# Classifying presentations of finite groups -- the case of dicyclic groups

## TL;DR

This paper classifies presentations of dicyclic groups up to isomorphism, identifying all equivalence classes with two generators and revealing new presentation types that depend on the group's order and parity.

## Contribution

It provides a complete classification of presentations of dicyclic groups with two generators, including new types based on group order and parity, expanding understanding of their structure.

## Key findings

- Finite number of presentation classes for dicyclic groups
- Identified all two-generator presentations including new types
- Results aid in characterizing group structure and properties

## Abstract

The problem of classifying equivalence classes of presentations up to isomorphism of Cayley graphs is considered in this article in the case of dicyclic groups. The number of equivalence classes of presentations is uniformly bounded - it is a "finite presentation type" case. We find all equivalence classes of presentations of dicyclic groups having two generators. For the dicyclic group of order $4n$ apart from the classical presentation with order multiset $\{\{2n,4\}\}$ for all $n$ there are presentations with order multiset $\{\{4,4\}\}$. If $n$ is odd there is an additional presentation having elements with order multiset $\{\{n,4\}\}$. These results may be used in characterizing group structure and properties.