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I want to calculate the number of all orders of elements in HS (Higman-Sims sporadic simple group). Is there any way of doing this with MAGMA or GAP? How I can determine orders of elements of a group with CharacterTable in GAP?

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  • $\begingroup$ Please reformulate "number of all orders" - it's not clear what you're asking (is it $\mid\{n \mid \exists g \in G : Order(g)=n \}\mid$? And yes, character table will answer this kind of questions. $\endgroup$ – Alexander Konovalov Sep 28 '14 at 14:38
  • $\begingroup$ Alternatively use the function ConjugacyClasses, and look at the sizes of the classes and the orders of their representative elements. $\endgroup$ – Derek Holt Sep 28 '14 at 19:58
  • $\begingroup$ @DerekHolt: indeed, I mean that for this particular group it's easier to fetch the precomputed character table from the library and then get the data about conjugacy classes and orders from it. For a general case, one should use ConjugacyClasses in the first instance. $\endgroup$ – Alexander Konovalov Sep 28 '14 at 20:02
  • $\begingroup$ You may also receive this information from the Atlas of Finite Groups, the relevant page for $HS$ is here $\endgroup$ – Alexander Konovalov Sep 29 '14 at 20:11
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In general, having a group in GAP, information about the orders of its elements could be derived from its conjugacy classes and orders of their representatives, for example:

gap> G:=Group([ (3,7,5)(4,8,6), (1,2,6)(3,4,8) ]);
Group([ (3,7,5)(4,8,6), (1,2,6)(3,4,8) ])
gap> List(ConjugacyClasses(G),Size);
[ 1, 56, 24, 24, 21, 42 ]
gap> List(ConjugacyClasses(G),c -> Order(Representative(c)));
[ 1, 3, 7, 7, 2, 4 ]

Clearly, for a large group having this information precomputed would be extremely useful, and due to the Atlas of Finite Group Representations, such opportunity exists. For example, you may see the information about conjugacy classes in the Atlas page on $HS$ here without using any computational algebra system.

The GAP Character Table Library derives some data from Atlas, and you may find information about conjugacy classes of $HS$ stored in its character table in the following way:

gap> t:=CharacterTable("HS");
CharacterTable( "HS" )
gap> SizesConjugacyClasses(t);
[ 1, 5775, 15400, 123200, 11550, 173250, 693000, 88704, 147840, 1774080, 
  1232000, 1848000, 6336000, 2772000, 2772000, 2772000, 2217600, 2217600, 
  4032000, 4032000, 3696000, 2956800, 2217600, 2217600 ]
gap> OrdersClassRepresentatives(t);
[ 1, 2, 2, 3, 4, 4, 4, 5, 5, 5, 6, 6, 7, 8, 8, 8, 10, 10, 11, 11, 12, 15, 
  20, 20 ]
gap> ClassNames(t);
[ "1a", "2a", "2b", "3a", "4a", "4b", "4c", "5a", "5b", "5c", "6a", "6b", 
  "7a", "8a", "8b", "8c", "10a", "10b", "11a", "11b", "12a", "15a", "20a", 
  "20b" ]
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