Groups whose orders factorise into at most four primes

TL;DR

This paper provides a new explicit method and algorithms for classifying and constructing all groups of order n where n's prime factorization involves at most four primes, with implementations in GAP.

Contribution

It introduces a novel explicit classification and effective algorithms for groups with orders factorizing into up to four primes, enhancing previous descriptions.

Findings

Explicit classification of such groups is achieved.

Algorithms for enumeration, construction, and identification are developed.

Implementation in GAP demonstrates practical applicability.

Abstract

The groups whose orders factorise into at most four primes have been described (up to isomorphism) in various papers. Given such an order n, this paper exhibits a new explicit and compact determination of the isomorphism types of the groups of order n together with effective algorithms to enumerate, construct, and identify these groups. The algorithms are implemented for the computer algebra system GAP.