Dear David,
for smallish groups you can directly compute the automorphism group
of a character table in GAP.
More pecisely, TableAutomorphisms.
gap> SmallGroupsInformation(81);
There are 15 groups of order 81.
They are sorted by their ranks.
1 is cyclic.
2 - 10 have rank 2.
11 - 14 have rank 3.
15 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.
gap> g81:=SmallGroup(81,15);
<pc group of size 81 with 4 generators>
gap> IsElementaryAbelian(g81); # an example of the group you are interested in
true
gap> t:=CharacterTable(g81);;
gap> a:=TableAutomorphisms(t,Irr(t),"closed");
<permutation group with 8 generators>
gap> OrbitLengths(a,[1..81]);
[ 1, 80 ]
(I think this illustrates the general picture - the trivial character
will be unique, in its own orbit, and the remaining characters are all
equivalent - which is not at all surprising, in view of
https://en.wikipedia.org/wiki/Pontryagin_duality)
Hope this helps,
Dima
On Sun, Jul 05, 2020 at 09:03:00AM +0000, David Musyoka wrote:
> Dear Dima and the GAP forum.
> Kindly appealing for your help in this;
> Given a group G and the group K=(q raised to the power of n) which is
> isomorphic to the Vector Space of Dimension n over GF(q),How do i compute the
> orbit lengths for the action of the group G on the set Irr(K) - the set of
> irreducible characters of K Using the GAP programme.
> Thanking you in advance.
> David.
> Sent from Yahoo Mail on Android
>
> On Mon, 29 Jun 2020 at 13:06, Dima Pasechnik<[email protected]>
> wrote: Dear all,
> > How can i generate a group e.g "Sp6(2)" as a matrix group by 7 * 7
> > matrices, then generate "S8" as a matrix group inside Sp6(2) by 7 * 7
> > matrices using GAP prog.
>
>
> gap> G:=GeneralOrthogonalGroup(7,2); # use the fact that Sp(6,2)=O(7,2)
> GO(0,7,2)
> gap> o:=Orbits(G, GF(2)^7,OnLines);;
> gap> List(o,Size); # in this representation S8 fixes a hyperplane, not a
> vector
> [ 1, 63, 63, 1 ]
> gap> Gt:=Group(List(GeneratorsOfGroup(G),TransposedMat));
> <matrix group with 2 generators>
> gap> ot:=Orbits(Gt, GF(2)^7,OnLines);;
> gap> List(ot,Length); # the stabiliser of a vector in 3rd orbit is S8
> [ 1, 63, 36, 28 ]
> gap> s8:=Stabilizer(Gt,ot[3][1],OnLines); # here it is.
> <matrix group of size 40320 with 3 generators>
>
> Hope this helps
> Dima
>
> On Mon, Jun 29, 2020 at 07:32:51AM +0000, David Musyoka wrote:
> > Dear Forum, Dear Alexander Hulpke,
> > Given a group G, and and a vector space V of dimension n over GF(q), i am
> > abe to compute the orbit Lengths of V under action of G using the following
> > GAP commands:
> > V:=FullRowSpace(GF(q),n);Orb:=OrbitLengths(G,V)
> > My question is, how then do i compute the corresponding point stabilizers
> > (which are subgroups of G) for the orbits using GAP.
> > Thank you team in advance.
> > David.
> > Sent from Yahoo Mail on Android
> >
> > On Sun, 21 Jun 2020 at 22:47, David Musyoka<[email protected]>
> >wrote: Dear Alexander Hulpke,Dear Forum.
> > Thank You very much.
> >
> > Sent from Yahoo Mail on Android
> >
> > On Sun, 21 Jun 2020 at 19:48,
> >Hulpke,Alexander<[email protected]> wrote: Dear Forum, Dear
> >Marc David Musyoka,
> >
> > On Jun 20, 2020, at 00:22, David Musyoka <[email protected]> wrote:
> > Deat all,
> > Kind request to this team, i am new to GAP and i wish to be assisted in the
> > following,
> > How can i generate a group e.g "Sp6(2)" as a matrix group by 7 * 7
> > matrices, then generate "S8" as a matrix group inside Sp6(2) by 7 * 7
> > matrices using GAP prog.
> >
> > I wish that am assisted on how to execute the same step by step and the
> > matrix generators for the two groups be listed.
> >
> >
> > Yes, I also often wish that someone would assist me in every step and
> > provide me with the full result.
> > Anyhow, in this case (the algorithm is exponential time and attempts will
> > fail if groups are too large, or if the subgroup to be embedded needs many
> > generators) `IsomorphicSubgroups` seems to do the trick in a few minutes.
> > gap> G:=SP(6,2);Sp(6,2)gap> emb:=IsomorphicSubgroups(G,SymmetricGroup(8));
> > # finds embeddings from S8 into G[ [ (1,8)(2,7)(3,5)(4,6), (1,8,4,7,3)(5,6)
> > ] -> [ <an immutable 6x6 matrix over GF2>, <an immutable 6x6 matrix over
> > GF2> ] ]gap> sub:=Image(emb[1]);<matrix group with 2 generators>
> > So there is one class of subgroups and `sub` is one (not necessarily the
> > nicest one) representative.
> > Of course this is computational overkill. The more sensible way would be to
> > produce the matrix representation (as reduced permutation representation),
> > find the form that it stabilizes, and then conjugate that form to the one
> > used for Sp.
> > Oh, here are the explicit matrix generators :-)
> > gap> GeneratorsOfGroup(sub);[ <an immutable 6x6 matrix over GF2>, <an
> > immutable 6x6 matrix over GF2> ]
> > (You can use `Print` or `Display` on each of them to see them with numbers.)
> > All the best,
> > Alexander Hulpke
> > -- Colorado State University, Department of Mathematics,
> > Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
> > email: [email protected],
> > http://www.math.colostate.edu/~hulpke
> >
> >
> >
> > _______________________________________________
> > Forum mailing list
> > [email protected]
> > https://mail.gap-system.org/mailman/listinfo/forum
>
> Dear all,
> > How can i generate a group e.g "Sp6(2)" as a matrix group by 7 * 7
> > matrices, then generate "S8" as a matrix group inside Sp6(2) by 7 * 7
> > matrices using GAP prog.
>
>
> gap> G:=GeneralOrthogonalGroup(7,2); # use the fact that Sp(6,2)=O(7,2)
> GO(0,7,2)
> gap> o:=Orbits(G, GF(2)^7,OnLines);;
> gap> List(o,Size); # in this representation S8 fixes a hyperplane, not a
> vector
> [ 1, 63, 63, 1 ]
> gap> Gt:=Group(List(GeneratorsOfGroup(G),TransposedMat));
> <matrix group with 2 generators>
> gap> ot:=Orbits(Gt, GF(2)^7,OnLines);;
> gap> List(ot,Length); # the stabiliser of a vector in 3rd orbit is S8
> [ 1, 63, 36, 28 ]
> gap> s8:=Stabilizer(Gt,ot[3][1],OnLines); # here it is.
> <matrix group of size 40320 with 3 generators>
>
> Hope this helps
> Dima
>
> On Mon, Jun 29, 2020 at 07:32:51AM +0000, David Musyoka wrote:
> > Dear Forum, Dear?Alexander Hulpke,
> > Given a group G, and and a vector space V of dimension n over GF(q), i am
> > abe to compute the orbit Lengths of V under action of G using the following
> > GAP commands:
> > V:=FullRowSpace(GF(q),n);Orb:=OrbitLengths(G,V)
> > My question is, how then do i compute the corresponding point stabilizers
> > (which?are subgroups of G)?for the orbits using GAP.
> > Thank you team in advance.
> > David.
> > Sent from Yahoo Mail on Android
> >
> > On Sun, 21 Jun 2020 at 22:47, David Musyoka<[email protected]>
> > wrote: Dear?Alexander Hulpke,Dear Forum.
> > Thank You very much.
> >
> > Sent from Yahoo Mail on Android
> >
> > On Sun, 21 Jun 2020 at 19:48,
> > Hulpke,Alexander<[email protected]> wrote: Dear Forum, Dear
> > Marc David Musyoka,
> >
> > On Jun 20, 2020, at 00:22, David Musyoka <[email protected]> wrote:
> > Deat all,
> > Kind request to this team, i am new to GAP and i wish to be assisted in the
> > following,
> > How can i generate a group e.g "Sp6(2)" as a matrix group by 7 * 7
> > matrices, then generate "S8" as a matrix group inside Sp6(2) by 7 * 7
> > matrices using GAP prog.
> >
> > I wish that am assisted on how to execute the same step by step and the
> > matrix generators for the two groups be listed.
> >
> >
> > Yes, I also often wish that someone would assist me in every step and
> > provide me with the full result.
> > Anyhow, in this case (the algorithm is exponential time and attempts will
> > fail if groups are too large, or if the subgroup to be embedded needs many
> > generators) `IsomorphicSubgroups` seems to do the trick in a few minutes.
> > gap> G:=SP(6,2);Sp(6,2)gap> emb:=IsomorphicSubgroups(G,SymmetricGroup(8));
> > # finds embeddings from S8 into G[ [ (1,8)(2,7)(3,5)(4,6), (1,8,4,7,3)(5,6)
> > ] ->? ? [ <an immutable 6x6 matrix over GF2>, <an immutable 6x6 matrix over
> > GF2> ] ]gap> sub:=Image(emb[1]);<matrix group with 2 generators>
> > So there is one class of subgroups and `sub` is one (not necessarily the
> > nicest one) representative.
> > Of course this is computational overkill. The more sensible way would be to
> > produce the matrix representation (as reduced permutation representation),
> > find the form that it stabilizes, and then conjugate that form to the one
> > used for Sp.
> > Oh, here are the explicit matrix generators :-)
> > gap> GeneratorsOfGroup(sub);[ <an immutable 6x6 matrix over GF2>, <an
> > immutable 6x6 matrix over GF2> ]
> > (You can use `Print` or `Display` on each of them to see them with numbers.)
> > All the best,
> > ? Alexander Hulpke
> > -- Colorado State University,?Department of Mathematics,
> > Weber Building, 1874 Campus?Delivery, Fort Collins, CO 80523-1874, USA
> > email:[email protected],?
> > http://www.math.colostate.edu/~hulpke
> >
> >
> >
> > _______________________________________________
> > Forum mailing list
> > [email protected]
> > https://mail.gap-system.org/mailman/listinfo/forum
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