Dear Marek, the group GO(+1,6,3) has the structure 2 x PGO(+1,6,3). The group in question is PGO(+1,6,3), which is called L4(3).2_2 in the ATLAS of Finite Groups. In GAP, you can construct the group for example as the permutation group obtained from the projective action of GO(+1,6,3), or as `PrimitiveGroup( 117, 3 )'.
The group GO(-1,6,3) has the structure 2.U_4(3).(2^2)_{122}, using ATLAS notation. The group in question is U_4(3).2_2. In GAP, it can be obtained for example as `PrimitiveGroup( 126, 8 )'. All the best, Thomas On Wed, Apr 10, 2013 at 11:42:05AM +0200, Marek Mitros wrote: > 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 > > ---------------------------------- > > > 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. _______________________________________________ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum