Mathieu groups
In [DK2] B. Külshammer and S. Danz determined
the vertices of the simple modules for the sporadic simple Mathieu groups M22,
M23 and M24 over a field F of
characteristic 2 and 3, respectively. Moreover, the
simple modules for the automorphism group M22:2 of M22 and those
for the covering groups and the bicyclic extensions of M22 have
been investigated.
For each simple module, besides its vertices, also its sources and its Green correspondent
with respect to the normalizer of some vertex have been determined. Most of the
results are due to computer calculations with the computer algebra system
MAGMA. The MAGMA
source code of our algorithms used to perform the vertex computation can be found
here.
In the following we summarize the results presented in
[DK2].
For each group G under consideration, we give a permutation representation.
Permutation representations of the vertices and their normalizers of the simple
FG-modules in characteristic 2 and 3, respectively, are given as well. Furthermore,
we provide representing matrices for the simple modules and, if computed, their Green
correspondents and
sources.
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M22 in characteristic 2
Blocks:
- one block, i.e. the principal block B1 of defect 7
Groups:
- A permutation representation of the group on 22 points is given
here.
- a Sylow 2-subgroup Q of M22;
NM 22(Q)=Q
Modules:
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M22:2 in characteristic 2
Blocks:
- one block, i.e. the principal block B1 of defect 8
Groups:
- A permutation representation of the group on 22 points is given
here.
- a Sylow 2-subgroup P of M22:2,
and a Sylow 2-subgroup Q ≤ P of M22
- NM 22(P)=P=NM 22(Q)
Modules:
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3.M22 in characteristic 2
Blocks:
- five blocks, i.e. the principal block B1 of defect 7, two faithful blocks
B2 and B3=B2* of defect 7, and two blocks of defect 0
Groups:
- A permutation representation of the group on 693 points is given
here.
- a Sylow 2-subgroup P of 3.M22;
N3.M 22(P) is
isomorphic to C3x P
Modules:
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3.M22:2 in characteristic 2
-- coming soon --
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M23 in characteristic 2
Blocks:
- the principal block B1 of defect 7, and two dual blocks of defect 0
Groups:
- A permutation representation of M23 on 23 points is given
here.
- a Sylow 2-subgroup P of M23; N
M23(P)=P
Modules:
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M24 in characteristic 2
Blocks:
- one block, i.e. the principal block B1 of defect 10
Groups:
- A permutation representation of the group on 24 points is given
here.
- a Sylow 2-subgroup P of M24;
NM24(P)=P
- a Sylow 2-subgroup Q of the commutator subgroup of the
maximal subgroup
26:(L3(2)xS3)< M24
- Q has order 512, NM24(Q) has order 3072,
and the quotient group NM24(Q)/Q is isomorphic to S3
Modules:
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M22 in characteristic 3
Blocks:
- the principal block B1 of defect 2 with elementary abelian defect groups
- the block B2 of defect 1, and
- the blocks B3, B4=B3*, B5 of defect 0
Groups:
- A permutation representation of the group on 22 points is given
here.
- a Sylow 3-subgroup P of M22;
NM22(P) has order 72
and is isomorphic to M9
- a defect group C of the block B2
- NM22(C) has order 72.
It has elementary abelian Sylow 3-subgroups of order 9
and dihedral Sylow 2-subgroups of order 8. The quotient group
NM22(C)/C is isomorphic to S4.
An explicit isomorphism is given here.
Modules:
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M22:2 in characteristic 3
Blocks:
- nine blocks, the ones of positive defect are the principal block B1 of defect 2,
and the blocks B2 and B3 of defect 1
Groups:
- A permutation representation of the group on 22 points is given
here.
- a Sylow 3-subgroup P of both M22 and
M22:2
- NM22:2(P)=
NM22(P):2 is isomorphic to
a split extension M9:2.
- a defect group C of the blocks B2 and
B3
- NM22:2(C) has order 144, and
the quotient group
NM22:2(C)/C is isomorphic to S4xC2. An explicit
isomorphism is given here.
Modules:
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2.M22 in characteristic 3
Blocks:
- nine blocks, the faithful ones are the block B3 of defect 2 with elementary
abelian defect groups, and the block B4 of defect 1
Groups:
- A permutation representation of the group on 660 points is
given here.
- a Sylow 3-subgroup P of 2.M22;
N2.M22(P) has order 144
- a defect group C of the block B4
- N2.M22(C) has order 144, and the
quotient group N2.M22(C)/C is isomorphic to S4xC2. An
explicit isomorphism is given here.
Modules:
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4.M22 in characteristic 3
Blocks:
- 15 blocks, the faithful ones are the blocks B5 and
B6=B5* of defect 2 with elementary abelian defect groups
Groups:
- A permutation representation of the group on 4928 points is given
here.
- a Sylow 3-subgroup P of 4.M22;
N4.M22(P) has order 288, and
is isomorphic to a split extension P:(C4:C8)
Modules:
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2.M22.2 in characteristic 3
Blocks:
13 blocks, the faithful ones of positive defect are the block B4 of defect 2 with
elementary abelian defect groups, and the blocks B5 and B6 of defect 1
Groups:
- a permutation representation of the group on 9240 points is given
here.
- a Sylow 3-subgroup P< 2.M22
of 2.M22.2=:G;
the normalizer NG(P) has order
288 and is an extension N2.M22(P).2.
- a common defect group C< 2.M22
of the blocks B5 and B6;
the normalizer
NG(C) has order
288 and is an extension N2.M22(C).2.
Modules:
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4.M22.2 in characteristic 3
-- coming soon --
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M23 in characteristic 3
Blocks:
- seven blocks, the ones of positive defect are the principal block B1 of
defect 2 with elementary abelian defect groups, and the block B4 of defect 1
Groups:
- A permutation representation of the group on 23 points is given
here.
- a Sylow 3-subgroup P of
M23; NM23(P) has
order 144 and
is isomorphic to a split extension M9:2. An explicit isomorphism is given here.
- a defect group C of the block B4
- NM23(C) and is isomorphic to a split extension (A5xC3):2.
The quotient group NM23(C)/C is isomorphic to S5.
Modules:
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M24 in characteristic 3
Blocks:
- six blocks: the principal block B1 of defect 3 with extraspecial defect group
of exponent 3, the blocks B2, B3=B2*,
B4 and B5 of defect 1, and the block B6 of defect 0
Groups:
- A permutation representation of the group of 24 points is given
here.
- a Sylow 3-subgroup P of M24;
NM24(P) has order 216, and
the quotient group NM24(P)/P is isomorphic to the dihedral
group of order 8
- a common defect group C3,1 of the blocks
B2, B3, B5
- NM24(C3,1) is
isomorphic
to L2(7)xS3
- a defect group C3,2 of the block
B4
- NM24(C3,2) is
isomorphic
to an extension 3.S6
- a subgroup Q of M24 of order 9
- NM24(Q) is isomorphic to the
automorphism group
of M9 which is a split extension M9:S3, and the
quotient group NM24(Q)/Q is isomorphic to GL(2,3). An explicit isomorphism is
given here.
Modules:
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References
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