Small group number 3 of order 125
G = E125 is Extraspecial 5-group of order 125 and exponent 5
G has 2 minimal generators, rank 2 and exponent 5.
The centre has rank 1.
The 6 maximal subgroups are:
V25 (6x).
There are 6 conjugacy classes of maximal
elementary abelian subgroups. Their ranks are:
2, 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 12 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, a nilpotent element
- x3 in degree 2
- x4 in degree 2
- w1 in degree 3, a nilpotent element
- w2 in degree 3, a nilpotent element
- s in degree 7, a nilpotent element
- r in degree 8
- q in degree 9, a nilpotent element
- p in degree 10, a regular element
There are 50 minimal relations:
- y22 =
0
- y1.y2 =
0
- y12 =
0
- y2.x4 =
y1.x3
- y2.x2 =
0
- y2.x1 =
y1.x2
- y1.x1 =
0
- x2.x4 =
y2.w1
- 2y1.w2
- x2.x3 =
- y2.w2
- x1.x4 =
y1.w1
- x1.x3 =
2y2.w1
- y1.w2
- x22 =
0
- x1.x2 =
0
- x12 =
0
- x4.w2 =
x3.w1
- 2y1.x3.x4
- 2y1.x32
- x2.w2 =
0
- x2.w1 =
x1.w2
- x1.w1 =
0
- w22 =
0
- w12 =
0
- y2.s =
- y2.x32.w2
+ 2y1.x3.x4.w1
+ 2y1.x32.w2
- 2y1.x32.w1
- y1.s =
2y1.x42.w1
- 2y1.x3.x4.w1
- y1.x32.w2
+ 2y1.x32.w1
- x4.s =
2x43.w1
- 2x3.x42.w1
+ 2x32.x4.w1
- x33.w1
+ 2y1.x3.x43
+ 2y1.x32.x42
+ y1.x33.x4
- y1.x34
- x3.s =
2x3.x42.w1
- 2x32.x4.w1
- x33.w2
+ 2x33.w1
+ 2y1.x3.x43
+ 2y1.x32.x42
+ 2y1.x33.x4
- y1.x34
- y2.r =
y1.x3.x43
- 2y1.x32.x42
- y1.x33.x4
+ y1.x34
- y1.r =
- 2y1.x3.x43
- y1.x32.x42
+ y1.x33.x4
+ y1.x34
- x2.s =
0
- x1.s =
0
- x4.r =
- 2x3.x44
- x32.x43
+ x33.x42
+ x34.x4
+ 2y1.x43.w1
- 2y1.x3.x42.w1
+ y1.x32.x4.w1
+ 2y1.x33.w2
- y1.x33.w1
- x3.r =
x3.x44
- 2x32.x43
- x33.x42
+ x34.x4
+ 2y2.x33.w2
+ 2y1.x3.x42.w1
- 2y1.x32.x4.w1
- y1.x33.w2
+ y1.x33.w1
- x2.r =
- y1.x3.x42.w1
+ 2y1.x32.x4.w1
- y1.x33.w2
+ y1.x33.w1
- x1.r =
- 2y1.x3.x42.w1
- y1.x32.x4.w1
+ y1.x33.w2
+ y1.x33.w1
- w2.s =
- y1.x3.x42.w1
- 2y1.x32.x4.w1
+ 2y1.x33.w2
- 2y1.x33.w1
- w1.s =
- 2y1.x3.x42.w1
- 2y1.x32.x4.w1
+ y1.x33.w2
- y1.x33.w1
- y2.q =
- 2y2.x33.w2
+ 2y1.x3.x42.w1
- y1.x32.x4.w1
+ 2y1.x33.w2
+ 2y1.x33.w1
- y1.q =
y1.x43.w1
- y1.x3.x42.w1
+ 2y1.x32.x4.w1
- y1.x33.w2
+ 2y1.x33.w1
- w2.r =
x3.x43.w1
- 2x32.x42.w1
- x33.x4.w1
+ x34.w1
+ 2y1.x32.x43
+ y1.x33.x42
+ y1.x35
- w1.r =
- 2x3.x43.w1
- x32.x42.w1
+ x33.x4.w1
+ x34.w1
- x4.q =
x44.w1
- x3.x43.w1
+ 2x32.x42.w1
+ 2x33.x4.w1
- x34.w1
- 2y1.x32.x43
+ y1.x33.x42
- y1.x34.x4
- x3.q =
2x3.x43.w1
- x32.x42.w1
+ 2x33.x4.w1
- 2x34.w2
+ 2x34.w1
+ y1.x32.x43
- 2y1.x33.x42
+ y1.x34.x4
- x2.q =
0
- x1.q =
0
- w2.q =
2y1.x32.x42.w1
+ 2y1.x34.w2
+ y1.x34.w1
- w1.q =
2y1.x32.x42.w1
- y1.x33.x4.w1
+ y1.x34.w1
- s2 =
0
- s.r =
- x33.x43.w1
+ 2x35.x4.w1
- 2x36.w1
+ 2y1.x34.x43
- y1.x37
- r2 =
- 2x34.x44
- x35.x43
- 2x36.x42
+ x37.x4
+ y1.x34.x42.w1
- y1.x35.x4.w1
+ y1.x36.w2
- s.q =
y1.x35.x4.w1
- 2y1.x36.w2
- r.q =
- 2x34.x43.w1
- x35.x42.w1
+ x36.x4.w1
- x37.w1
- y1.x35.x43
- y1.x37.x4
- y1.x38
- q2 =
0
A minimal Gröbner basis for the relations ideal
consists of this minimal generating set, together with the
following redundant relations:
- y2.x3.w1 =
y1.x3.w2
- y2.w1.w2 =
0
- y1.w1.w2 =
0
- x3.w1.w2 =
2y1.x32.w2
+ 2y1.x32.w1
- y1.x3.x44 =
y1.x35
- x3.x45 =
x35.x4
- y1.x3.x43.w1 =
y1.x34.w2
- x3.x44.w1 =
x35.w1
Essential ideal:
There are 4 minimal generators:
-
y1.x2
-
y2.w1
- y1.w2
-
x1.w2
-
w1.w2
- 2y1.x3.w2
- 2y1.x3.w1
Nilradical:
There are 8 minimal generators:
This cohomology ring was obtained from a calculation
out to degree 18. The cohomology ring approximation
is stable from degree 18 onwards, and
Carlson's tests detect stability from degree 18
onwards.
This cohomology ring has dimension 2 and depth 1.
Here is a homogeneous system of parameters:
- h1 =
p
in degree 10
- h2 =
x42
- x3.x4
+ 2x32
in degree 4
The first
term h1 forms
a regular sequence of maximum length.
The remaining
term h2 is
annihilated by the class
y1.x2.
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 4 as a module over the polynomial algebra
on h1.
These free generators are:
- G1 =
y1.x2
in degree 3
- G2 =
y2.w1
- y1.w2
in degree 4
- G3 =
x1.w2
in degree 5
- G4 =
w1.w2
- 2y1.x3.w2
- 2y1.x3.w1
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 14.
-
1
in degree 0
-
y2
in degree 1
-
y1
in degree 1
-
x4
in degree 2
-
x3
in degree 2
-
x2
in degree 2
-
x1
in degree 2
-
w2
in degree 3
-
w1
in degree 3
-
y2.x3
in degree 3
-
y1.x4
in degree 3
-
y1.x3
in degree 3
-
y1.x2
in degree 3
-
x3.x4
in degree 4
-
x32
in degree 4
-
y2.w2
in degree 4
-
y2.w1
in degree 4
-
y1.w2
in degree 4
-
y1.w1
in degree 4
-
x4.w1
in degree 5
-
x3.w2
in degree 5
-
x3.w1
in degree 5
-
y1.x3.x4
in degree 5
-
y1.x32
in degree 5
-
x1.w2
in degree 5
-
x32.x4
in degree 6
-
x33
in degree 6
-
w1.w2
in degree 6
-
y1.x3.w2
in degree 6
-
y1.x3.w1
in degree 6
-
s
in degree 7
-
x32.w2
in degree 7
-
x32.w1
in degree 7
-
y1.x32.x4
in degree 7
-
y1.x33
in degree 7
-
r
in degree 8
-
x33.x4
in degree 8
-
x34
in degree 8
-
y1.x32.w2
in degree 8
-
y1.x32.w1
in degree 8
-
q
in degree 9
-
x33.w2
in degree 9
-
x33.w1
in degree 9
-
y1.x33.x4
in degree 9
-
y1.x34
in degree 9
-
x34.x4
in degree 10
-
x35
in degree 10
-
y1.x33.w2
in degree 10
-
y1.x33.w1
in degree 10
-
x34.w2
in degree 11
-
x34.w1
in degree 11
-
y1.x34.w1
in degree 12
A basis for AnnR/(h1)(h2) is as follows.
Carlson's Koszul condition stipulates that this must
be confined to degrees less than 10.
-
y1.x2
in degree 3
-
y2.w1
- y1.w2
in degree 4
-
x1.w2
in degree 5
-
w1.w2
- 2y1.x3.w2
- 2y1.x3.w1
in degree 6
Restriction to maximal subgroup number 1, which is V25
- y1 restricts to
0
- y2 restricts to
y1
- x1 restricts to
0
- x2 restricts to
- y1.y2
- x3 restricts to
x1
- x4 restricts to
0
- w1 restricts to
0
- w2 restricts to
y2.x1
- y1.x2
- s restricts to
- y2.x13
+ y1.x12.x2
- r restricts to
2y1.y2.x13
- q restricts to
- 2y2.x14
+ 2y1.x13.x2
- p restricts to
x25
- x14.x2
+ 2y1.y2.x14
Restriction to maximal subgroup number 2, which is V25
- y1 restricts to
y1
- y2 restricts to
0
- x1 restricts to
y1.y2
- x2 restricts to
0
- x3 restricts to
0
- x4 restricts to
x1
- w1 restricts to
y2.x1
- y1.x2
- w2 restricts to
0
- s restricts to
2y2.x13
- 2y1.x12.x2
- r restricts to
2y1.y2.x13
- q restricts to
y2.x14
- y1.x13.x2
- p restricts to
x25
- x14.x2
+ 2y1.y2.x14
Restriction to maximal subgroup number 3, which is V25
- y1 restricts to
- y1
- y2 restricts to
y1
- x1 restricts to
- y1.y2
- x2 restricts to
- y1.y2
- x3 restricts to
x1
- x4 restricts to
- x1
- w1 restricts to
- y2.x1
+ y1.x2
- 2y1.x1
- w2 restricts to
y2.x1
- y1.x2
+ 2y1.x1
- s restricts to
- 2y2.x13
+ 2y1.x12.x2
- y1.x13
- r restricts to
x14
- 2y1.y2.x13
- q restricts to
y2.x14
- y1.x13.x2
+ y1.x14
- p restricts to
x25
- x14.x2
- x15
+ 2y1.y2.x14
Restriction to maximal subgroup number 4, which is V25
- y1 restricts to
2y1
- y2 restricts to
y1
- x1 restricts to
2y1.y2
- x2 restricts to
- y1.y2
- x3 restricts to
x1
- x4 restricts to
2x1
- w1 restricts to
2y2.x1
- 2y1.x2
- y1.x1
- w2 restricts to
y2.x1
- y1.x2
+ y1.x1
- s restricts to
y2.x13
- y1.x12.x2
+ 2y1.x13
- r restricts to
- 2x14
- q restricts to
- y2.x14
+ y1.x13.x2
- y1.x14
- p restricts to
x25
- x14.x2
- 2x15
+ y1.y2.x14
Restriction to maximal subgroup number 5, which is V25
- y1 restricts to
y1
- y2 restricts to
y1
- x1 restricts to
y1.y2
- x2 restricts to
- y1.y2
- x3 restricts to
x1
- x4 restricts to
x1
- w1 restricts to
y2.x1
- y1.x2
+ 2y1.x1
- w2 restricts to
y2.x1
- y1.x2
- 2y1.x1
- s restricts to
y2.x13
- y1.x12.x2
+ y1.x13
- r restricts to
- x14
+ 2y1.y2.x13
- q restricts to
- 2y2.x14
+ 2y1.x13.x2
- y1.x14
- p restricts to
x25
- x14.x2
+ x15
- y1.y2.x14
Restriction to maximal subgroup number 6, which is V25
- y1 restricts to
- 2y1
- y2 restricts to
y1
- x1 restricts to
- 2y1.y2
- x2 restricts to
- y1.y2
- x3 restricts to
x1
- x4 restricts to
- 2x1
- w1 restricts to
- 2y2.x1
+ 2y1.x2
+ y1.x1
- w2 restricts to
y2.x1
- y1.x2
- y1.x1
- s restricts to
y2.x13
- y1.x12.x2
+ y1.x13
- r restricts to
x14
+ y1.y2.x13
- q restricts to
2y2.x14
- 2y1.x13.x2
+ y1.x14
- p restricts to
x25
- x14.x2
- 2y1.y2.x14
Restriction to maximal elementary abelian number 1, which is V25
- y1 restricts to
y2
+ y1
- y2 restricts to
0
- x1 restricts to
- y1.y2
- x2 restricts to
0
- x3 restricts to
0
- x4 restricts to
x2
+ x1
- w1 restricts to
- y2.x1
+ y1.x2
- w2 restricts to
0
- s restricts to
- 2y2.x1.x22
+ y2.x12.x2
- 2y2.x13
+ 2y1.x23
- y1.x1.x22
+ 2y1.x12.x2
- r restricts to
- 2y1.y2.x23
- y1.y2.x1.x22
- y1.y2.x12.x2
- 2y1.y2.x13
- q restricts to
- y2.x1.x23
+ 2y2.x12.x22
+ 2y2.x13.x2
- y2.x14
+ y1.x24
- 2y1.x1.x23
- 2y1.x12.x22
+ y1.x13.x2
- p restricts to
- x1.x24
+ x12.x23
- x13.x22
+ x14.x2
- 2y1.y2.x24
+ 2y1.y2.x1.x23
- 2y1.y2.x12.x22
+ 2y1.y2.x13.x2
- 2y1.y2.x14
Restriction to maximal elementary abelian number 2, which is V25
- y1 restricts to
2y2
- y2 restricts to
y2
- x1 restricts to
- 2y1.y2
- x2 restricts to
y1.y2
- x3 restricts to
x2
- x4 restricts to
2x2
- w1 restricts to
- y2.x2
- 2y2.x1
+ 2y1.x2
- w2 restricts to
y2.x2
- y2.x1
+ y1.x2
- s restricts to
2y2.x23
- y2.x1.x22
+ y1.x23
- r restricts to
- 2x24
- q restricts to
- y2.x24
+ y2.x1.x23
- y1.x24
- p restricts to
- 2x25
- x1.x24
+ x15
- y1.y2.x24
Restriction to maximal elementary abelian number 3, which is V25
- y1 restricts to
0
- y2 restricts to
y2
- x1 restricts to
0
- x2 restricts to
y1.y2
- x3 restricts to
x2
- x4 restricts to
0
- w1 restricts to
0
- w2 restricts to
- y2.x1
+ y1.x2
- s restricts to
y2.x1.x22
- y1.x23
- r restricts to
- 2y1.y2.x23
- q restricts to
2y2.x1.x23
- 2y1.x24
- p restricts to
- x1.x24
+ x15
- 2y1.y2.x24
Restriction to maximal elementary abelian number 4, which is V25
- y1 restricts to
y2
- y2 restricts to
y2
- x1 restricts to
- y1.y2
- x2 restricts to
y1.y2
- x3 restricts to
x2
- x4 restricts to
x2
- w1 restricts to
2y2.x2
- y2.x1
+ y1.x2
- w2 restricts to
- 2y2.x2
- y2.x1
+ y1.x2
- s restricts to
y2.x23
- y2.x1.x22
+ y1.x23
- r restricts to
- x24
- 2y1.y2.x23
- q restricts to
- y2.x24
+ 2y2.x1.x23
- 2y1.x24
- p restricts to
x25
- x1.x24
+ x15
+ y1.y2.x24
Restriction to maximal elementary abelian number 5, which is V25
- y1 restricts to
- y2
- y2 restricts to
y2
- x1 restricts to
y1.y2
- x2 restricts to
y1.y2
- x3 restricts to
x2
- x4 restricts to
- x2
- w1 restricts to
- 2y2.x2
+ y2.x1
- y1.x2
- w2 restricts to
2y2.x2
- y2.x1
+ y1.x2
- s restricts to
- y2.x23
+ 2y2.x1.x22
- 2y1.x23
- r restricts to
x24
+ 2y1.y2.x23
- q restricts to
y2.x24
- y2.x1.x23
+ y1.x24
- p restricts to
- x25
- x1.x24
+ x15
- 2y1.y2.x24
Restriction to maximal elementary abelian number 6, which is V25
- y1 restricts to
- 2y2
- y2 restricts to
y2
- x1 restricts to
2y1.y2
- x2 restricts to
y1.y2
- x3 restricts to
x2
- x4 restricts to
- 2x2
- w1 restricts to
y2.x2
+ 2y2.x1
- 2y1.x2
- w2 restricts to
- y2.x2
- y2.x1
+ y1.x2
- s restricts to
y2.x23
- y2.x1.x22
+ y1.x23
- r restricts to
x24
- y1.y2.x23
- q restricts to
y2.x24
- 2y2.x1.x23
+ 2y1.x24
- p restricts to
- x1.x24
+ x15
+ 2y1.y2.x24
Restriction to the greatest central elementary abelian, which is C5
- y1 restricts to
0
- y2 restricts to
0
- x1 restricts to
0
- x2 restricts to
0
- x3 restricts to
0
- x4 restricts to
0
- w1 restricts to
0
- w2 restricts to
0
- s restricts to
0
- r restricts to
0
- q restricts to
0
- p restricts to
2x5
(1 + 2t + 4t2
+ 6t3 + 6t4 + 6t5
+ 5t6 + 4t7 + 4t8
+ 4t9 + 3t10 + 2t11
+ t12) /
(1 - t4) (1 - t10)
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