Visualizing conjugacy classes

[balopat]

I'm a big fan of visualizing mathematics, and just ran into the group explorer which I find very cool.

Roughly two and a half years ago I created a prototype for visualizing conjugacy classes in groups using Mathematica after my first graduate abstract algebra class: https://refactorium.com/blog/2021/10/20/visualizing-commutatitivity/

It's close to a multiplication table, but instead of the result of the multiplication of the two group elements, cells can have one of two colors depending on the commutativity between the two elements. Elements are also grouped by their conjugacy classes, represented by groups of rows (columns) between yellow highlighted lines. This way single row (column) conjugacy classes can be easily seen to be the center of the group (maybe we could organize the conjugacy classes by their size? then the center would be together).

Here's an example for the Pauli group:

If there is interest, I could put aside some time in the coming months and contribute it to the project.