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Quaternionic Representation of Snub 24-Cell and its Dual Polytope Derived From E_8 Root System

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.