Small group number 8 of order 32
G is the group 32gp8
The Hall-Senior number of this group is 48.
G has 2 minimal generators, rank 2 and exponent 8.
The centre has rank 1.
The 3 maximal subgroups are:
Q8xC2, 16gp6 (2x).
There is one conjugacy class of maximal elementary abelian
subgroups. Each maximal elementary abelian has rank 2.
This cohomology ring calculation is complete.
Ring structure
| Completion information
| Koszul information
| Restriction information
| Poincaré series
The cohomology ring has 9 generators:
- y1 in degree 1, a nilpotent element
- y2 in degree 1, a nilpotent element
- x1 in degree 2, a nilpotent element
- x2 in degree 2
- w in degree 3, a nilpotent element
- 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.y2 =
0
- y12 =
0
- y1.x2 =
y23
- y1.x1 =
0
- x1.x2 =
y2.w
- x12 =
y22.x1
- y1.w =
y22.x1
- x1.w =
y22.w
- y23.x2 =
0
- y1.u2 =
y23.w
- w2 =
y2.x2.w
+ y22.x22
+ y23.w
- y2.u1 =
0
- y1.u1 =
y23.w
- x2.u1 =
y22.u2
- x1.u2 =
y2.t
+ y22.x2.w
- x1.u1 =
0
- y1.t =
0
- w.u2 =
x2.t
+ y2.x22.w
+ y22.t
- w.u1 =
y22.t
- x1.t =
y22.t
- w.t =
y2.x2.t
+ y22.x2.u2
- u22 =
x25
+ y2.x22.u2
+ y22.x24
+ y22.r
- u1.u2 =
y22.x24
- u12 =
0
- u2.t =
x24.w
+ y2.x1.r
- u1.t =
y22.x23.w
- t2 =
y2.x24.w
+ y22.x25
+ y22.x22.t
+ y22.x1.r
A minimal Gröbner basis for the relations ideal
consists of this minimal generating set, together with the
following redundant relations:
- y24 =
0
- y23.x1 =
0
- y23.u2 =
0
- y23.t =
0
Essential ideal:
There are 2 minimal generators:
Nilradical:
There are 6 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 2 and depth 1.
Here is a homogeneous system of parameters:
- h1 =
r
in degree 8
- h2 =
x2
in degree 2
The first
term h1 forms
a regular sequence of maximum length.
The remaining
term h2 is
annihilated by the class
y23.
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
free of rank 3 as a module over the polynomial algebra
on h1.
These free generators are:
- G1 =
y23
in degree 3
- G2 =
y22.x1
in degree 4
- G3 =
y23.w
in degree 6
The essential ideal squares to zero.
A basis for R/(h1, h2) is as follows.
Carlson's Koszul condition stipulates that this must
be confined to degrees less than 10.
-
1
in degree 0
-
y2
in degree 1
-
y1
in degree 1
-
x1
in degree 2
-
y22
in degree 2
-
w
in degree 3
-
y2.x1
in degree 3
-
y22.x1
in degree 4
-
u2
in degree 5
-
u1
in degree 5
-
t
in degree 6
-
y2.u2
in degree 6
-
y2.t
in degree 7
-
y22.t
in degree 8
A basis for AnnR/(h1)(h2) is as follows.
Carlson's Koszul condition stipulates that this must
be confined to degrees less than 8.
-
y23
in degree 3
-
y23.w
in degree 6
Restriction to maximal subgroup number 1, which is 16gp12
- y1 restricts to
0
- y2 restricts to
y2
+ y1
- x1 restricts to
y12
- x2 restricts to
y32
+ y2.y3
+ y1.y3
- w restricts to
y2.y32
+ y1.y2.y3
+ y12.y3
- u1 restricts to
y1.y2.y33
- u2 restricts to
y35
+ y2.v
+ y1.v
+ y1.y2.y33
+ y12.y2.y32
- t restricts to
y2.y35
+ y12.v
+ y1.y2.y34
+ y12.y34
- r restricts to
v2
+ y34.v
+ y38
+ y12.y36
+ y12.y2.y35
Restriction to maximal subgroup number 2, which is 16gp6
- y1 restricts to
y1
- y2 restricts to
0
- x1 restricts to
y1.y2
- x2 restricts to
y22
+ y1.y2
- w restricts to
w
- u1 restricts to
y1.v
- u2 restricts to
y25
+ y1.v
- t restricts to
y23.w
+ y1.y2.v
- r restricts to
y28
+ v2
Restriction to maximal subgroup number 3, which is 16gp6
- y1 restricts to
y1
- y2 restricts to
y1
- x1 restricts to
y1.y2
- x2 restricts to
y22
+ y1.y2
- w restricts to
w
- u1 restricts to
y1.v
- u2 restricts to
y25
+ y22.w
- t restricts to
y23.w
+ y1.y2.v
- r restricts to
y28
+ v2
Restriction to maximal elementary abelian number 1, which is V4
- y1 restricts to
0
- y2 restricts to
0
- x1 restricts to
0
- x2 restricts to
y22
- w restricts to
0
- u1 restricts to
0
- u2 restricts to
y25
- t restricts to
0
- r restricts to
y28
+ y14.y24
+ y18
Restriction to the greatest central elementary abelian, which is C2
- y1 restricts to
0
- y2 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 + 2t + 2t2
+ 2t3 + t4 + t5
+ 2t6 + t7) /
(1 - t2) (1 - t8)
Back to the groups of order 32