# Amusing Permutation Representations of Group Extensions

## TL;DR

This paper explores permutation representations of finite group extensions, especially subgroups of SU(2), providing visual diagrams and injective homomorphisms into semi-direct products to better understand their structure.

## Contribution

It introduces novel permutation diagram representations for various finite subgroups of SU(2), enhancing visualization and understanding of their extensions and quotient structures.

## Key findings

- Permutation diagrams for quaternion and polyhedral groups are developed.
- Quotients as subgroups of permutation groups are clearly identified.
- Injective homomorphisms into semi-direct products are constructed.

## Abstract

Semi-direct products of finite groups have permutation representations that are constructed from the permutation representations of their constituents. One can envision these in a metaphoric sense in which a rope is made from a bundle of threads. In this way, subgroups and quotients are easily visualized. The general idea is applied to the finite subgroups of the special unitary group of $(2\times 2)$-matrices. Amusing diagrams are developed that describe the unit quaternions, the binary tetrahedral, octahedral, and icosahedral group as well as the dicyclic groups. In all cases, the quotients as subgroups of the permutation group are readily apparent. These permutation representations lead to injective homomorphisms into semi-direct products.