[GAP Forum] A request for finding a group

[Stefan Kohl]

Dear Zohreh,

First observe that we are looking for perfect groups of order 2^6 * |PSL(2,8)| = 64 * 504 = 32256. Since 18^2*98 = 31752 is not much smaller than 32256, some scepticism regarding the existence of such groups with 98 characters of degree 18 seems in order. Anyway, let's continue. -- We can check how many perfect groups of this order there are, up to isomorphism:

    gap> NrPerfectGroups(32256); # Order 32256 is covered by the data
    2                            # in GAP's Perfect Groups Library

So we know that there are precisely two groups we need to have a look at. We get them from the said library, represented as permutation groups (this being most convenient here):

    gap> G := PerfectGroup(IsPermGroup,32256,1);
    L2(8) 2^6
    gap> H := PerfectGroup(IsPermGroup,32256,2);
    L2(8) N 2^6

Now we can compute the lists of normal subgroups of both groups ...

    gap> normsG := NormalSubgroups(G);
    [ Group(()), <permutation group of size 64 with 6 generators>, L2(8) 2^6 ]
    gap> normsH := NormalSubgroups(H);
    [ Group(()), <permutation group of size 64 with 6 generators>, L2(8) N 2^6 ]

... and have a look at the quotients:

    gap> StructureDescription(G/normsG[2]);
    "PSL(2,8)"
    gap> StructureDescription(H/normsH[2]);
    "PSL(2,8)"

So far, both groups fulfil our criteria. -- But now let's compute the character degrees of our two groups:

    gap> CharacterDegrees(G);
    [ [ 1, 1 ], [ 7, 4 ], [ 8, 1 ], [ 9, 3 ], [ 63, 8 ] ]
    gap> CharacterDegrees(H);
    [ [ 1, 1 ], [ 7, 4 ], [ 8, 1 ], [ 9, 3 ], [ 63, 8 ] ]

As we see, none of the groups has an irreducible character of degree 18. Therefore -- unless I have misread the conditions -- a group with the desired properties does not exist.

Hope this helps,

    Stefan


________________________________
From: zohreh sayanjali <zohrehsayanjali at gmail.com>
Sent: Friday, March 15, 2019 3:22:09 PM
To: GAP Forum
Subject: [GAP Forum] A request for finding a group

Dear  GAP Forum,


I am trying  to figure out  if  there is  any perfect group  G whose normal
minimal subgroup, say  N, is an elementary abelian 2-group of order  2^6,
G/N is isomorphic to L2(8) and cd(G) = {1, 18, 9, 8, 7}, where the number
of irreducible characters whose degrees are 18 is 98.

Unfortunately, I do not know how to construct such a group and figure out
about its character degrees. I would really appreciate it if you help me to
find them, if there exists any.
Regards,
zohreh sayanjali.
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