} $. In particular, it has been shown that the dual polytopes 600-cell with 120 vertices and 120-cell with 600 vertices decompose as 120=24+96 and 600=24+96+192+288 respectively under the group $W(D_{ 4}):C_{ 3} $. The dual polytope of the \textit{ snub 24-cell} is explicitly constructed. Decompositions of the Archimedean $W(H_{ 4})$ polytopes under the symmetry of the group $W(D_{ 4}):C_{ 3} $ are given in the appendix.
Categories: math-ph math.MP