# [GAP Forum] Retrieving group name

Stefan Kohl kohl at mathematik.uni-stuttgart.de
Tue Jun 29 15:25:43 BST 2004

```Dear Forum,

Igor Schein wrote:

> I am struggling to find the way to retrieve canonical group names in
> GAP4 ( I've done it once in GAP3 ).  Basically, I need a function
> which will output "Q8" if the input is SmallGroup(8,4), for example.

Currently there is no function which returns a canonical name for
a given group, but you can get information about the isomorphism types
of the groups of a given order by the function 'SmallGroupsInformation',
e.g.

----------------------------------------------------------------------

gap> SmallGroupsInformation(8);

There are 5 groups of order 8.
1 is of type c8.
2 is of type 2x4.
3 is of type D8.
4 is of type Q8.
5 is of type 2^3.

The groups whose order factorises in at most 3 primes
have been classified by O. Hoelder. This classification is
used in the SmallGroups library.

This size belongs to layer 1 of the SmallGroups library.
IdSmallGroup is available for this size.

----------------------------------------------------------------------

For example the line `4 is of type Q8.' tells you that
SmallGroup(8,4) is the quaternion group of order 8.

However there are no canonical names for larger groups --
thus for these, the function `SmallGroupsInformation' provides
only partial information -- e.g.

----------------------------------------------------------------------

gap> SmallGroupsInformation(64);

There are 267 groups of order 64.
They are sorted by their ranks.
1 is cyclic.
2 - 54 have rank 2.
55 - 191 have rank 3.
192 - 259 have rank 4.
260 - 266 have rank 5.
267 is elementary abelian.

For the selection functions the values of the following attributes
are precomputed and stored:
IsAbelian, PClassPGroup, RankPGroup, FrattinifactorSize and
FrattinifactorId.

This size belongs to layer 2 of the SmallGroups library.
IdSmallGroup is available for this size.

----------------------------------------------------------------------

Note in this context that distinguishing groups of orders with
`many' equal prime factors is often quite difficult, and that one
would need a very elaborate and complicate naming scheme for them
if one would like to have descriptive names.

Best wishes,

Stefan Kohl

```

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