# What’s the logic behind this labelling of conjugacy classes of groups which seems common?

On the website GroupNames here that lists groups with orders below $$500$$ as well as the ATLAS of finite group representations (and other places I’m sure I’ve stumbled upon and forgotten about), conjugacy classes are often labelled as a number and then sometimes a letter. For example, here GroupNames labels $$S_4$$’s conjugacy classes as $$1$$, $$2A$$, $$2B$$, $$3$$, $$4$$. For a much longer example, here is ATLAS doing such a labelling on the baby monster.

What’s the meaning of the labels here? By the looks of it, it seems like the number corresponds to the largest algebraic degree of the entries of the character table for the column corresponding to that conjugacy class. Is that correct?

Furthermore, what’s the meaning of the letters? It seems the letters correspond to the size of the conjugacy class. For example, the baby monster link I provided shows it has conjugacy classes labelled $$6A, \dots, 6K$$ with $$6A$$ larger than $$6B$$, $$6B$$ larger than $$6C$$ and so on. Is this indeed the case? What if there were two conjugacy classes with the same number and same size - how are the letters chosen? (This can indeed happen, consider $$A_5$$, found here on GroupNames.)

And perhaps the most important question: (why) is such a labelling useful?

• The number is the order of elements in the class. I am not sure what is the convention for the ordering of the letters. Nov 20 at 22:32
• The labeling is useful as a common standard for refering to the conjugacy class.
– Ted
Nov 21 at 3:11
• The classes of elements of the same order are ordered by increasing size of class (or equivalently by decreasing order of centralizer). This does not distinguish between classes of elements of the same order with the same length (an in particular classes that are fused by an outer automorphism of the group). Nov 21 at 6:06