FischerGroupFi23

FischerGroupFi23[]

represents the sporadic simple Fischer group .

Details

  • The default permutation representation of FischerGroupFi23[] acts on points {1,,31671}.

Background & Context

  • FischerGroupFi23[] represents the Fischer group , which is a group of order TemplateBox[{2, 18}, Superscript].TemplateBox[{3, 13}, Superscript].TemplateBox[{5, 2}, Superscript].7.11.13.17.23. It is one of the 26 sporadic simple groups of finite order. The default representation of FischerGroupFi23 is as a permutation group on the symbols having two generators.
  • The Fischer group is the sixth largest of the sporadic finite simple groups. It was introduced by Bernd Fischer in the 1970s and first defined in terms of a rank-3 action on the graph of vertices corresponding to its 3-transpositions. In addition to its default permutation representation, which has as its point stabilizer the double cover of the Fischer group , FischerGroupFi23 has a second rank-3 group action on points. It also has an irreducible representation of dimension 253 over the field with three elements. Along with the other sporadic simple groups, the Fischer groups played a foundational role in the monumental (and complete) classification of finite simple groups.
  • The usual group theoretic functions may be applied to FischerGroupFi23[], including GroupOrder, GroupGenerators, GroupElements and so on. However, due its large order, a number of such group theoretic functions may return unevaluated when applied to it. A number of precomputed properties of the Fischer group are available via FiniteGroupData[{"Fisher",23},"prop"].
  • FischerGroupFi23 is related to a number of other symbols. FischerGroupFi23 is one of the eight groups (along with FischerGroupFi22, FischerGroupFi24Prime, HeldGroupHe, HaradaNortonGroupHN, ThompsonGroupTh, BabyMonsterGroupB and MonsterGroupM) collectively referred to as the "third generation" of sporadic finite simple groups. It is also one of 20 so-called "happy" sporadic groups, which all appear as a subquotient of the monster group.

Examples

Basic Examples  (3)

Order of the Fischer group :

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Number of points moved by the generators of a permutation representation of the Fischer group :

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Order of a pseudorandom element of the Fischer group :

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See Also

PermutationGroup  FischerGroupFi22  FischerGroupFi24Prime

Tutorials

Introduced in 2010
(8.0)