This is a file the includes all of the data obtained from MAGMA for the paper
“A generalization of a theorem of Huppert and Manz” by Mark L. Lewis that appeared in the Journal of Algebra and its Applications.
--------------------------------------------------------------------------------
Here is the code I used
a7 := AlternatingGroup (7);
cta7 := CharacterTable (a7);
lat := SubgroupLattice (a7);
t := [ ];
s := [ ];
for i in [2..9] do
ttemp := [ ];
stemp := [ ];
chi := cta7[i];
for j in [1..39] do
temp := lat[j];
tct := CharacterTable (temp);
tchi := Restriction (chi,temp);
ttemp[j] := InnerProduct (tchi,tct[1]);
if ttemp[j] gt 0 then
Append(~stemp,j);
end if;
end for;
t[i] := ttemp;
s[i] := stemp;
end for;
--------------------------------------------------------------------------------
This gave the following results: (The numbers correspond to subgroups of
The lattice for A7.
> t;
[
undef,
[ 6, 4, 4, 2, 2, 0, 3, 3, 2, 2, 3, 2, 2, 2, 0, 2, 1, 3, 1, 1, 1, 1, 2, 2, 1,
1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 0, 0, 1 ],
[ 10, 4, 4, 4, 2, 1, 1, 1, 2, 2, 1, 1, 2, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0,
0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 10, 4, 4, 4, 2, 1, 1, 1, 2, 2, 1, 1, 2, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0,
0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 14, 8, 4, 6, 2, 2, 5, 5, 4, 2, 3, 4, 2, 2, 2, 3, 3, 1, 3, 1, 1, 3, 2, 2,
1, 2, 1, 1, 2, 2, 1, 1, 0, 1, 1, 0, 1, 1, 0 ],
[ 14, 8, 6, 4, 2, 2, 5, 5, 4, 4, 4, 3, 2, 2, 0, 3, 1, 3, 1, 3, 2, 1, 3, 2,
1, 1, 2, 2, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0 ],
[ 15, 7, 7, 5, 3, 3, 3, 3, 3, 3, 3, 2, 3, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1,
0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0 ],
[ 21, 11, 5, 7, 5, 3, 6, 6, 5, 3, 3, 4, 1, 3, 1, 3, 2, 0, 2, 2, 1, 2, 2, 1,
1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0 ],
[ 35, 17, 11, 11, 7, 5, 8, 8, 9, 5, 5, 5, 3, 3, 1, 4, 2, 2, 2, 2, 3, 2, 2,
1, 2, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0 ]
]
>
> s;\
[
undef,
[ 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22,
23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 39 ],
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 17, 18, 19, 20, 21, 22, 31
],
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 17, 18, 19, 20, 21, 22, 31
],
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21,
22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 37, 38 ],
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22,
23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 35, 36 ],
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21,
22, 23, 24, 27, 31, 33 ],
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22,
23, 24, 25, 26, 28, 29, 30, 34 ],
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21,
22, 23, 24, 25, 26, 27, 28, 29, 30, 32 ]
]
>
--------------------------------------------------------------------------------
Here is the lattice for A7:
[ 1] Order 1
Length 1 Maximal
Subgroups:
---
[ 2] Order
2 Length 105 Cyclic
Maximal Subgroups: 1
[ 3] Order
3 Length 35 Cyclic
Maximal Subgroups: 1
[ 4] Order
3 Length 140 Cyclic
Maximal Subgroups: 1
[ 5] Order
5 Length 126 Cyclic
Maximal Subgroups: 1
[ 6] Order
7 Length 120 Cyclic
Maximal Subgroups: 1
---
[ 7] Order
4 Length 35 Elem Abel
Maximal Subgroups: 2
[ 8] Order
4 Length 105 Elem Abel
Maximal Subgroups: 2
[ 9] Order
4 Length 315 Cyclic
Maximal Subgroups: 2
[10] Order
6 Length 105 Cyclic
Maximal Subgroups: 2 3
[11] Order
6 Length 210 Soluble
Maximal Subgroups: 2 3
[12] Order
6 Length 420 Soluble
Maximal Subgroups: 2 4
[13] Order
9 Length 70 Elem Abel
Maximal Subgroups: 3 4
[14] Order
10 Length 126 Soluble
Maximal Subgroups: 2 5
[15] Order
21 Length 120 Soluble
Maximal Subgroups: 4 6
---
[16] Order
8 Length 315 Nilpotent
Maximal Subgroups: 7 8 9
[17] Order
12 Length 35 Soluble
Maximal Subgroups: 4 7
[18] Order
12 Length 35 Soluble
Maximal Subgroups: 3 7
[19] Order
12 Length 35 Soluble
Maximal Subgroups: 4 7
[20] Order
12 Length 35 Abelian Maximal Subgroups: 7 10
[21] Order
12 Length 105 Soluble
Maximal Subgroups: 9 10
[22] Order
12 Length 105 Soluble
Maximal Subgroups: 4 8
[23] Order
12 Length 105 Soluble
Maximal Subgroups: 8 10 11
[24] Order
18 Length 70 Soluble
Maximal Subgroups: 11 12 13
[25] Order
20 Length 126 Soluble
Maximal Subgroups: 9 14
---
[26] Order
24 Length 105 Soluble
Maximal Subgroups: 12 16 17
[27] Order
24 Length 105 Soluble
Maximal Subgroups: 11 16 18
[28] Order
24 Length 105
Soluble Maximal Subgroups: 16
20 21 23
[29] Order
24 Length 105 Soluble
Maximal Subgroups: 12 16 22
[30] Order
24 Length 105 Soluble
Maximal Subgroups: 12 16 19
[31] Order
36 Length 35 Soluble
Maximal Subgroups: 13 17 18 19 20
[32] Order
36 Length 70 Soluble
Maximal Subgroups: 9 24
[33] Order
60 Length 21 Simple
Maximal Subgroups: 11 14 18
[34] Order
60 Length 42 Simple
Maximal Subgroups: 12 14 22
---
[35] Order
72 Length 35 Soluble
Maximal Subgroups: 24 26 27 28 30 31
[36] Order
120 Length 21 Nonsolvable
Maximal Subgroups: 23 25 27 33
[37] Order
168 Length 15 Simple
Maximal Subgroups: 15 26 29
[38] Order
168 Length 15 Simple
Maximal Subgroups: 15 29 30
---
[39] Order
360 Length 7 Simple
Maximal Subgroups: 27 29 32 33 34
---
[40] Order
2520 Length 1 Simple
Maximal Subgroups: 35 36 37 38 39
--------------------------------------------------------------------------------
Finally, we include the character table for A7:
Character Table of Group a7
---------------------------
---------------------------------------------
Class | 1 2 3 4 5 6 7 8 9
Size | 1 105 70 280 630 504 210 360 360
Order | 1 2 3 3 4 5 6 7 7
---------------------------------------------
p = 2 1 1 3 4 2 6 3 8 9
p = 3 1 2 1 1 5 6 2 9 8
p = 5 1 2 3 4 5 1 7 9 8
p = 7 1 2 3 4 5 6 7 1 1
---------------------------------------------
X.1 + 1 1 1 1 1 1 1 1 1
X.2 + 6 2 3 0 0 1 -1 -1 -1
X.3 0 10 -2 1 1 0 0 1 Z1 Z1#3
X.4 0 10 -2 1 1 0 0 1 Z1#3 Z1
X.5 + 14 2 -1 2 0 -1 -1 0 0
X.6 + 14 2 2 -1 0 -1 2 0 0
X.7 + 15 -1 3 0 -1 0 -1 1 1
X.8 + 21 1 -3 0 -1 1 1 0 0
X.9 + 35 -1 -1 -1 1 0 -1 0 0
Explanation of Symbols:
-----------------------
# denotes algebraic conjugation, that is,
#k indicates replacing the root of unity w by w^k
Z1 = (CyclotomicField(7:Sparse := true)) ! [ RationalField() | 0, 1, 1, 0,
1, 0 ]