Polyhedral Groups in $G_2(\mathbb{C})$

TL;DR

This paper classifies how certain finite groups, specifically $A_4$, $S_4$, and $A_5$, can be embedded into the complex Lie group $G_2( ext{C})$, providing a detailed understanding of their conjugacy classes.

Contribution

It provides a complete classification of embeddings of $A_4$, $S_4$, and $A_5$ into $G_2( ext{C})$ up to conjugation, which was previously unknown.

Findings

Identified all conjugacy classes of embeddings for $A_4$, $S_4$, and $A_5$ in $G_2( ext{C})$

Established the structure and properties of these embeddings

Enhanced understanding of finite subgroup structures within complex Lie groups.

Abstract

We classify embeddings of the finite groups $A_4$, $S_4$ and $A_5$ in the Lie group $G_2(\mathbb{C})$ up to conjugation.