FischerGroupFi22

represents the sporadic simple Fischer group .

Details • The default permutation representation of acts on points {1,,3510}.

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 default representation of FischerGroupFi22 is as a permutation group on the symbols having two generators.
• The Fischer group is the tenth 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 permutation representation, can be defined in terms of generators and relations as  , where centralizes an element of order 3 in the baby monster group. In addition to its default permutation representation, it has a pair of rank-3 group actions on points. 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, 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",22},"prop"].
• FischerGroupFi22 is related to a number of other symbols. FischerGroupFi22 is one of the eight groups (along with FischerGroupFi23, 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 :

 In:= Out= Number of points moved by the generators of a permutation representation of :

 In:= Out= Order of a pseudorandom element of the Fischer group :

 In:= Out= Introduced in 2010
(8.0)