retrieve a group from the database of small groups
SmallGroup( n, d )
SmallGroup( n, d, f )
a positive integer
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 pk, for a prime number p, and for k≤4, as well as groups whose order is 4⁢p. In addition, the SmallGroup command can construct groups of order p⁢q, 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.
G ≔ SmallGroup⁡113,3:
G ≔ SmallGroup⁡54,10,'form'=fpgroup:
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.
Download Help Document