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GroupTheory[AllSmallGroups]

Calling Sequence

AllSmallGroups( r )

AllSmallGroups( r, f )

Parameters

r

-

a positive integer, an integer range, or a small group ID range

f

-

optional equation: form=permgroup (default) or form=fpgroup

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.

• 

In its simplest form, the command AllSmallGroups( r ) returns a list of all the small groups in the small groups library of order r, where r is a positive integer less than 512.

• 

If r is a range of the form m .. n, where m and n are positive integers, then AllSmallGroups( r ) returns a list of all the groups whose order lies in the range m .. n.

• 

More generally, r may be a "range" of the form [ln, lk] .. [un, uk], where ln and un are positive integer less than 512, and where lk is a positive integer in the range 1 .. NumGroups( ln ), and uk is an integer in the range 1 .. NumGroups( un ). In this case, AllSmallGroups( r ) returns a list of the groups whose orders lie in the range ln .. un, beginning with the lk-th group of order ln, and ending with the uk-th group of order un. Think of the groups of each order as forming a "row" of a "ragged" matrix, and the first operand of the range r specifies a first position  in this matrix, while the second operand of r specifies a second position in the matrix, so that the range r selects all the groups occurring between these two positions, where the matrix is traversed in row-major order, from the first to the second position.

Examples

withGroupTheory:

AllSmallGroups6

1,23,64,5,1,3,42,5,6,1,2,4,6,5,3

(1)

AllSmallGroups6,form=fpgroup

_a,_b_a2,_b3,_a-1_b_a_b-2,g0g06

(2)

AllSmallGroups6,form=permgroup

1,23,64,5,1,3,42,5,6,1,2,4,6,5,3

(3)

AllSmallGroups4..6

1,2,4,3,1,23,4,1,32,4,1,2,4,5,3,1,23,64,5,1,3,42,5,6,1,2,4,6,5,3

(4)

AllSmallGroups6,2..8,3

1,2,4,6,5,3,1,2,4,6,7,5,3,1,2,4,6,8,7,5,3,1,2,5,34,6,8,7,1,42,63,75,8,1,23,74,65,8,1,32,54,86,7,1,42,63,85,7

(5)

AllSmallGroups6,2..8,3,'form'=fpgroup

g7g76,g8g87,_a_a8,_a,_b_b2,_a4,_a-1_b-1_a_b,_a,_b,_c_a2,_b2,_c2,_c-1_a-1_c_a,_c-1_b-1_c_b,_b-1_a-1_b_a_c-1

(6)

See Also

GroupTheory[NumGroups], GroupTheory[SmallGroup]


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