Dear GAP-forum,

I suspect that there is a bug in routine FpGroup when applied to
a permutation group. In the manual FpGroup is mentioned only for
AgGroups, but since it occurs also in the operations record of
a permutation group I thought it would be safe to use it that case,
too. The problem arises with

p := Group( (2,7)(3,6)(4,5), (1,8)(3,5)(4,6), (1,8)(3,5,6,4) );

A correct set of relators for the three generators of this group is

r:= [ [ 1, 1 ], [ 2, 2 ], [ 2, 3, 2, 3 ], [ 1, 2, 1, 2 ], [ 1, 3, 1, -3 ],
  [ 3, 3, 3, 3 ] ];

A first indication of the problem is that FpGroup(p) typically returns
a group on which Size fails (more than 64000 cosets, but Size(p)=16!).

I became aware of this problem when debugging routines which compute
space groups from a given point group g. I first compute a set of
inequivalent group extensions of g by Z^n, and then identify those
which are equivalent as space groups. In 4D the results do not yet
always agree with what is in the GAP group library. Particularly
wierd is the behaviour for the transpose of MatGroupZClass(4,13,7,2),
from which I have constructed above permutation group. On different
invocations I randomly get either 32 or 64 inequivalent group
extensions. Correct seems to be 64. For the computation I need a
set of relators for the point group, which I determine with FpGroup
and PresentationFpGroup. These relators are the only thing that
varies. I am quite confident that my group extensions code works
correctly, but, of course, one can never be completely sure. If
you really trust FpGroup, I would be happy to send you my code,
but the forum is probably not the right place for that.

Following are a few samples of relator sets I obtained with
FpGroup and PresentationFpGroup from p. Relator sets from the
first list are lacking at least one commutator relator, and
yield 32 inequivalent group extensions. Relator sets in the
second list are lacking a relator [3,3,3,3]. These yield 64
inequivalent group extensions.

[ # 32 inequivalent group extensions
  [ [ 1, 1 ], [ 2, 2 ], [ 2, 1, 2, 1 ], [ 3, 2, 3, 2 ], [ 3, 3, 3, 3 ] ],
  [ [ 1, 1 ], [ 2, 2 ], [ 2, 3, 2, 3 ], [ 2, 1, 2, 1 ] ]
]

[ # 64 inequivalent group extensions
  [ [ 1, 1 ], [ 2, 2 ], [ 1, 3, 1, -3 ], [ 2, 1, 2, 1 ], [ 3, 2, 3, 2 ] ],
  [ [ 1, 1 ], [ 2, 2 ], [ 2, -3, 2, -3 ], [ 2, 1, 2, 1 ],
    [ 3, 1, 3, 3, 1, 3 ] ],
  [ [ 1, 1 ], [ 2, 2 ], [ 2, 1, 2, 1 ], [ 3, 2, 3, 2 ],
    [ 2, 3, 1, 2, 3, 1 ] ],
  [ [ 1, 1 ], [ 2, 2 ], [ 2, 1, 2, 1 ], [ 3, 2, 3, 2 ],
    [ 3, 3, 1, 3, 1, 3 ] ]
]

I am running GAP 3.4 at patch level 3, and I think I have applied all
bugfixes, in particular bugfix04. Perhaps I should try the newly
announced PresentationViaCosetTable!

With kind regards,

Franz Gaehler