Small group number 50 of order 32
G = E32- is Extraspecial 2-group of order 32 and type -
The Hall-Senior number of this group is 43.
G has 4 minimal generators, rank 2 and exponent 4.
The centre has rank 1.
The 15 maximal subgroups are:
Q8xC2 (5x), 16gp13 (10x).
There are 5 conjugacy classes of maximal
elementary abelian subgroups. Their ranks are:
2, 2, 2, 2, 2.
This cohomology ring calculation is complete.
Ring structure
| Completion information
| Koszul information
| Restriction information
| Poincaré series
The cohomology ring has 5 generators:
- y1 in degree 1
- y2 in degree 1
- y3 in degree 1
- y4 in degree 1
- r in degree 8, a regular element
There are 3 minimal relations:
- y3.y4 =
y22
+ y1.y4
+ y1.y2
+ y12
- y22.y4 =
y22.y3
+ y1.y2.y4
+ y1.y2.y3
+ y12.y4
+ y12.y3
+ y13
- y13.y42 =
y13.y32
+ y13.y22
+ y14.y4
+ y14.y3
+ y14.y2
A minimal Gröbner basis for the relations ideal
consists of this minimal generating set, together with the
following redundant relations:
- y22.y32 =
y24
+ y1.y2.y32
+ y1.y22.y3
+ y12.y32
+ y12.y2.y3
+ y12.y22
- y13.y33 =
0
- y13.y24 =
y16.y4
+ y16.y3
+ y16.y2
+ y17
Essential ideal:
Zero ideal
Nilradical:
There are 4 minimal generators:
-
y2.y4
+ y22
-
y2.y3
+ y22
+ y1.y2
-
y1.y4
+ y12
-
y1.y3
This cohomology ring was obtained from a calculation
out to degree 12. The cohomology ring approximation
is stable from degree 8 onwards, and
Carlson's tests detect stability from degree 10
onwards.
This cohomology ring has dimension 2 and depth 2.
Here is a homogeneous system of parameters:
- h1 =
r
in degree 8
- h2 =
y42
+ y32
+ y2.y3
in degree 2
The first
2 terms h1, h2 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) is as follows.
Carlson's Koszul condition stipulates that this must
be confined to degrees less than 10.
-
1
in degree 0
-
y4
in degree 1
-
y3
in degree 1
-
y2
in degree 1
-
y1
in degree 1
-
y32
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.y32
in degree 3
-
y22.y3
in degree 3
-
y23
in degree 3
-
y1.y32
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
-
y13
in degree 3
-
y24
in degree 4
-
y1.y2.y32
in degree 4
-
y1.y22.y3
in degree 4
-
y1.y23
in degree 4
-
y12.y32
in degree 4
-
y12.y2.y4
in degree 4
-
y12.y2.y3
in degree 4
-
y12.y22
in degree 4
-
y13.y4
in degree 4
-
y13.y3
in degree 4
-
y13.y2
in degree 4
-
y14
in degree 4
-
y1.y24
in degree 5
-
y12.y2.y32
in degree 5
-
y12.y22.y3
in degree 5
-
y12.y23
in degree 5
-
y13.y32
in degree 5
-
y13.y2.y4
in degree 5
-
y13.y22
in degree 5
-
y14.y4
in degree 5
-
y14.y3
in degree 5
-
y14.y2
in degree 5
-
y15
in degree 5
-
y12.y24
in degree 6
-
y13.y23
in degree 6
-
y14.y2.y4
in degree 6
-
y14.y22
in degree 6
-
y15.y4
in degree 6
-
y15.y3
in degree 6
-
y15.y2
in degree 6
-
y16
in degree 6
-
y15.y2.y4
in degree 7
-
y15.y22
in degree 7
-
y16.y2
in degree 7
-
y17
in degree 7
-
y17.y2
in degree 8
Restriction to maximal subgroup number 1, which is 16gp12
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
y3
- y4 restricts to
0
- r restricts to
v2
+ y34.v
Restriction to maximal subgroup number 2, which is 16gp13
- y1 restricts to
0
- y2 restricts to
y3
+ y2
- y3 restricts to
y3
+ y1
- y4 restricts to
y3
- r restricts to
y17.y3
+ v2
Restriction to maximal subgroup number 3, which is 16gp13
- y1 restricts to
y3
+ y1
- y2 restricts to
y2
+ y1
- y3 restricts to
y2
- y4 restricts to
y3
+ y1
- r restricts to
y38
+ y2.y37
+ y17.y2
+ v2
Restriction to maximal subgroup number 4, which is 16gp13
- y1 restricts to
y2
- y2 restricts to
0
- y3 restricts to
y2
+ y1
- y4 restricts to
y3
- r restricts to
y2.y37
+ y17.y3
+ v2
Restriction to maximal subgroup number 5, which is 16gp13
- y1 restricts to
y2
- y2 restricts to
y3
- y3 restricts to
y3
+ y1
- y4 restricts to
y3
- r restricts to
y2.y37
+ y16.y2.y3
+ y17.y3
+ v2
Restriction to maximal subgroup number 6, which is 16gp13
- y1 restricts to
y2
+ y1
- y2 restricts to
y2
+ y1
- y3 restricts to
y2
- y4 restricts to
y3
+ y1
- r restricts to
y2.y37
+ y17.y3
+ y17.y2
+ v2
Restriction to maximal subgroup number 7, which is 16gp13
- y1 restricts to
y3
+ y2
- y2 restricts to
y2
- y3 restricts to
y1
- y4 restricts to
y3
- r restricts to
y38
+ y16.y2.y3
+ y17.y3
+ v2
Restriction to maximal subgroup number 8, which is 16gp13
- y1 restricts to
y3
+ y1
- y2 restricts to
y3
+ y2
- y3 restricts to
0
- y4 restricts to
y3
+ y2
+ y1
- r restricts to
y18
+ v2
Restriction to maximal subgroup number 9, which is 16gp12
- y1 restricts to
y1
- y2 restricts to
y3
+ y2
- y3 restricts to
y3
+ y1
- y4 restricts to
y3
+ y1
- r restricts to
v2
+ y34.v
+ y1.y37
+ y12.y36
Restriction to maximal subgroup number 10, which is 16gp12
- y1 restricts to
y2
+ y1
- y2 restricts to
y1
- y3 restricts to
y2
+ y1
- y4 restricts to
y3
+ y2
- r restricts to
v2
+ y34.v
+ y2.y37
+ y1.y37
+ y1.y2.y36
+ y12.y2.y35
Restriction to maximal subgroup number 11, which is 16gp13
- y1 restricts to
y3
+ y1
- y2 restricts to
y1
- y3 restricts to
y2
- y4 restricts to
y3
+ y2
+ y1
- r restricts to
y38
+ y2.y37
+ y17.y2
+ v2
Restriction to maximal subgroup number 12, which is 16gp13
- y1 restricts to
y3
+ y2
- y2 restricts to
y2
+ y1
- y3 restricts to
y2
+ y1
- y4 restricts to
y3
+ y2
+ y1
- r restricts to
y38
+ y2.y37
+ y16.y2.y3
+ y17.y3
+ y17.y2
+ v2
Restriction to maximal subgroup number 13, which is 16gp12
- y1 restricts to
y3
+ y2
- y2 restricts to
y3
+ y2
+ y1
- y3 restricts to
y2
- y4 restricts to
y3
+ y1
- r restricts to
v2
+ y34.v
+ y2.y37
+ y12.y36
Restriction to maximal subgroup number 14, which is 16gp13
- y1 restricts to
y3
+ y2
- y2 restricts to
y3
+ y1
- y3 restricts to
y2
+ y1
- y4 restricts to
y3
+ y2
+ y1
- r restricts to
y2.y37
+ y16.y2.y3
+ y17.y2
+ v2
Restriction to maximal subgroup number 15, which is 16gp12
- y1 restricts to
y3
- y2 restricts to
y2
+ y1
- y3 restricts to
y2
- y4 restricts to
y3
+ y1
- r restricts to
v2
+ y34.v
+ y38
+ y2.y37
+ y1.y2.y36
+ y12.y2.y35
Restriction to maximal elementary abelian number 1, which is V4
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
y2
- y4 restricts to
0
- r restricts to
y14.y24
+ y18
Restriction to maximal elementary abelian number 2, which is V4
- y1 restricts to
0
- y2 restricts to
y2
- y3 restricts to
y2
- y4 restricts to
y2
- r restricts to
y14.y24
+ y18
Restriction to maximal elementary abelian number 3, which is V4
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
0
- y4 restricts to
y2
+ y1
- r restricts to
y14.y24
Restriction to maximal elementary abelian number 4, which is V4
- y1 restricts to
y2
- y2 restricts to
y2
- y3 restricts to
0
- y4 restricts to
y2
- r restricts to
y14.y24
+ y18
Restriction to maximal elementary abelian number 5, which is V4
- y1 restricts to
y2
- y2 restricts to
0
- y3 restricts to
0
- y4 restricts to
y2
- 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
- y3 restricts to
0
- y4 restricts to
0
- r restricts to
y8
(1 + 4t + 8t2
+ 11t3 + 12t4 + 11t5
+ 8t6 + 4t7 + t8) /
(1 - t2) (1 - t8)
Back to the groups of order 32