# Re: [GAP Forum] Can GAP3 program construct the character table of projective general linear groups?

```On Sun, May 31, 2015 at 04:48:59PM +0800, 杨高 wrote:

Do you work with GAP3?
Is it really necessary for you?```
```
>
> Is this the correct and most concise way to construct PGL(2,3) with GAP?
In GAP4 you can just do
gap> ProjectiveGeneralLinearGroup(2,3);
Group([ (3,4), (1,2,4) ])

>
>
> > W:=GeneralLinearGroup(2,3);;
> > H:=Subgroup(W,[[[Z(3),0*Z(3)],[0*Z(3),Z(3)]]]);;
> > B:=W/H;;
>
>
> Morevoer, I want to construct the character table of this group. But there is
> ian error.
>
>
> >Display(CharTable(B));
> Error, sorry, I don't know how to construct the table of <grp>
>  in
> CharTable( B ) called from
> main loop

well, GAP4 has no difficulty here:
gap> H:=Subgroup(W,[[[Z(3),0*Z(3)],[0*Z(3),Z(3)]]]);;
gap> B:=W/H;;
gap> Display(CharacterTable(B));
CT2

2  3  2  .  3  2
3  1  .  1  .  .

1a 2a 3a 2b 4a
2P 1a 1a 3a 1a 2b
3P 1a 2a 1a 2b 4a

X.1     1  1  1  1  1
X.2     1 -1  1  1 -1
X.3     2  . -1  2  .
X.4     3 -1  . -1  1
X.5     3  1  . -1 -1
gap>

> So there are question I would like to ask. For constructing character tables,
> what kind of groups can GAP deal with? Can GAP construct the character table
> of PGL(n,m)? If the answer is yes, then how can I do this with it?
You can contruct the character table of a given group, but it gets slower as
the
size etc grows. How far do you need to go with n and m?

There are some infinite families of groups for which GAP can construct
character tables directly,
see
http://www.gap-system.org/Manuals/pkg/ctbllib/doc/chap4.html#X81E3F9A384365282
E.g. for n=2 one can have tables of the PSL groups.

HTH,
Dmitrii

>
>
> Thanks to everyone.
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