# JankoGroupJ4

represents the sporadic simple Janko group .

# Details

• No permutation representation is implemented for .

# Background & Context

• represents the Janko group , a group of order . It is one of the 26 sporadic simple groups of finite order.
• The Janko group is the fourth largest of the sporadic finite simple groups. It was discovered by mathematician Zvonimir Janko in the mid 1970s, making it the last of the sporadic simple groups to be found. JankoGroupJ4 has a number of permutation representations, the smallest of which is on symbols with an -related point stabilizer, which may be identified with a specific 112-dimensional faithful representation over the field of two elements. 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. 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 Janko group are available via FiniteGroupData[{"Janko",4},"prop"].
• JankoGroupJ4 is related to a number of other symbols. Along with JankoGroupJ1, JankoGroupJ3, LyonsGroupLy, ONanGroupON and RudvalisGroupRu (but not JankoGroupJ2), JankoGroupJ4 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 group :

 In[1]:=
 Out[1]=