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Of course we can compute conjugacy classes by first converting to a permutation group, computing it there, and converting back (and this might very well be a first implementation).
But in practice one ideally wants specific representatives (not random ones), and perhaps even a certain order of conjugacy classes, and "parametrized"
As to what is desired one might want to ask @frankluebeck
In the meantime some perhaps relevant links:
- http://www.numdam.org/article/CM_1972__25_1_1_0.pdf
- https://link.springer.com/chapter/10.1007/BFb0081548
- https://arxiv.org/abs/2012.11015
- and of course one could look at SageMath, Magma, CHEVIA (?), ...