# A Database of Groups with Equivalent Character Tables

## TL;DR

This paper presents a comprehensive database of finite groups classified by their character tables, utilizing high-throughput computing and graph isomorphism algorithms to efficiently identify groups with identical character tables.

## Contribution

The authors created a large database of groups with equivalent character tables for all groups under 2000 order, excluding order 1024, using novel algorithmic and computational techniques.

## Key findings

- Successfully classified over 408 million groups of order 1536 by character table.
- Developed a new algorithmic approach using graph isomorphism for character table comparison.
- Demonstrated the feasibility of large-scale classification of finite groups based on character tables.

## Abstract

Two groups are said to have the same character table if a permutation of the rows and a permutation of the columns of one table produces the other table. The problem of determining when two groups have the same character table is computationally intriguing. We have constructed a database containing for all finite groups of order less than 2000 (excluding those of order 1024), a partitioning of groups into classes having the same character table. To handle the 408,641,062 groups of order 1536 and other orders with a large number of groups we utilized high-throughput computing together with a new algorithmic approach to the problem. Our approach involved using graph isomorphism software to construct canoncial graphs that correspond to the character table of a group and then hashing the graphs.