Small group number 239 of order 64
G = Q8xQ8 is Direct product Q8 x Q8
The Hall-Senior number of this group is 156.
G has 4 minimal generators, rank 2 and exponent 4.
The centre has rank 2.
The 15 maximal subgroups are:
32gp26 (6x), 32gp35 (9x).
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 6 generators:
- y1 in degree 1, a nilpotent element
- y2 in degree 1, a nilpotent element
- y3 in degree 1, a nilpotent element
- y4 in degree 1, a nilpotent element
- v1 in degree 4, a regular element
- v2 in degree 4, a regular element
There are 4 minimal relations:
- y42 =
y2.y4
+ y22
+ y1.y2
+ y12
- y32 =
y1.y3
+ y12
- y23 =
0
- y13 =
0
This minimal generating set constitutes a Gröbner
basis for the relations ideal.
Essential ideal:
There are 4 minimal generators:
-
y1.y22.y3.y4
-
y12.y2.y3.y4
-
y12.y22.y4
-
y12.y22.y3
Nilradical:
There are 4 minimal generators:
This cohomology ring was obtained from a calculation
out to degree 8. The cohomology ring approximation
is stable from degree 4 onwards, and
Carlson's tests detect stability from degree 8
onwards.
This cohomology ring has dimension 2 and depth 2.
Here is a homogeneous system of parameters:
- h1 =
v1
in degree 4
- h2 =
v2
in degree 4
The first
2 terms h1, h2 form
a regular sequence of maximum length.
The first
2 terms h1, h2 form
a complete Duflot regular sequence.
That is, their restrictions to the greatest central elementary abelian
subgroup form a regular sequence of maximal length.
The ideal of essential classes is
free of rank 5 as a module over the polynomial algebra
on h1, h2.
These free generators are:
- G1 =
y1.y22.y3.y4
in degree 5
- G2 =
y12.y2.y3.y4
in degree 5
- G3 =
y12.y22.y4
in degree 5
- G4 =
y12.y22.y3
in degree 5
- G5 =
y12.y22.y3.y4
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 8.
-
1
in degree 0
-
y4
in degree 1
-
y3
in degree 1
-
y2
in degree 1
-
y1
in degree 1
-
y3.y4
in degree 2
-
y2.y4
in degree 2
-
y2.y3
in degree 2
-
y22
in degree 2
-
y1.y4
in degree 2
-
y1.y3
in degree 2
-
y1.y2
in degree 2
-
y12
in degree 2
-
y2.y3.y4
in degree 3
-
y22.y4
in degree 3
-
y22.y3
in degree 3
-
y1.y3.y4
in degree 3
-
y1.y2.y4
in degree 3
-
y1.y2.y3
in degree 3
-
y1.y22
in degree 3
-
y12.y4
in degree 3
-
y12.y3
in degree 3
-
y12.y2
in degree 3
-
y22.y3.y4
in degree 4
-
y1.y2.y3.y4
in degree 4
-
y1.y22.y4
in degree 4
-
y1.y22.y3
in degree 4
-
y12.y3.y4
in degree 4
-
y12.y2.y4
in degree 4
-
y12.y2.y3
in degree 4
-
y12.y22
in degree 4
-
y1.y22.y3.y4
in degree 5
-
y12.y2.y3.y4
in degree 5
-
y12.y22.y4
in degree 5
-
y12.y22.y3
in degree 5
-
y12.y22.y3.y4
in degree 6
Restriction to maximal subgroup number 1, which is 32gp26
- y1 restricts to
0
- y2 restricts to
y2
- y3 restricts to
y3
+ y1
- y4 restricts to
y1
- v1 restricts to
x2
- v2 restricts to
v
Restriction to maximal subgroup number 2, which is 32gp26
- y1 restricts to
y2
+ y1
- y2 restricts to
0
- y3 restricts to
y2
- y4 restricts to
y3
+ y2
- v1 restricts to
v
- v2 restricts to
x2
Restriction to maximal subgroup number 3, which is 32gp35
- y1 restricts to
y1
- y2 restricts to
y1
- y3 restricts to
y3
+ y2
- y4 restricts to
y2
- v1 restricts to
v
+ x2
+ y12.x
- v2 restricts to
v
Restriction to maximal subgroup number 4, which is 32gp26
- y1 restricts to
y3
+ y1
- y2 restricts to
y2
- y3 restricts to
0
- y4 restricts to
y3
+ y2
- v1 restricts to
x2
- v2 restricts to
v
Restriction to maximal subgroup number 5, which is 32gp26
- y1 restricts to
y3
+ y1
- y2 restricts to
y1
- y3 restricts to
y3
+ y1
- y4 restricts to
y3
+ y2
+ y1
- v1 restricts to
x2
- v2 restricts to
v
Restriction to maximal subgroup number 6, which is 32gp35
- y1 restricts to
y2
+ y1
- y2 restricts to
y1
- y3 restricts to
y1
- y4 restricts to
y3
- v1 restricts to
v
- v2 restricts to
v
+ x2
+ y12.x
Restriction to maximal subgroup number 7, which is 32gp35
- y1 restricts to
y2
+ y1
- y2 restricts to
y1
- y3 restricts to
y2
- y4 restricts to
y3
- v1 restricts to
v
- v2 restricts to
v
+ x2
+ y12.x
Restriction to maximal subgroup number 8, which is 32gp35
- y1 restricts to
y1
- y2 restricts to
y2
+ y1
- y3 restricts to
y3
+ y2
+ y1
- y4 restricts to
0
- v1 restricts to
v
+ x2
+ y12.x
- v2 restricts to
v
Restriction to maximal subgroup number 9, which is 32gp26
- y1 restricts to
y2
- y2 restricts to
y3
+ y1
- y3 restricts to
y1
- y4 restricts to
y2
- v1 restricts to
v
- v2 restricts to
x2
Restriction to maximal subgroup number 10, which is 32gp35
- y1 restricts to
y1
- y2 restricts to
y3
+ y2
+ y1
- y3 restricts to
y2
- y4 restricts to
y3
+ y2
+ y1
- v1 restricts to
v
- v2 restricts to
v
+ x2
+ y12.x
Restriction to maximal subgroup number 11, which is 32gp26
- y1 restricts to
y1
- y2 restricts to
y3
+ y1
- y3 restricts to
y2
- y4 restricts to
y3
- v1 restricts to
v
- v2 restricts to
x2
Restriction to maximal subgroup number 12, which is 32gp35
- y1 restricts to
y2
- y2 restricts to
y3
+ y2
- y3 restricts to
y2
+ y1
- y4 restricts to
y2
+ y1
- v1 restricts to
v
- v2 restricts to
v
+ x2
+ y12.x
+ y12.y2.y3
Restriction to maximal subgroup number 13, which is 32gp35
- y1 restricts to
y2
+ y1
- y2 restricts to
y3
+ y2
+ y1
- y3 restricts to
y1
- y4 restricts to
y2
- v1 restricts to
v
- v2 restricts to
v
+ x2
+ y12.x
+ y12.y2.y3
Restriction to maximal subgroup number 14, which is 32gp35
- y1 restricts to
y3
+ y2
+ y1
- y2 restricts to
y2
+ y1
- y3 restricts to
y3
+ y2
- y4 restricts to
y3
+ y1
- v1 restricts to
v
+ x2
+ y12.x
- v2 restricts to
v
+ y12.y2.y3
Restriction to maximal subgroup number 15, which is 32gp35
- y1 restricts to
y3
+ y2
- y2 restricts to
y2
+ y1
- y3 restricts to
y1
- y4 restricts to
y3
- v1 restricts to
v
+ x2
+ y12.x
- v2 restricts to
v
+ y12.y2.y3
Restriction to maximal elementary abelian number 1, which is V4
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
0
- y4 restricts to
0
- v1 restricts to
y24
- v2 restricts to
y14
Restriction to the greatest central elementary abelian, which is V4
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
0
- y4 restricts to
0
- v1 restricts to
y24
- v2 restricts to
y14
(1 + 4t + 8t2
+ 10t3 + 8t4 + 4t5
+ t6) /
(1 - t4)2
Back to the groups of order 64