ConwayGroupCo2

ConwayGroupCo2[]

represents the sporadic simple Conway group Co2.

Details

  • By default, ConwayGroupCo2[] is represented as a permutation group acting on points {1,…,2300}.

Background & Context

  • ConwayGroupCo2[] represents the Conway group , which is a group of order TemplateBox[{2, 18}, Superscript].TemplateBox[{3, 6}, Superscript].TemplateBox[{5, 3}, Superscript].7.11.23. It is one of the 26 sporadic simple groups of finite order. The default representation of ConwayGroupCo2 is as a permutation group on the symbols having two generators.
  • The Conway group is the eleventh largest of the sporadic finite simple groups. It was introduced by John Horton Conway in the late 1960s. ConwayGroupCo2 is a subgroup of ConwayGroupCo1 that stabilizes a sublattice of the so-called Leech lattice. In addition to its permutation representation, ConwayGroupCo2 also has a 22-dimensional faithful representation over the field of two elements, which is the smallest faithful representation over any field. It can also be defined in terms of generators and relations as , where and . Along with the other sporadic simple groups, the Conway groups played a foundational role in the monumental (and complete) classification of finite simple groups.
  • The usual group theoretic functions may be applied to ConwayGroupCo2[], including GroupOrder, GroupGenerators, GroupElements and so on. However, due its large order, a number of such group theoretic functions may return unevaluated when applied to it. A number of precomputed properties of the Conway group are available via FiniteGroupData[{"Conway",2},"prop"].
  • ConwayGroupCo2 is related to a number of other symbols. ConwayGroupCo2 is one of the seven groups (along with ConwayGroupCo1, ConwayGroupCo3, JankoGroupJ2, HigmanSimsGroupHS, McLaughlinGroupMcL and SuzukiGroupSuz) collectively referred to as the "second generation" of sporadic finite simple groups. It is also one of 20 so-called "happy" sporadic groups, which all appear as a subquotient of the monster group.

Examples

Basic Examples  (3)

Order of the group Co2:

Number of points moved by each generator of a permutation representation:

Order of a pseudorandom element of the group Co2:

Wolfram Research (2010), ConwayGroupCo2, Wolfram Language function, https://reference.wolfram.com/language/ref/ConwayGroupCo2.html.

Text

Wolfram Research (2010), ConwayGroupCo2, Wolfram Language function, https://reference.wolfram.com/language/ref/ConwayGroupCo2.html.

CMS

Wolfram Language. 2010. "ConwayGroupCo2." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ConwayGroupCo2.html.

APA

Wolfram Language. (2010). ConwayGroupCo2. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ConwayGroupCo2.html

BibTeX

@misc{reference.wolfram_2024_conwaygroupco2, author="Wolfram Research", title="{ConwayGroupCo2}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/ConwayGroupCo2.html}", note=[Accessed: 18-November-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_conwaygroupco2, organization={Wolfram Research}, title={ConwayGroupCo2}, year={2010}, url={https://reference.wolfram.com/language/ref/ConwayGroupCo2.html}, note=[Accessed: 18-November-2024 ]}