GroupTheory
SmallGroup
retrieve a group from the database of small groups
Calling Sequence
Parameters
Description
Examples
Compatibility
SmallGroup( n, d )
SmallGroup( n, d, f )
n

a positive integer
d
f
optional equation: form=fpgroup or form=permgroup (the default)
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.
$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$
$\mathrm{SmallGroup}\left(6\,2\right)$
$\u27e8\left({1}{\,}{2}{\,}{4}{\,}{6}{\,}{5}{\,}{3}\right)\u27e9$
$\mathrm{SmallGroup}\left(6\,2\,\mathrm{form}=\mathrm{permgroup}\right)$
$\mathrm{SmallGroup}\left(6\,2\,\mathrm{form}=\mathrm{fpgroup}\right)$
$\u27e8{}{\mathrm{g1}}{}{\mid}{}{{\mathrm{g1}}}^{{6}}{}\u27e9$
$G\u2254\mathrm{SmallGroup}\left({11}^{3}\,3\right)\:$
$\mathrm{type}\left(G\,'\mathrm{PermutationGroup}'\right)$
${\mathrm{true}}$
$\mathrm{GroupOrder}\left(G\right)$
${1331}$
$G\u2254\mathrm{SmallGroup}\left({5}^{4}\,10\,'\mathrm{form}'=''fpgroup''\right)\:$
$\mathrm{type}\left(G\,'\mathrm{FPGroup}'\right)$
${625}$
$\mathrm{IsAbelian}\left(G\right)$
${\mathrm{false}}$
The GroupTheory[SmallGroup] command was introduced in Maple 17.
For more information on Maple 17 changes, see Updates in Maple 17.
The GroupTheory[SmallGroup] command was updated in Maple 2020.
See Also
GroupTheory[AllSmallGroups]
GroupTheory[NumGroups]
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