# LyonsGroupLy

represents the sporadic simple Lyons group .

# Details

• No permutation representation is implemented for .

# Background & Context

• represents the Lyons group , a group of order . It is one of the 26 sporadic simple groups of finite order. The Lyons group is also referred to as the LyonsSims group and denoted .
• The Lyons group is the eighth largest of the sporadic finite simple groups. It was discovered by mathematician Richard Lyons and constructed explicitly by Charles Sims in the early 1970s. LyonsGroupLy was first described as the unique possible group having an involution with centralizer isomorphic to the nontrivial central extension of AlternatingGroup[11] by CyclicGroup[2]. The Lyons group has a number of different representations, including a pair of rank-5 permutation representations on and symbols (the prior of which is faithful), as well as a modular representation of dimension 111 over the field with five elements. 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 Lyons group are available via FiniteGroupData["Lyons","prop"].
• LyonsGroupLy is related to a number of other symbols. Along with JankoGroupJ1, JankoGroupJ3, JankoGroupJ4, ONanGroupON and RudvalisGroupRu, LyonsGroupLy 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 Lyons group :

 In[1]:=
 Out[1]=