# GAP:AllSmallGroups

## Contents

INTERNAL AD: To get summary information on groups of a given order on this wiki, type "groups of order <a>" where a is the order, into the search box. For instance, groups of order 8 gives information on the groups of order 8.

## Definition

### Function type

AllSmallGroups is a GAP function that takes as input a natural number and outputs a list of groups.

### Behavior

The function is supposed to return a list of all the groups whose order is the given natural number. This list is based on GAP's in-built library and the groups always appear in the same sequence in the list. GAP does not compute these groups on the spot.

The following caveats should be noted:

• For a finite solvable group, the group is stored as a PcGroup: in other words, it is stored in terms of a polycyclic series for the group. Thus, if the group is solvable, the command SmallGroup returns a polycyclic series.
• For a finite group that is not solvable, the group is stored as a permutation group.

Error types:

• If the groups of order equal to the input are not stored in the library, GAP returns an error stating that the library of groups of order $a$ is not available.
• If the input is not a positive integer, GAP returns a usage error.

### Typical use

`AllSmallGroups(n);`

where $n$ is a natural number.

## Related functions

• GAP:SmallGroup: This takes as input an ordered pair of natural numbers $(a,b)$, and returns the $b^{th}$ group of order $a$.
• GAP:OneSmallGroup: This returns only one group of order equal to the given natural number, namely, the first member of the list returned by AllSmallGroups.
• GAP:SmallGroupsInformation: This provides verbal information on the groups of a given order and how they are stored in GAP's library.