Small group number 134 of order 64
G = Syl2(M12) is Sylow 2-subgroup of M_12
The Hall-Senior number of this group is 261.
G has 3 minimal generators, rank 3 and exponent 8.
The centre has rank 1.
The 7 maximal subgroups are:
32gp11 (2x), 32gp34, 32gp43 (2x), E32+, 32gp7.
There are 5 conjugacy classes of maximal
elementary abelian subgroups. Their ranks are:
3, 3, 3, 3, 3.
This cohomology ring calculation is complete.
Ring structure
| Completion information
| Koszul information
| Restriction information
| Poincaré series
The cohomology ring has 7 generators:
- y1 in degree 1
- y2 in degree 1
- y3 in degree 1
- x1 in degree 2
- x2 in degree 2
- w in degree 3
- v in degree 4, a regular element
There are 9 minimal relations:
- y2.y3 =
0
- y1.y3 =
0
- y3.x2 =
0
- y1.x1 =
0
- x22 =
y22.x2
+ y1.y2.x2
- x1.x2 =
y2.w
- y1.w =
0
- x2.w =
y22.w
- w2 =
y3.x1.w
+ y33.w
+ y2.x1.w
+ y32.v
This minimal generating set constitutes a Gröbner
basis for the relations ideal.
Essential ideal:
Zero ideal
Nilradical:
Zero ideal
This cohomology ring was obtained from a calculation
out to degree 10. The cohomology ring approximation
is stable from degree 6 onwards, and
Carlson's tests detect stability from degree 8
onwards.
This cohomology ring has dimension 3 and depth 3.
Here is a homogeneous system of parameters:
- h1 =
v
in degree 4
- h2 =
y32
+ y22
in degree 2
- h3 =
x1
+ y12
in degree 2
The first
3 terms h1, h2, h3 form
a regular sequence of maximum length.
The first
term h1 forms
a complete Duflot regular sequence.
That is, its restriction to the greatest central elementary abelian
subgroup forms a regular sequence of maximal length.
The ideal of essential classes is
the zero ideal.
The essential ideal squares to zero.
A basis for R/(h1, h2, h3) is as follows.
Carlson's Koszul condition stipulates that this must
be confined to degrees less than 8.
-
1
in degree 0
-
y3
in degree 1
-
y2
in degree 1
-
y1
in degree 1
-
x2
in degree 2
-
y22
in degree 2
-
y1.y2
in degree 2
-
y12
in degree 2
-
w
in degree 3
-
y2.x2
in degree 3
-
y1.x2
in degree 3
-
y12.y2
in degree 3
-
y3.w
in degree 4
-
y1.y2.x2
in degree 4
-
y12.x2
in degree 4
-
y12.y2.x2
in degree 5
Restriction to maximal subgroup number 1, which is 32gp49
- y1 restricts to
y3
- y2 restricts to
y4
+ y2
+ y1
- y3 restricts to
0
- x1 restricts to
y2.y4
+ y1.y2
- x2 restricts to
y42
+ y2.y3
+ y22
+ y1.y4
+ y1.y2
- w restricts to
y2.y42
+ y22.y4
+ y1.y2.y3
- v restricts to
y44
+ y23.y3
+ y24
+ y1.y43
+ y1.y23
+ y12.y42
+ y12.y2.y3
+ y12.y22
+ y13.y4
+ y13.y2
+ v
Restriction to maximal subgroup number 2, which is 32gp34
- y1 restricts to
0
- y2 restricts to
y2
+ y1
- y3 restricts to
y3
+ y2
+ y1
- x1 restricts to
y22
+ y12
+ x2
+ x1
- x2 restricts to
y22
+ y1.y2
- w restricts to
y23
+ y3.x2
+ y2.x1
+ y1.x2
- v restricts to
y24
+ y13.y2
+ y32.x2
+ y22.x2
+ y22.x1
+ y1.y2.x2
+ y1.y2.x1
+ x1.x2
Restriction to maximal subgroup number 3, which is 32gp7
- y1 restricts to
y1
- y2 restricts to
y2
+ y1
- y3 restricts to
y1
- x1 restricts to
x1
- x2 restricts to
x2
+ y22
- w restricts to
w2
+ y2.x1
- v restricts to
v1
+ x12
+ y2.w2
+ y24
+ v2
Restriction to maximal subgroup number 4, which is 32gp43
- y1 restricts to
y3
- y2 restricts to
0
- y3 restricts to
y1
- x1 restricts to
y22
+ y1.y2
- x2 restricts to
y2.y3
- w restricts to
w
+ y1.y22
+ y12.y2
- v restricts to
y2.w
+ y1.w
+ y12.y22
+ y13.y2
+ v
Restriction to maximal subgroup number 5, which is 32gp11
- y1 restricts to
y2
+ y1
- y2 restricts to
y1
- y3 restricts to
y1
- x1 restricts to
x2
- x2 restricts to
x1
- w restricts to
w
- v restricts to
v
Restriction to maximal subgroup number 6, which is 32gp43
- y1 restricts to
y3
- y2 restricts to
y3
- y3 restricts to
y1
- x1 restricts to
y22
+ y1.y2
- x2 restricts to
y2.y3
- w restricts to
w
- v restricts to
y2.w
+ y2.y33
+ y1.w
+ v
Restriction to maximal subgroup number 7, which is 32gp11
- y1 restricts to
y2
+ y1
- y2 restricts to
y2
- y3 restricts to
y1
- x1 restricts to
x2
- x2 restricts to
x1
- w restricts to
w
- v restricts to
v
+ y22.x1
Restriction to maximal elementary abelian number 1, which is V8
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
y2
- x1 restricts to
y32
+ y2.y3
- x2 restricts to
0
- w restricts to
y1.y22
+ y12.y2
- v restricts to
y1.y2.y32
+ y1.y22.y3
+ y1.y23
+ y12.y32
+ y12.y2.y3
+ y14
Restriction to maximal elementary abelian number 2, which is V8
- y1 restricts to
y2
- y2 restricts to
y3
- y3 restricts to
0
- x1 restricts to
0
- x2 restricts to
0
- w restricts to
0
- v restricts to
y1.y2.y32
+ y1.y22.y3
+ y12.y32
+ y12.y2.y3
+ y12.y22
+ y14
Restriction to maximal elementary abelian number 3, which is V8
- y1 restricts to
y2
- y2 restricts to
y3
- y3 restricts to
0
- x1 restricts to
0
- x2 restricts to
y32
+ y2.y3
- w restricts to
0
- v restricts to
y34
+ y2.y33
+ y1.y2.y32
+ y1.y22.y3
+ y12.y32
+ y12.y2.y3
+ y12.y22
+ y14
Restriction to maximal elementary abelian number 4, which is V8
- y1 restricts to
0
- y2 restricts to
y2
- y3 restricts to
0
- x1 restricts to
y32
+ y2.y3
- x2 restricts to
0
- w restricts to
0
- v restricts to
y1.y2.y32
+ y1.y22.y3
+ y12.y32
+ y12.y2.y3
+ y12.y22
+ y14
Restriction to maximal elementary abelian number 5, which is V8
- y1 restricts to
0
- y2 restricts to
y3
- y3 restricts to
0
- x1 restricts to
y2.y3
+ y22
- x2 restricts to
y32
- w restricts to
y2.y32
+ y22.y3
- v restricts to
y34
+ y1.y2.y32
+ y1.y22.y3
+ y12.y32
+ y12.y2.y3
+ y12.y22
+ y14
Restriction to the greatest central elementary abelian, which is C2
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
0
- x1 restricts to
0
- x2 restricts to
0
- w restricts to
0
- v restricts to
y4
(1 + 3t + 4t2
+ 4t3 + 3t4 + t5) /
(1 - t2)2 (1 - t4)
Back to the groups of order 64