Erhard Aichinger asked:

I am using
TwoGroup (32, 6)
to get the 6th group of order 32. Do you
know whether the numbering of groups in the twogroup-package is
identical to the numbering in the book
Thomas, A.D., and Wood G.V., Group Tables, Shiva Publishing Ltd. ?

The answer is : no, the numbering schemes are non-identical.

According to their preface, Thomas and Wood follow the numbering in
Hall/Senior (The groups of order 2^n, n<=6), except for abelian
groups.

The numbering in the two-groups library of GAP is the one by Newman
and O'Brien which is determined by their specific p-group generation
algorithm. (see remark and references in the description of the
two-group library in the chapter 'Group Libraries' of the GAP manual).

You can get some information about different namings by using the
function 'GroupId', described in the chapter 'Groups' in the manual.

So for the group obtained in GAP by TwoGroup (32, 6) the function
GroupId gives (i.a.) 'catalogue := [32, 46]' and the group has indeed
number 46 both in Hall/Senior and in Thomas/Wood.

Note however that for abelian groups there are still little
differences in the numbering by Hall/Senior and the number given by
the entry 'catalogue'. Therefore for abelian groups you should best
use the function 'AbelianInvariants', also described in the chapter
'Groups' in the manual, to see what you have got. This function
should, however, ONLY BE APPLIED TO ABELIAN GROUPS, as said in the
manual. If you are not sure if your group is abelian you can first use
the function 'IsAbelian' to test this.

WARNING: At present the function 'AbelianInvariants' applied to a
nonabelian group will sometimes (e.g. for finitely presented groups)
determine the abelian invariants of its commutator factor group, but
sometimes it may produce some rather ununderstandable error message
and, worst of all, for nonabelian ag groups it may produce a
syntactically innocent looking wrong result. We intend to change the
function so that it will always produce the abelian invariants of the
commutator factor group, but this will only be in a forthcoming
release.

Sorry for the inconvenience, but agreeing about a naming scheme, such
as in Chemistry, is too difficult for mathematicians.

Joachim Neubueser