SmallGroup - Maple Help

GroupTheory

 SmallGroup
 retrieve a group from the database of small groups

 Calling Sequence SmallGroup( n, d ) SmallGroup( n, d, f )

Parameters

 n - a positive integer d - a positive integer f - optional equation: form=fpgroup or form=permgroup (the default)

Description

 • The small groups library contains all groups of small orders up to $511$. The groups are sorted by their orders and they are listed up to isomorphism; that is, for each of the available orders a complete and irredundant list of isomorphism type representatives of groups is given. These groups are available as permutation groups and as groups defined by generators and relations.
 • The SmallGroup( n, d ) command returns the $d$-th group of order $n$ in the small groups library. The value of the parameter n must be at most $511$, and the value of the parameter d must be less than or equal to the number of groups of order $n$.
 • The SmallGroup command can construct groups of certain orders greater than $511$ of particular forms. It can produce the groups whose order is of the form ${p}^{k}$, for a prime number $p$, and for $k\le 4$, as well as groups whose order is $4p$. In addition, the SmallGroup command can construct groups of order $pq$, for distinct primes $p$ and $q$.
 • Use the NumGroups command to determine the number of groups of order equal to $n$ in the small groups library.
 • The numbering used is consistent with that used by GAP, a de facto standard, so that you can refer to a specific group by its number, e.g., SmallGroup( 60, 5 ) refers to the fifth group of order 60, which happens to be isomorphic to the alternating group of degree 5.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $\mathrm{SmallGroup}\left(6,2\right)$
 $⟨\left({1}{,}{2}\right)\left({3}{,}{4}{,}{5}\right)⟩$ (1)
 > $\mathrm{SmallGroup}\left(6,2,\mathrm{form}=\mathrm{permgroup}\right)$
 $⟨\left({1}{,}{2}\right)\left({3}{,}{4}{,}{5}\right)⟩$ (2)
 > $\mathrm{SmallGroup}\left(6,2,\mathrm{form}=\mathrm{fpgroup}\right)$
 $⟨{}{g}{}{\mid }{}{{g}}^{{6}}{}⟩$ (3)
 > $G≔\mathrm{SmallGroup}\left({11}^{3},3\right):$
 > $\mathrm{type}\left(G,'\mathrm{PermutationGroup}'\right)$
 ${\mathrm{true}}$ (4)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${1331}$ (5)
 > $G≔\mathrm{SmallGroup}\left({5}^{4},10,'\mathrm{form}'="fpgroup"\right):$
 > $\mathrm{type}\left(G,'\mathrm{FPGroup}'\right)$
 ${\mathrm{true}}$ (6)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${625}$ (7)
 > $\mathrm{IsAbelian}\left(G\right)$
 ${\mathrm{false}}$ (8)

Compatibility

 • The GroupTheory[SmallGroup] command was introduced in Maple 17.