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.

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$

$\mathrm{SmallGroup}\left(6\,2\right)$

$⟨\left(1\,2\right)\left(3\,4\,5\right)⟩$

$\mathrm{SmallGroup}\left(6\,2\,\mathrm{form}=\mathrm{permgroup}\right)$

$⟨\left(1\,2\right)\left(3\,4\,5\right)⟩$

$\mathrm{SmallGroup}\left(6\,2\,\mathrm{form}=\mathrm{fpgroup}\right)$

$⟨g\mid g^6⟩$

$G≔\mathrm{SmallGroup}\left({11}^3\,3\right)\:$

$\mathrm{type}\left(G\,'\mathrm{PermutationGroup}'\right)$

$\mathrm{true}$

$\mathrm{GroupOrder}\left(G\right)$

$1331$

$G≔\mathrm{SmallGroup}\left(5^4\,10\,'\mathrm{form}'=''fpgroup''\right)\:$

$\mathrm{type}\left(G\,'\mathrm{FPGroup}'\right)$

$\mathrm{true}$

$\mathrm{GroupOrder}\left(G\right)$

$625$

$\mathrm{IsAbelian}\left(G\right)$

$\mathrm{false}$

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.