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GroupGenerators[group]

returns a list of generators of group.

Details
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Examples  
Basic Examples  
Scope  
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GroupGenerators

GroupGenerators[group]

returns a list of generators of group.

Details

  • The type of generators will depend on the representation type of the group.
  • For an explicit permutation representation, GroupGenerators[PermutationGroup[{g1,…,gn}]] returns the list {g1,…,gn}.
  • For a named abstract group, the generators of a default representation are returned.

Examples

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Basic Examples  (2)

Generators of a permutation group:

Wolfram Language code: PermutationGroup[{Cycles[{{1, 2}}], Cycles[{{1, 3}, {2, 4}}]}]//GroupGenerators

A symmetric group is generated by default with two permutations, a transposition, and a shift:

Wolfram Language code: GroupGenerators[SymmetricGroup[8]]

Scope  (3)

Generators for a group defined by generators:

Wolfram Language code: GroupGenerators[PermutationGroup[{Cycles[{{1, 2}, {3, 4}}], Cycles[{{1, 3}, {2, 4}}]}]]

Generators for a named group, using a default representation:

Wolfram Language code: GroupGenerators[DihedralGroup[4]]

Support lengths of the default permutation generators of a sporadic simple group:

Wolfram Language code: PermutationLength /@ GroupGenerators[FischerGroupFi23[]]
Wolfram Research (2010), GroupGenerators, Wolfram Language function, https://reference.wolfram.com/language/ref/GroupGenerators.html.

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_groupgenerators, author="Wolfram Research", title="{GroupGenerators}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/GroupGenerators.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_groupgenerators, organization={Wolfram Research}, title={GroupGenerators}, year={2010}, url={https://reference.wolfram.com/language/ref/GroupGenerators.html}, note=[Accessed: 12-June-2026]}