Small group number 14 of order 625
G = E125*C25 is Central product E125 * C_25
G has 3 minimal generators, rank 2 and exponent 25.
The centre has rank 1.
The 31 maximal subgroups are:
Ab(25,5) (6x), E125, M125 (24x).
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 9 generators:
- y1 in degree 1, a nilpotent element
- y2 in degree 1, a nilpotent element
- y3 in degree 1, a nilpotent element
- x1 in degree 2
- x2 in degree 2
- v in degree 4, a nilpotent element
- t in degree 6, a nilpotent element
- r in degree 8
- p in degree 10, a regular element
There are 22 minimal relations:
- y32 =
0
- y22 =
0
- y12 =
0
- y2.x2 =
y1.x1
- y2.v =
0
- y1.v =
0
- x2.v =
0
- x1.v =
0
- y2.t =
0
- y1.t =
0
- x2.t =
0
- x1.t =
0
- v2 =
0
- y2.r =
y1.x1.x23
- y1.x12.x22
+ y1.x13.x2
+ 2y1.x14
- y1.r =
- y1.x1.x23
+ y1.x12.x22
+ 2y1.x13.x2
+ y1.x14
- x2.r =
- x1.x24
+ x12.x23
+ 2x13.x22
+ x14.x2
- x1.r =
x1.x24
- x12.x23
+ x13.x22
+ 2x14.x2
- v.t =
0
- v.r =
0
- t2 =
0
- t.r =
0
- r2 =
- 2x14.x24
+ 2x15.x23
+ 2x16.x22
+ 2x17.x2
A minimal Gröbner basis for the relations ideal
consists of this minimal generating set, together with the
following redundant relations:
- y1.y2.x1 =
0
- y1.x1.x24 =
y1.x15
- x1.x25 =
x15.x2
Essential ideal:
There are 3 minimal generators:
Nilradical:
There are 5 minimal generators:
This cohomology ring was obtained from a calculation
out to degree 16. The cohomology ring approximation
is stable from degree 16 onwards, and
Carlson's tests detect stability from degree 16
onwards.
This cohomology ring has dimension 2 and depth 1.
Here is a homogeneous system of parameters:
- h1 =
p
in degree 10
- h2 =
x22
- x1.x2
+ x12
in degree 4
The first
term h1 forms
a regular sequence of maximum length.
The remaining
term h2 is
annihilated by the class
y1.y2.
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 =
y1.y2.y3
in degree 3
- G2 =
y3.v
in degree 5
- G3 =
y3.t
in degree 7
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
-
y3
in degree 1
-
y2
in degree 1
-
y1
in degree 1
-
x2
in degree 2
-
x1
in degree 2
-
y2.y3
in degree 2
-
y1.y3
in degree 2
-
y1.y2
in degree 2
-
y3.x2
in degree 3
-
y3.x1
in degree 3
-
y2.x1
in degree 3
-
y1.x2
in degree 3
-
y1.x1
in degree 3
-
y1.y2.y3
in degree 3
-
x1.x2
in degree 4
-
x12
in degree 4
-
v
in degree 4
-
y2.y3.x1
in degree 4
-
y1.y3.x2
in degree 4
-
y1.y3.x1
in degree 4
-
y3.x1.x2
in degree 5
-
y3.x12
in degree 5
-
y1.x1.x2
in degree 5
-
y1.x12
in degree 5
-
y3.v
in degree 5
-
x12.x2
in degree 6
-
x13
in degree 6
-
t
in degree 6
-
y1.y3.x1.x2
in degree 6
-
y1.y3.x12
in degree 6
-
y3.x12.x2
in degree 7
-
y3.x13
in degree 7
-
y1.x12.x2
in degree 7
-
y1.x13
in degree 7
-
y3.t
in degree 7
-
r
in degree 8
-
x13.x2
in degree 8
-
x14
in degree 8
-
y1.y3.x12.x2
in degree 8
-
y1.y3.x13
in degree 8
-
y3.r
in degree 9
-
y3.x13.x2
in degree 9
-
y3.x14
in degree 9
-
y1.x13.x2
in degree 9
-
y1.x14
in degree 9
-
x14.x2
in degree 10
-
x15
in degree 10
-
y1.y3.x13.x2
in degree 10
-
y1.y3.x14
in degree 10
-
y3.x14.x2
in degree 11
-
y3.x15
in degree 11
-
y1.x15
in degree 11
-
y1.y3.x15
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.y2
in degree 2
-
y1.y2.y3
in degree 3
-
v
in degree 4
-
y3.v
in degree 5
-
t
in degree 6
-
y3.t
in degree 7
Restriction to maximal subgroup number 1, which is 125gp2
- y1 restricts to
0
- y2 restricts to
y2
- y3 restricts to
y1
- x1 restricts to
x1
- x2 restricts to
0
- v restricts to
0
- t restricts to
0
- r restricts to
0
- p restricts to
x25
- x14.x2
Restriction to maximal subgroup number 2, which is 125gp2
- y1 restricts to
y2
- y2 restricts to
0
- y3 restricts to
y1
- x1 restricts to
0
- x2 restricts to
x1
- v restricts to
0
- t restricts to
0
- r restricts to
0
- p restricts to
x25
- x14.x2
Restriction to maximal subgroup number 3, which is 125gp2
- y1 restricts to
- y2
- y2 restricts to
y2
- y3 restricts to
y1
- x1 restricts to
x1
- x2 restricts to
- x1
- v restricts to
0
- t restricts to
0
- r restricts to
x14
- p restricts to
x25
- x14.x2
- 2x15
Restriction to maximal subgroup number 4, which is 125gp2
- y1 restricts to
2y2
- y2 restricts to
y2
- y3 restricts to
y1
- x1 restricts to
x1
- x2 restricts to
2x1
- v restricts to
0
- t restricts to
0
- r restricts to
x14
- p restricts to
x25
- x14.x2
Restriction to maximal subgroup number 5, which is 125gp2
- y1 restricts to
y2
- y2 restricts to
y2
- y3 restricts to
y1
- x1 restricts to
x1
- x2 restricts to
x1
- v restricts to
0
- t restricts to
0
- r restricts to
- 2x14
- p restricts to
x25
- x14.x2
Restriction to maximal subgroup number 6, which is 125gp2
- y1 restricts to
- 2y2
- y2 restricts to
y2
- y3 restricts to
y1
- x1 restricts to
x1
- x2 restricts to
- 2x1
- v restricts to
0
- t restricts to
0
- r restricts to
- x14
- p restricts to
x25
- x14.x2
+ 2x15
Restriction to maximal subgroup number 7, which is 125gp3
- y1 restricts to
y2
+ y1
- y2 restricts to
- 2y2
+ 2y1
- y3 restricts to
0
- x1 restricts to
2x4
- 2x3
- x2 restricts to
x4
+ x3
- v restricts to
2y2.w1
- 2y1.w2
- t restricts to
2w1.w2
+ y1.x3.w2
+ y1.x3.w1
- r restricts to
r
- x44
+ x3.x43
+ x34
- 2y2.x32.w2
- 2y1.x42.w1
+ 2y1.x3.x4.w1
+ y1.x32.w2
- y1.x32.w1
- p restricts to
- x45
+ x3.x44
- x33.x42
+ 2x34.x4
- p
+ 2y2.x33.w2
+ 2y1.x43.w1
- 2y1.x3.x42.w1
+ 2y1.x32.x4.w1
+ 2y1.x33.w2
- 2y1.x33.w1
Restriction to maximal subgroup number 8, which is 125gp4
- y1 restricts to
- 2y1
- y2 restricts to
- y2
- y1
- y3 restricts to
2y1
- x1 restricts to
- x
+ y1.y2
- x2 restricts to
2y1.y2
- v restricts to
- y2.w
- t restricts to
2y2.u
- r restricts to
- y2.s
- p restricts to
- 2p
+ 2y2.q
Restriction to maximal subgroup number 9, which is 125gp4
- y1 restricts to
- y1
- y2 restricts to
- 2y2
- y3 restricts to
2y1
- x1 restricts to
- 2x
- x2 restricts to
y1.y2
- v restricts to
- y2.w
- t restricts to
2y2.u
- r restricts to
- y2.s
- p restricts to
- 2p
+ 2y2.q
Restriction to maximal subgroup number 10, which is 125gp4
- y1 restricts to
- 2y1
- y2 restricts to
y2
+ y1
- y3 restricts to
- 2y1
- x1 restricts to
x
- y1.y2
- x2 restricts to
2y1.y2
- v restricts to
- y2.w
- t restricts to
- 2y2.u
- r restricts to
- y2.s
- p restricts to
2p
- 2y2.q
Restriction to maximal subgroup number 11, which is 125gp4
- y1 restricts to
2y1
- y2 restricts to
2y2
- 2y1
- y3 restricts to
- y1
- x1 restricts to
2x
+ 2y1.y2
- x2 restricts to
- 2y1.y2
- v restricts to
y2.w
- t restricts to
y2.u
- r restricts to
- y2.s
- p restricts to
p
- y2.q
Restriction to maximal subgroup number 12, which is 125gp4
- y1 restricts to
y2
- y1
- y2 restricts to
y1
- y3 restricts to
- y1
- x1 restricts to
- y1.y2
- x2 restricts to
x
+ y1.y2
- v restricts to
y2.w
- t restricts to
y2.u
- r restricts to
- y2.s
- p restricts to
p
- y2.q
Restriction to maximal subgroup number 13, which is 125gp4
- y1 restricts to
y2
+ y1
- y2 restricts to
- y2
+ 2y1
- y3 restricts to
2y1
- x1 restricts to
- x
- 2y1.y2
- x2 restricts to
x
- y1.y2
- v restricts to
- y2.w
- t restricts to
2y2.u
- r restricts to
x4
- y2.s
- p restricts to
2x5
- 2p
+ 2y2.q
Restriction to maximal subgroup number 14, which is 125gp4
- y1 restricts to
y2
+ y1
- y2 restricts to
- 2y2
+ y1
- y3 restricts to
2y1
- x1 restricts to
- 2x
- y1.y2
- x2 restricts to
x
- y1.y2
- v restricts to
- y2.w
- t restricts to
2y2.u
- r restricts to
x4
- y2.s
- p restricts to
- 2p
+ 2y2.q
Restriction to maximal subgroup number 15, which is 125gp4
- y1 restricts to
y2
+ y1
- y2 restricts to
y2
- 2y1
- y3 restricts to
- 2y1
- x1 restricts to
x
+ 2y1.y2
- x2 restricts to
x
- y1.y2
- v restricts to
- y2.w
- t restricts to
- 2y2.u
- r restricts to
- 2x4
- y2.s
- p restricts to
2p
- 2y2.q
Restriction to maximal subgroup number 16, which is 125gp4
- y1 restricts to
y2
- y2 restricts to
2y2
+ y1
- y3 restricts to
- y1
- x1 restricts to
2x
- y1.y2
- x2 restricts to
x
- v restricts to
y2.w
- t restricts to
y2.u
- r restricts to
- x4
- y2.s
- p restricts to
- x5
+ p
- y2.q
Restriction to maximal subgroup number 17, which is 125gp4
- y1 restricts to
2y2
- y1
- y2 restricts to
- 2y1
- y3 restricts to
- y1
- x1 restricts to
2y1.y2
- x2 restricts to
2x
+ y1.y2
- v restricts to
y2.w
- t restricts to
y2.u
- r restricts to
- y2.s
- p restricts to
p
- y2.q
Restriction to maximal subgroup number 18, which is 125gp4
- y1 restricts to
2y2
+ y1
- y2 restricts to
- y2
- y1
- y3 restricts to
y1
- x1 restricts to
- x
+ y1.y2
- x2 restricts to
2x
- y1.y2
- v restricts to
y2.w
- t restricts to
- y2.u
- r restricts to
- x4
- y2.s
- p restricts to
- 2x5
- p
+ y2.q
Restriction to maximal subgroup number 19, which is 125gp4
- y1 restricts to
2y2
+ y1
- y2 restricts to
- 2y2
- y3 restricts to
- 2y1
- x1 restricts to
- 2x
- x2 restricts to
2x
- y1.y2
- v restricts to
- y2.w
- t restricts to
- 2y2.u
- r restricts to
x4
- y2.s
- p restricts to
- x5
+ 2p
- 2y2.q
Restriction to maximal subgroup number 20, which is 125gp4
- y1 restricts to
2y2
+ 2y1
- y2 restricts to
y2
- 2y1
- y3 restricts to
y1
- x1 restricts to
x
+ 2y1.y2
- x2 restricts to
2x
- 2y1.y2
- v restricts to
y2.w
- t restricts to
- y2.u
- r restricts to
x4
- y2.s
- p restricts to
- p
+ y2.q
Restriction to maximal subgroup number 21, which is 125gp4
- y1 restricts to
2y2
+ y1
- y2 restricts to
2y2
+ 2y1
- y3 restricts to
- 2y1
- x1 restricts to
2x
- 2y1.y2
- x2 restricts to
2x
- y1.y2
- v restricts to
- y2.w
- t restricts to
- 2y2.u
- r restricts to
- 2x4
- y2.s
- p restricts to
2p
- 2y2.q
Restriction to maximal subgroup number 22, which is 125gp4
- y1 restricts to
- y2
+ y1
- y2 restricts to
- 2y1
- y3 restricts to
- 2y1
- x1 restricts to
2y1.y2
- x2 restricts to
- x
- y1.y2
- v restricts to
- y2.w
- t restricts to
- 2y2.u
- r restricts to
- y2.s
- p restricts to
2p
- 2y2.q
Restriction to maximal subgroup number 23, which is 125gp4
- y1 restricts to
- y2
+ y1
- y2 restricts to
- y2
- 2y1
- y3 restricts to
2y1
- x1 restricts to
- x
+ 2y1.y2
- x2 restricts to
- x
- y1.y2
- v restricts to
- y2.w
- t restricts to
2y2.u
- r restricts to
- 2x4
- y2.s
- p restricts to
- 2p
+ 2y2.q
Restriction to maximal subgroup number 24, which is 125gp4
- y1 restricts to
- y2
- y1
- y2 restricts to
- 2y2
- y3 restricts to
2y1
- x1 restricts to
- 2x
- x2 restricts to
- x
+ y1.y2
- v restricts to
- y2.w
- t restricts to
2y2.u
- r restricts to
- x4
- y2.s
- p restricts to
x5
- 2p
+ 2y2.q
Restriction to maximal subgroup number 25, which is 125gp4
- y1 restricts to
- y2
+ 2y1
- y2 restricts to
y2
+ y1
- y3 restricts to
- 2y1
- x1 restricts to
x
- y1.y2
- x2 restricts to
- x
- 2y1.y2
- v restricts to
- y2.w
- t restricts to
- 2y2.u
- r restricts to
x4
- y2.s
- p restricts to
- 2x5
+ 2p
- 2y2.q
Restriction to maximal subgroup number 26, which is 125gp4
- y1 restricts to
- y2
- 2y1
- y2 restricts to
2y2
+ y1
- y3 restricts to
2y1
- x1 restricts to
2x
- y1.y2
- x2 restricts to
- x
+ 2y1.y2
- v restricts to
- y2.w
- t restricts to
2y2.u
- r restricts to
x4
- y2.s
- p restricts to
- 2p
+ 2y2.q
Restriction to maximal subgroup number 27, which is 125gp4
- y1 restricts to
- 2y2
- y2 restricts to
- 2y1
- y3 restricts to
y1
- x1 restricts to
2y1.y2
- x2 restricts to
- 2x
- v restricts to
y2.w
- t restricts to
- y2.u
- r restricts to
- y2.s
- p restricts to
- p
+ y2.q
Restriction to maximal subgroup number 28, which is 125gp4
- y1 restricts to
- 2y2
+ y1
- y2 restricts to
- y2
- y3 restricts to
- y1
- x1 restricts to
- x
- x2 restricts to
- 2x
- y1.y2
- v restricts to
y2.w
- t restricts to
y2.u
- r restricts to
x4
- y2.s
- p restricts to
p
- y2.q
Restriction to maximal subgroup number 29, which is 125gp4
- y1 restricts to
- 2y2
+ 2y1
- y2 restricts to
- 2y2
+ y1
- y3 restricts to
- 2y1
- x1 restricts to
- 2x
- y1.y2
- x2 restricts to
- 2x
- 2y1.y2
- v restricts to
- y2.w
- t restricts to
- 2y2.u
- r restricts to
- 2x4
- y2.s
- p restricts to
2p
- 2y2.q
Restriction to maximal subgroup number 30, which is 125gp4
- y1 restricts to
- 2y2
- 2y1
- y2 restricts to
y2
- y3 restricts to
- 2y1
- x1 restricts to
x
- x2 restricts to
- 2x
+ 2y1.y2
- v restricts to
- y2.w
- t restricts to
- 2y2.u
- r restricts to
- x4
- y2.s
- p restricts to
2x5
+ 2p
- 2y2.q
Restriction to maximal subgroup number 31, which is 125gp4
- y1 restricts to
- 2y2
- y1
- y2 restricts to
2y2
- 2y1
- y3 restricts to
- y1
- x1 restricts to
2x
+ 2y1.y2
- x2 restricts to
- 2x
+ y1.y2
- v restricts to
y2.w
- t restricts to
y2.u
- r restricts to
x4
- y2.s
- p restricts to
x5
+ p
- y2.q
Restriction to maximal elementary abelian number 1, which is V25
- y1 restricts to
y2
- y2 restricts to
y2
- y3 restricts to
0
- x1 restricts to
x2
- x2 restricts to
x2
- v restricts to
0
- t restricts to
0
- r restricts to
- 2x24
- p restricts to
x1.x24
- x15
Restriction to maximal elementary abelian number 2, which is V25
- y1 restricts to
y2
+ y1
- y2 restricts to
0
- y3 restricts to
0
- x1 restricts to
0
- x2 restricts to
x2
+ x1
- v restricts to
0
- t restricts to
0
- r restricts to
0
- p restricts to
x1.x24
- x12.x23
+ x13.x22
- x14.x2
Restriction to maximal elementary abelian number 3, which is V25
- y1 restricts to
0
- y2 restricts to
y2
- y3 restricts to
0
- x1 restricts to
x2
- x2 restricts to
0
- v restricts to
0
- t restricts to
0
- r restricts to
0
- p restricts to
x1.x24
- x15
Restriction to maximal elementary abelian number 4, which is V25
- y1 restricts to
2y2
- y2 restricts to
y2
- y3 restricts to
0
- x1 restricts to
x2
- x2 restricts to
2x2
- v restricts to
0
- t restricts to
0
- r restricts to
x24
- p restricts to
x1.x24
- x15
Restriction to maximal elementary abelian number 5, which is V25
- y1 restricts to
- y2
- y2 restricts to
y2
- y3 restricts to
0
- x1 restricts to
x2
- x2 restricts to
- x2
- v restricts to
0
- t restricts to
0
- r restricts to
x24
- p restricts to
- 2x25
+ x1.x24
- x15
Restriction to maximal elementary abelian number 6, which is V25
- y1 restricts to
- 2y2
- y2 restricts to
y2
- y3 restricts to
0
- x1 restricts to
x2
- x2 restricts to
- 2x2
- v restricts to
0
- t restricts to
0
- r restricts to
- x24
- p restricts to
2x25
+ x1.x24
- x15
Restriction to the greatest central elementary abelian, which is C5
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
0
- x1 restricts to
0
- x2 restricts to
0
- v restricts to
0
- t restricts to
0
- r restricts to
0
- p restricts to
- x5
(1 + 3t + 5t2
+ 6t3 + 6t4 + 5t5
+ 4t6 + 4t7 + 4t8
+ 4t9 + 3t10 + 2t11
+ t12) /
(1 - t4) (1 - t10)
Back to the groups of order 625