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Let $G$ be a finite group. How could we obtain all conjugacy classes of element centralizers of $G$ by GAP?

(By the centralizer of an element $g$ in $G$, I mean the subgroup $C_G(g):=\{x\in G | xg=gx \}$ of $G$ and by the conjugacy class of $C_G(g)$ I mean the set $\{x^{-1}C_G(g)x | x\in G\}$).

Thank you so much!

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  • $\begingroup$ I assume you are aware of the naive way to do this and is looking for a more efficient one than just applying the obvious functions to all elements? $\endgroup$ – Tobias Kildetoft Apr 2 '16 at 18:19
  • $\begingroup$ Dear Tobias Yes! I want to apply a more efficient and shorter way to obtain them. Any suggestion please? $\endgroup$ – sebastian Apr 2 '16 at 18:26
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If you compute ConjugacyClasses you get a list of classes, each having a Representative and a Centralizer, these centralizers are (with duplicates if two classes have the same centralizer, e.g. Galois-conjugate elements) the different possible centralizers in $G$:

List(ConjugacyClasses(G),Centralizer);
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  • $\begingroup$ Dear ahulpke thank you for your useful answer! $\endgroup$ – sebastian Apr 3 '16 at 5:53
  • $\begingroup$ How could we compute the centralizer of a subgroup in the whole group in GAP? $\endgroup$ – M. R. Nov 12 '19 at 8:22

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