JankoGroupJ3

JankoGroupJ3[]

represents the sporadic simple Janko group .

Details

  • By default, JankoGroupJ3[] is represented as a permutation group acting on points {1,,6156}.

Background & Context

  • JankoGroupJ3[] represents the Janko group , a group of order TemplateBox[{2, 7}, Superscript].TemplateBox[{3, 5}, Superscript].5.17.19, which is one of the 26 sporadic simple groups of finite order. The default representation of JankoGroupJ3 is as a permutation group on the symbols having two generators.
  • The Janko group is the eighth smallest of the sporadic finite simple groups. It was discovered (along with JankoGroupJ1 and JankoGroupJ2) by mathematician Zvonimir Janko in the mid 1900s, making these groups tied for second in chronological order of discovery among sporadic groups. JankoGroupJ3 has a modular representation of dimension 18 over the finite field with nine elements. In addition to its permutation representation, can be defined in terms of generators and relations as , where , , and . 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 JankoGroupJ3[], including GroupOrder, GroupGenerators, GroupElements and so on. A number of precomputed properties of the Janko group are available via FiniteGroupData[{"Janko",3},"prop"].
  • JankoGroupJ3 is related to a number of other symbols. Along with JankoGroupJ1, JankoGroupJ4, LyonsGroupLy, ONanGroupON and RudvalisGroupRu (but not JankoGroupJ2), JankoGroupJ3 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  (3)

Order of the group :

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Number of points moved by the generators of a permutation representation of :

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Order of a pseudorandom element of the group :

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See Also

PermutationGroup  JankoGroupJ1  JankoGroupJ2  JankoGroupJ4

Tutorials

Introduced in 2010
(8.0)