HigmanSimsGroupHS

HigmanSimsGroupHS[]

represents the sporadic simple Higman–Sims group .

Details

  • By default, HigmanSimsGroupHS[] is represented as a permutation group acting on points {1,…,100}.

Background & Context

  • HigmanSimsGroupHS[] represents the Higman–Sims group , which is a group of order TemplateBox[{2, 9}, Superscript].TemplateBox[{3, 2}, Superscript].TemplateBox[{5, 3}, Superscript].7.11. It is one of the 26 sporadic simple groups of finite order. The default representation of HigmanSimsGroupHS is as a permutation group on the symbols having two generators.
  • The Higman–Sims group is the seventh smallest of the sporadic finite simple groups. It was introduced by mathematicians Donald G. Higman and Charles Sims in the late 1960s. HigmanSimsGroupHS is the simple index-2 subgroup of the automorphism group of the so-called Higman–Sims graph and contains a one-point stabilizer isomorphic to the Mathieu group . In addition to its numerous permutation representations, can be defined in terms of generators and relations as , with , , and . It can also be found as subgroups of various permutation groups such as ConwayGroupCo2 and ConwayGroupCo3, related to the so-called Leech lattice. 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 HigmanSimsGroupHS[], 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 Higman–Sims group are available via FiniteGroupData["HigmanSims","prop"].
  • HigmanSimsGroupHS is related to a number of other symbols. HigmanSimsGroupHS is one of the seven groups (along with ConwayGroupCo1, ConwayGroupCo2, ConwayGroupCo3, JankoGroupJ2, 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  (2)

Order of the group :

Generators of a permutation representation of the group :

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

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2023_higmansimsgrouphs, author="Wolfram Research", title="{HigmanSimsGroupHS}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/HigmanSimsGroupHS.html}", note=[Accessed: 20-April-2024 ]}

BibLaTeX

@online{reference.wolfram_2023_higmansimsgrouphs, organization={Wolfram Research}, title={HigmanSimsGroupHS}, year={2010}, url={https://reference.wolfram.com/language/ref/HigmanSimsGroupHS.html}, note=[Accessed: 20-April-2024 ]}