represents the sporadic simple O'Nan group .


  • No permutation representation is implemented for ONanGroupON[].

Background & Context

  • ONanGroupON[] represents the O'Nan group , which is a group of order TemplateBox[{2, 9}, Superscript].TemplateBox[{3, 4}, Superscript].5.TemplateBox[{7, 3}, Superscript].11.19.31. It is one of the 26 sporadic simple groups of finite order and is sometimes known as the O'NanSims group.
  • The O'Nan group is the thirteenth largest of the sporadic finite simple groups. It was discovered by mathematician Michael O'Nan and constructed explicitly by Charles Sims in the mid-1970s. ONanGroupON was first found while investigating groups having so-called Alperin-type Sylow 2-subgroups. The O'Nan group has a permutation representation on symbols, a triple cover with a pair of 45-dimensional modular representations over field with seven elements, and a number of maximal subgroups, including the Janko group , Mathieu group and the alternating group . 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 ONanGroupON[], including GroupOrder, GroupGenerators, GroupElements and so on. However, while ONanGroupON[] 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 O'Nan group are available via FiniteGroupData["ONan","prop"].
  • ONanGroupON is related to a number of other symbols. Along with JankoGroupJ1, JankoGroupJ3, JankoGroupJ4, LyonsGroupLy and RudvalisGroupRu, ONanGroupON is one of six sporadic simple groups referred to as "pariahs" as a consequence of their failure to occur as subquotients of the monster group.


Basic Examples  (1)

Order of the O'Nan group :

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Introduced in 2010