Small group number 102 of order 64
G is the group 64gp102
The Hall-Senior number of this group is 120.
G has 3 minimal generators, rank 3 and exponent 8.
The centre has rank 1.
The 7 maximal subgroups are:
32gp11 (4x), 32gp24, 32gp37, 32gp48.
There are 2 conjugacy classes of maximal
elementary abelian subgroups. Their ranks are:
3, 3.
This cohomology ring calculation is complete.
Ring structure
| Completion information
| Koszul information
| Restriction information
| Poincaré series
The cohomology ring has 10 generators:
- y1 in degree 1, a nilpotent element
- y2 in degree 1
- y3 in degree 1
- x1 in degree 2, a nilpotent element
- x2 in degree 2
- w in degree 3
- u1 in degree 5, a nilpotent element
- u2 in degree 5
- t in degree 6, a nilpotent element
- r in degree 8, a regular element
There are 27 minimal relations:
- y1.y3 =
0
- y12 =
0
- y3.x2 =
y22.y3
+ y1.x2
- y1.x1 =
0
- y3.w =
y23.y3
+ y22.x1
+ y1.w
- x1.x2 =
y22.x1
+ y1.w
+ y1.y2.x2
- x12 =
0
- x1.w =
y23.x1
+ y1.y2.w
- y1.y22.x2 =
0
- w2 =
y24.x2
+ y1.y2.x22
+ y1.y22.w
- y3.u1 =
y1.u1
- y1.u2 =
y1.y2.x22
+ y1.y22.w
+ y1.u1
- y1.x2.w =
y1.u1
- x2.u2 =
x22.w
+ y2.x23
+ y22.u2
+ y24.w
+ y25.x2
+ y22.u1
+ y1.x23
+ y1.y2.u1
- x1.u2 =
y3.t
+ y24.y3.x1
- x1.u1 =
y1.y2.u1
- y1.t =
0
- w.u2 =
y2.x22.w
+ y23.u2
+ y25.w
+ y22.t
+ y26.x1
+ y1.y2.x23
+ y1.y24.w
+ y1.x2.u1
- w.u1 =
x2.t
+ y22.t
+ y23.u1
+ y1.y24.w
- x1.t =
0
- w.t =
y23.t
+ y24.u1
+ y1.y25.w
+ y1.y2.x2.u1
- u22 =
y2.y34.u2
+ y22.x24
+ y22.y33.u2
+ y24.x23
+ y24.y3.u2
+ y25.y35
+ y26.x22
+ y26.y34
+ y27.y33
+ y28.x2
+ y28.y32
+ y32.r
+ y22.y32.t
+ y25.y33.x1
+ y26.y32.x1
+ y28.x1
+ u12
- u1.u2 =
x22.t
+ y2.x22.u1
+ y24.t
+ y25.u1
+ y1.y26.w
+ y1.x22.u1
- y1.y2.x24 =
u12
- u2.t =
y2.x22.t
+ y2.y34.t
+ y22.y33.t
+ y24.x2.u1
+ y25.t
+ y26.u1
+ y27.y32.x1
+ y28.y3.x1
+ y1.y27.w
+ y3.x1.r
+ y1.y2.x22.u1
- u1.t =
y1.y2.x22.u1
- t2 =
0
A minimal Gröbner basis for the relations ideal
consists of this minimal generating set, together with the
following redundant relations:
- y1.y22.u1 =
0
- y2.u12 =
0
- y1.u12 =
0
- u14 =
0
Essential ideal:
Zero ideal
Nilradical:
There are 4 minimal generators:
This cohomology ring was obtained from a calculation
out to degree 12. The cohomology ring approximation
is stable from degree 12 onwards, and
Carlson's tests detect stability from degree 12
onwards.
This cohomology ring has dimension 3 and depth 2.
Here is a homogeneous system of parameters:
- h1 =
r
in degree 8
- h2 =
x2
+ y32
+ y22
in degree 2
- h3 =
x2
+ y32
in degree 2
The first
2 terms h1, h2 form
a regular sequence of maximum length.
The remaining
term h3 is
annihilated by the class
y1.y22.
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 12.
-
1
in degree 0
-
y3
in degree 1
-
y2
in degree 1
-
y1
in degree 1
-
y32
in degree 2
-
y2.y3
in degree 2
-
x1
in degree 2
-
y1.y2
in degree 2
-
w
in degree 3
-
y2.y32
in degree 3
-
y3.x1
in degree 3
-
y2.x1
in degree 3
-
y2.w
in degree 4
-
y2.y3.x1
in degree 4
-
y1.w
in degree 4
-
u2
in degree 5
-
u1
in degree 5
-
y1.y2.w
in degree 5
-
y3.u2
in degree 6
-
y2.u2
in degree 6
-
t
in degree 6
-
y2.u1
in degree 6
-
y2.y3.u2
in degree 7
-
y3.t
in degree 7
-
y2.t
in degree 7
-
y32.t
in degree 8
-
y2.y3.t
in degree 8
-
y2.y32.t
in degree 9
A basis for AnnR/(h1, h2)(h3) is as follows.
Carlson's Koszul condition stipulates that this must
be confined to degrees less than 10.
-
y1.y22
in degree 3
-
y1.y23
in degree 4
-
y1.u1
in degree 6
-
y1.y2.u1
in degree 7
Restriction to maximal subgroup number 1, which is 32gp48
- y1 restricts to
0
- y2 restricts to
y3
+ y2
+ y4
- y3 restricts to
y3
- x1 restricts to
y1.y3
+ y12
- x2 restricts to
y32
+ y12
+ y42
- w restricts to
y33
+ y1.y32
+ y12.y2
+ y13
+ y32.y4
+ y22.y4
+ y3.y42
+ y1.y42
+ y43
- u1 restricts to
y1.y22.y42
+ y13.y42
- u2 restricts to
y14.y2
+ y3.v
+ y34.y4
+ y24.y4
+ y1.y33.y4
+ y13.y2.y4
+ y23.y42
+ y1.y32.y42
+ y12.y2.y42
+ y13.y42
+ y32.y43
+ y1.y3.y43
+ y12.y43
+ y1.y44
- t restricts to
y1.y35
+ y16
+ y1.y3.v
+ y1.y34.y4
+ y12.v
+ y15.y4
+ y1.y23.y42
+ y13.y2.y42
+ y1.y32.y43
+ y12.y2.y43
+ y1.y3.y44
+ y1.y2.y44
- r restricts to
y18
+ y34.v
+ y37.y4
+ y1.y33.v
+ y1.y36.y4
+ y13.y2.v
+ y14.v
+ y17.y4
+ v2
+ y33.y4.v
+ y1.y25.y42
+ y32.y42.v
+ y35.y43
+ y22.y42.v
+ y1.y3.y42.v
+ y1.y34.y43
+ y14.y2.y43
+ y15.y43
+ y24.y44
+ y1.y23.y44
+ y44.v
+ y23.y45
+ y1.y32.y45
+ y1.y22.y45
+ y32.y46
+ y22.y46
+ y12.y46
+ y2.y47
+ y1.y47
+ y48
Restriction to maximal subgroup number 2, which is 32gp37
- y1 restricts to
y1
- y2 restricts to
y3
- y3 restricts to
0
- x1 restricts to
y1.y2
- x2 restricts to
y22
+ y1.y2
- w restricts to
y2.y32
+ w
+ y1.y2.y3
+ y1.y32
- u1 restricts to
y22.w
+ y1.v
+ y32.w
+ y1.y2.y33
- u2 restricts to
y24.y3
+ y23.y32
+ y22.y33
+ y2.y34
+ y22.w
+ y1.v
+ y1.y2.y33
+ y1.y34
- t restricts to
y1.y2.v
+ y2.y32.w
+ y1.y3.v
+ y33.w
- r restricts to
y28
+ v2
+ y22.y32.v
+ y25.y33
+ y24.y34
+ y34.v
+ y23.y32.w
+ y22.y33.w
+ y1.y33.v
+ y35.w
+ y1.y2.y36
Restriction to maximal subgroup number 3, which is 32gp24
- y1 restricts to
y1
- y2 restricts to
y3
- y3 restricts to
y1
- x1 restricts to
y22
+ y1.y2
- x2 restricts to
x
- w restricts to
w
+ y3.x
+ y2.y32
+ y2.x
+ y23
- u1 restricts to
y2.y32.x
+ y1.v
+ y2.x2
- u2 restricts to
y32.w
+ x.w
+ y2.y34
+ y2.y32.x
+ y2.x2
+ y1.x2
- t restricts to
y2.x.w
+ y2.y33.x
+ y2.y3.x2
+ y1.y2.v
+ y22.x2
- r restricts to
v2
+ y3.x2.w
+ y32.x.v
+ y34.x2
+ x2.v
+ y32.x3
+ x4
+ y2.y32.x.w
+ y2.y35.x
+ y2.y33.x2
+ y22.x3
Restriction to maximal subgroup number 4, which is 32gp11
- y1 restricts to
y1
- y2 restricts to
0
- y3 restricts to
y2
- x1 restricts to
x1
- x2 restricts to
x2
- w restricts to
w
- u1 restricts to
x2.w
+ y1.v
- u2 restricts to
y2.v
+ x2.w
+ y1.x22
+ y1.v
- t restricts to
x1.v
- r restricts to
x24
+ v2
Restriction to maximal subgroup number 5, which is 32gp11
- y1 restricts to
y1
- y2 restricts to
y1
- y3 restricts to
y2
- x1 restricts to
x1
- x2 restricts to
x2
- w restricts to
w
+ y1.x2
- u1 restricts to
x2.w
+ y1.x22
+ y1.v
- u2 restricts to
y2.v
+ x2.w
+ y1.x22
+ y1.v
- t restricts to
x1.v
- r restricts to
x24
+ v2
Restriction to maximal subgroup number 6, which is 32gp11
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
y2
- x1 restricts to
x1
- x2 restricts to
x2
+ y22
- w restricts to
y23
+ w
+ y2.x1
+ y1.x2
- u1 restricts to
x2.w
+ y1.x22
+ y1.v
- u2 restricts to
y2.v
+ x2.w
+ y1.x22
+ y1.v
- t restricts to
y24.x1
+ x1.v
- r restricts to
x24
+ y24.v
+ v2
+ y22.x1.v
Restriction to maximal subgroup number 7, which is 32gp11
- y1 restricts to
y1
- y2 restricts to
y2
+ y1
- y3 restricts to
y2
- x1 restricts to
x1
- x2 restricts to
x2
+ y22
- w restricts to
y23
+ w
+ y2.x1
- u1 restricts to
x2.w
+ y1.v
- u2 restricts to
y2.v
+ x2.w
+ y1.x22
+ y1.v
- t restricts to
y24.x1
+ x1.v
- r restricts to
x24
+ y24.v
+ v2
+ y22.x1.v
Restriction to maximal elementary abelian number 1, which is V8
- y1 restricts to
0
- y2 restricts to
y2
- y3 restricts to
0
- x1 restricts to
0
- x2 restricts to
y32
- w restricts to
y22.y3
- u1 restricts to
0
- u2 restricts to
y2.y34
+ y22.y33
+ y23.y32
+ y24.y3
- t restricts to
0
- r restricts to
y38
+ y23.y35
+ y24.y34
+ y12.y22.y34
+ y12.y24.y32
+ y14.y34
+ y14.y22.y32
+ y14.y24
+ y18
Restriction to maximal elementary abelian number 2, which is V8
- y1 restricts to
0
- y2 restricts to
y2
- y3 restricts to
y3
- x1 restricts to
0
- x2 restricts to
y22
- w restricts to
y23
- u1 restricts to
0
- u2 restricts to
y23.y32
+ y24.y3
+ y12.y33
+ y14.y3
- t restricts to
0
- r restricts to
y24.y34
+ y25.y33
+ y26.y32
+ y28
+ y12.y2.y35
+ y12.y22.y34
+ y12.y24.y32
+ y14.y34
+ y14.y2.y33
+ y14.y22.y32
+ y14.y24
+ y18
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
- u1 restricts to
0
- u2 restricts to
0
- t restricts to
0
- r restricts to
y8
(1 + 3t + 4t2
+ 4t3 + 3t4 + 2t5
+ 3t6 + 3t7 + t8) /
(1 - t2)2 (1 - t8)
Back to the groups of order 64