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GroupTheory

  

SmallGroup

  

retrieve a group from the database of small groups

 

Calling Sequence

Parameters

Description

Examples

Compatibility

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 pk, for a prime number p, and for k4, 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

withGroupTheory:

SmallGroup6,2

1,2,4,6,5,3

(1)

SmallGroup6,2,form=permgroup

1,2,4,6,5,3

(2)

SmallGroup6,2,form=fpgroup

g1g16

(3)

GSmallGroup113,3:

typeG,PermutationGroup

true

(4)

GroupOrderG

1331

(5)

GSmallGroup54,10,form=fpgroup:

typeG,FPGroup

true

(6)

GroupOrderG

625

(7)

IsAbelianG

false

(8)

Compatibility

• 

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]