So if you look here:
http://web.mat.bham.ac.uk/atlas/v2.0/spor/Th/
they provide matrices, $a$ and $b$, which generate the Thompson sporadic group. They also give a representative for each conjugacy class in terms of these generators. However for certain pairs of classes, e.g. 39A/B, they only provide one representative. How is this possible?
More generally, is there an easy way to test which conjugacy class a matrix belongs to? The way I've been doing it is finding the order of the matrix and the trace and checking this against the representation theory of Th (in particular, the 248 dimensional irrep) but this is not enough to determine the class for e.g. 39A/B since the traces and order are the same.
Thank you!
The webpage
http://brauer.maths.qmul.ac.uk/Atlas/v3/spor/Th/
seems to have a bit more information.
I am not certain of this because I haven't really used the Atlas much before, but if we look at the conjugacy classes $39A$, $39B$, it seems like $39A^7$ is $39B$ and $39B^7$ is $39A$, and so once you know a representative for one, then you know a representative for the other. Then at the bottom, it lists the word in $\{a,b\}$ that creates one of them.