FischerGroupFi24Prime

represents the sporadic simple Fischer group .

Details

• No permutation representation is implemented for .

Background & Context

• represents the Fischer group , which is a group of order . It is one of the 26 sporadic simple groups of finite order.
• The Fischer group is the third 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 Fischer group , FischerGroupFi24Prime has an irreducible representation of dimension 781 over the field with three elements and has a finite cover that centralizes an element of order three in the monster group. 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 , including GroupOrder, GroupGenerators, GroupElements and so on. However, while is a permutation group, due its large order, an explicit permutation representation is impractical for direct implementation. As a result, 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",24},"prop"].
• FischerGroupFi24Prime is related to a number of other symbols. FischerGroupFi24Prime is one of the eight groups (along with FischerGroupFi22, FischerGroupFi23, 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(1)

Order of the Fischer group :

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