I know that elements of order 3 in A4 group split into two separate conjugacy classes, due to lack of odd elements to form a unique class like happens in the full S4.
Since A4 is the rotation symmetry group of the Tetrahedron, I was wondering if these two split conjugacy classes have a direct geometrical meaning. While I can see a direct meaning of transpositions, clearly distinguishing them from 3-cycles, I cannot see any apparent related split of them from a geometrical perspective.
Nevertheless, my feeling is this binding makes sense, hiding somewhere from my view.
Would you point me to the right direction, please?
Thanks in advance
For me, it is helpful to see the conjugacy classes of $A_4$ listed out:
Now, let there be a tetrahedron sitting on a desk. Label the three vertices on the desk $1,2,3$, going clockwise, and label the vertex at the top as $4$. Now, if you rotate the tetrahedron $120^\circ$ degrees clockwise, that would be like the cycle $(123)$. Rotate it counterclockwise, however, and that would be like the cycle $(132)$.
Now, turn the tetrahedron so that the vertex $1$ is now on top. You will now notice that, in clockwise order, the vertices on the desk are $2,4,3$. Thus, rotate the tetrahedron $120^\circ$ degrees clockwise, and that would be like the cycle $(243)$. Rotate it counterclockwise, and that would be like the cycle $(234)$.
Again, turn the tetrahedron so that the vertex $2$ is now on top. In clockwise order, the vertices on the desk are now $1,3,4$. Thus, rotate the tetrahedron $120^\circ$ degrees clockwise, and that would be like the cycle $(134)$. Rotate it counterclockwise, and that would be like the cycle $(143)$.
Hopefully, you can guess what will happen if you turn the tetrahedron so that the vertex $3$ is on top. Anyway, I hope you can now see how the conjugacy class with $(123)$ corresponds to clockwise rotations when looking from above while the conjugacy class with $(132)$ corresponds to counter-clockwise rotations when looking from above.