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HaradaNortonGroupHN

HaradaNortonGroupHN[]

represents the sporadic simple Harada–Norton group .

Details

No permutation representation is implemented for HaradaNortonGroupHN[].

Background & Context

HaradaNortonGroupHN[] represents the Harada–Norton group , which is a group of order . It is one of the 26 sporadic simple groups of finite order.
The Harada–Norton group is the ninth largest of the sporadic finite simple groups. It was introduced independently by mathematicians Simon Norton and Koichiro Harada in the late 1970s and centralizes an element of order five in the monster group. It thus acts on a 133-dimensional algebra having a commutative but nonassociative product over the field . Along with the other sporadic simple groups, played a foundational role in the monumental (and complete) classification of finite simple groups.
The usual group theoretic functions may be applied to HaradaNortonGroupHN[], including GroupOrder, GroupGenerators, GroupElements and so on. However, while HaradaNortonGroupHN[] 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 Harada–Norton group are available via FiniteGroupData["HaradaNorton","prop"].
HaradaNortonGroupHN is related to a number of other symbols. HaradaNortonGroupHN is one of the eight groups (along with FischerGroupFi22, FischerGroupFi23, FischerGroupFi24Prime, HeldGroupHe, 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 Harada–Norton group :

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