Currently, the search for conjugacy classes is pretty slow for groups of large order (we use random search).
This could be improved dramatically.
The most accessible method seems now the Butler1994 reference below. It needs the computation of Sylow subgroups, which could be implemented following Holt2005, chapter 4.
See references:
G. Butler, “An Inductive Schema for Computing Conjugacy Classes in Permutation Groups,” Mathematics of Computation, vol. 62, no. 205, pp. 363–383, 1994, doi: 10.2307/2153415.
A. Hulpke, “Computing conjugacy classes of elements in matrix groups,” Journal of Algebra, vol. 387, pp. 268–286, Aug. 2013, doi: 10.1016/j.jalgebra.2013.02.043.
J. Cannon and B. Souvignier, “On the computation of conjugacy classes in permutation groups,” in Proceedings of the 1997 international symposium on Symbolic and algebraic computation, Kihei, Maui, Hawaii, USA, Jul. 1997, pp. 392–399, doi: 10.1145/258726.258855.
D. F. Holt, B. Eick, and E. A. O’Brien, Handbook of Computational Group Theory. CRC Press, 2005.
Currently, the search for conjugacy classes is pretty slow for groups of large order (we use random search).
This could be improved dramatically.
The most accessible method seems now the Butler1994 reference below. It needs the computation of Sylow subgroups, which could be implemented following Holt2005, chapter 4.
See references:
G. Butler, “An Inductive Schema for Computing Conjugacy Classes in Permutation Groups,” Mathematics of Computation, vol. 62, no. 205, pp. 363–383, 1994, doi: 10.2307/2153415.
A. Hulpke, “Computing conjugacy classes of elements in matrix groups,” Journal of Algebra, vol. 387, pp. 268–286, Aug. 2013, doi: 10.1016/j.jalgebra.2013.02.043.
J. Cannon and B. Souvignier, “On the computation of conjugacy classes in permutation groups,” in Proceedings of the 1997 international symposium on Symbolic and algebraic computation, Kihei, Maui, Hawaii, USA, Jul. 1997, pp. 392–399, doi: 10.1145/258726.258855.
D. F. Holt, B. Eick, and E. A. O’Brien, Handbook of Computational Group Theory. CRC Press, 2005.