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Vertices of the 4-dimensional semi-regular polytope,  \textit{snub 24-cell} and its symmetry group $W(D_{4}):C_{3} $ of order 576are represented in terms of quaternions with unit norm. It follows from theicosian representation of \textbf{$E_{8} $} root system. The quaternionic rootsystem of $H_{4} $ splits as the vertices of 24-cell and the \textit{snub24-cell} under the symmetry group of the \textit{snub 24-cell} which is one ofthe 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 ofthe 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 analyzethe 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 structureof the snub 24-cell has been explicitly analyzed with quaternions by using thesubgroups of the group $W(D_{4}):C_{3} $. In particular, it has been shown thatthe dual polytopes 600-cell with 120 vertices and 120-cell with 600 verticesdecompose as 120=24+96 and 600=24+96+192+288 respectively under the group$W(D_{4}):C_{3} $. The dual polytope of the \textit{snub 24-cell} is explicitlyconstructed. Decompositions of the Archimedean $W(H_{4})$ polytopes under thesymmetry of the group $W(D_{4}):C_{3} $ are given in the appendix.

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