Complete Isocategorical Classification of Groups of Order 64 via GAP

TL;DR

This paper presents a complete classification of groups of order 64 under monoidal equivalence, utilizing a novel computational approach with GAP to identify exactly two pairs of non-isomorphic isocategorical groups.

Contribution

The authors develop an original computational method with GAP to fully classify groups of order 64 under monoidal equivalence, filling a gap in existing classifications.

Findings

Exactly two pairs of non-isomorphic isocategorical groups of order 64 identified.

Methodology extends classification techniques to order 64 groups.

Provides a computational framework for future classifications in quantum group theory.

Abstract

The classification of finite groups under monoidal equivalence is a fundamental topic in the study of finite quantum groups. While a complete classification has been established for all groups of order strictly less than 64, the case for order 64 has remained limited to the construction of specific examples. In this study, we achieve the complete classification for groups of order 64 by developing an original computational approach using GAP. We describe our methodology and demonstrate that there exist exactly two pairs of non-isomorphic isocategorical groups of this order.