# JankoGroupJ1

represents the sporadic simple Janko group .

# Details

• By default, is represented as a permutation group acting on points {1,,266}.

# Background & Context

• represents the Janko group , which is a group of order . It is one of the 26 sporadic simple groups of finite order. The default representation of JankoGroupJ1 is as a permutation group on the symbols having two generators.
• The Janko group is the third smallest of the sporadic finite simple groups. It was discovered (along with JankoGroupJ2 and JankoGroupJ3) by mathematician Zvonimir Janko in the mid 1900s, making these groups tied for second in chronological order of discovery among sporadic groups. JankoGroupJ1 is the unique simple group with an abelian Sylow-2 subgroup and with an involution whose centralizer is the direct product of CyclicGroup[2] and . In addition to its permutation representation, can be defined in terms of generators and relations as and is contained in the O'Nan group as the subgroup of elements fixed by an outer automorphism of order 2. Along with the other sporadic simple groups, the Janko 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. A number of precomputed properties of the Janko group are available via FiniteGroupData[{"Janko",1},"prop"].
• JankoGroupJ1 is related to a number of other symbols. Along with JankoGroupJ3, JankoGroupJ4, LyonsGroupLy, ONanGroupON and RudvalisGroupRu (but not JankoGroupJ2), JankoGroupJ1 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(2)

Order of the group :

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Generators of a permutation representation of the group :

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# Tutorials

Introduced in 2010
(8.0)