# Determine the class equation of the tetrahedral group

This is a question from Artin's algebra textbook.

The tetrahedral group of rotations has 1 element of order 1, 8 elements of order 3 (rotations of $$120^°$$ around a vertex), and 3 elements of order 2 (rotations of $$180^o$$ around the axis through the midpoints of opposite edges).

Is there a way to find the class equation without using brute force on $$A_4$$?

The conjugacy classes in $$S_n$$ are easy to describe. And you can prove that each conjugacy class in $$S_n$$ which consists of even permutations (i.e., which is contained in $$A_n$$) will either stay a conjugacy class in $$A_n$$ or split in half (see here). For $$A_4$$, there's only one conjugacy class (of size 8) that even has the potential to split in half. You can just check if that happens.
• Your wording is confusing, because $S_n$ has even and odd permutations – J. W. Tanner Feb 11 at 21:25
• Thanks -- I guess the parentheses don't do what I wanted them to do. I mean the ones which consist of even permutations, i.e., the only ones which are actually in $A_n$. I'll make an edit. – csprun Feb 11 at 22:23
The class equation of $$A_4$$ is well known to be $$12=1+4+4+3$$, since the cycle structure is preserved by conjugation. In this case, the class of $$3$$-cycles of size $$8$$ splits.
The subgroup of the tetrahedral group you refer to is also well known to be $$A_4$$.