[GAP Forum] 3-transposition groups

[Marek Mitros]

Dear Forum,

I have read in wikipedia that there are following 3-transposition groups.
See below quote from wikipedia article on "3-transposition groups". I have
no problem to obtain in GAP groups SO(1,n,2), Omega(1,n,2), PSU(n,2),
Sp(n,2). I can also define Fischer groups Fi22, Fi23, Fi24 using Atlas
package. I have only trouble with O+-(2n,3) case. E.g. neither SO(1,6,3)
nor Omega (1,6,3) is 3-transposition group. What is the way to define the
3-transposition group defined under case  Oμ, π(*n*, 3) below.


Regards,
Marek

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Suppose that *G* is a group that is generated by conjugacy class of
3-transpositions and such that the 2 and 3
cores<http://en.wikipedia.org/wiki/P-core>
*O*2(*G*) and *O*3(*G*) are both contained in the center *Z*(*G*) of
*G*and the derived group of
*G* is perfect. Then Fischer
(1971<http://en.wikipedia.org/wiki/3-transposition_group#CITEREFFischer1971>)
proved that up to isomorphism *G*/*Z*(*G*) is one of the following groups
and *D* is the image of the given conjugacy class:

   - *G*/*Z*(*G*) is a symmetric group *Sn*, and *D* is the class of
   transpositions.
   - *G*/*Z*(*G*) is a symplectic group Sp(2*n*, 2) over the field of order
   2, and *D* is the class of transvections
   - *G*/*Z*(*G*) is a projective special unitary group PSU(*n*, 2),
and *D*is the class of transvections
   - *G*/*Z*(*G*) is an orthogonal group Oμ(2*n*, 2), and *D* is the class
   of transvections
   - *G*/*Z*(*G*) is an index 2 subgroup Oμ, π(*n*, 3) of the orthogonal
   group Oμ(*n*, 3) generated by the class *D* of reflections of norm π
   vectors, where μ and π can be 1 or -1.
   - *G*/*Z*(*G*) is one of the three Fischer
groups<http://en.wikipedia.org/wiki/Fischer_group>Fi
   22, Fi23, Fi24.

If the condition that the derived group of *G* is perfect is dropped there
are two extra cases:

   - *G*/*Z*(*G*) is one of two groups containing on orthogonal group O+(8,
   2) or O-(8, 3) with index 3.