# ONanGroupON

represents the sporadic simple O'Nan group .

# Details

• No permutation representation is implemented for .

# Background & Context

• represents the O'Nan group , which is a group of order . 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 , 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 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.

# Examples

## Basic Examples(1)

Order of the O'Nan group :

 In[1]:=
 Out[1]=

# Tutorials

Introduced in 2010
(8.0)