Small group number 3 of order 625
G is the group 625gp3
G has 2 minimal generators, rank 3 and exponent 25.
The centre has rank 2.
The 6 maximal subgroups are:
Ab(25,5) (5x), V125.
There is one conjugacy class of maximal elementary abelian
subgroups. Each maximal elementary abelian has rank 3.
This cohomology ring calculation is complete.
Ring structure
| Completion information
| Koszul information
| Restriction information
| Poincaré series
The cohomology ring has 16 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, a regular element
- w1 in degree 3, a nilpotent element
- w2 in degree 3, a nilpotent element
- v in degree 4, a nilpotent element
- u in degree 5, a nilpotent element
- t in degree 6, a nilpotent element
- s in degree 7, a nilpotent element
- r in degree 8, a nilpotent element
- q in degree 9, a nilpotent element
- p1 in degree 10, a nilpotent element
- p2 in degree 10, a regular element
There are 90 minimal relations:
- y22 =
0
- y1.y2 =
0
- y12 =
0
- y1.x3 =
0
- y2.x2 =
2y1.x2
- y2.x1 =
y1.x2
- y1.x1 =
0
- x2.x3 =
y2.w2
+ 2y1.w2
- x1.x3 =
2y1.w2
- x22 =
0
- x1.x2 =
0
- x12 =
0
- y2.w1 =
2y1.w2
- y1.w1 =
0
- x3.w1 =
0
- x2.w2 =
- 2y2.v
+ y1.x2.x4
- x2.w1 =
y2.v
+ 2y1.x2.x4
- x1.w2 =
y2.v
+ 2y1.x2.x4
- x1.w1 =
0
- y1.v =
0
- x3.v =
y2.u
+ y2.x3.w2
- w22 =
0
- w1.w2 =
2y2.u
- 2y1.x4.w2
- w12 =
0
- x2.v =
0
- x1.v =
0
- y1.u =
0
- x3.u =
2y2.x32.x4
- w2.v =
2y2.t
+ 2y2.x4.v
+ 2y1.x2.x42
- w1.v =
0
- x2.u =
y2.t
- y1.x2.x42
- x1.u =
0
- y1.t =
0
- x3.t =
y2.s
- 2y2.x32.w2
+ 2y2.x4.u
+ y2.x3.x4.w2
- 2y2.x42.w2
- 2y1.x42.w2
- v2 =
0
- w2.u =
- y2.s
- y2.x3.x4.w2
+ 2y2.x42.w2
+ 2y1.x42.w2
- w1.u =
0
- x2.t =
0
- x1.t =
0
- y1.s =
2y1.x42.w2
- x3.s =
x32.x4.w2
- 2y2.x33.x4
+ 2x3.x42.w2
+ 2y2.x32.x42
- y2.x3.x43
- v.u =
0
- w2.t =
- 2y2.r
+ y2.x4.t
- y2.x42.v
- 2y1.x2.x43
- w1.t =
0
- x2.s =
y2.r
+ y2.x4.t
+ y2.x42.v
+ y1.x2.x43
- x1.s =
2y2.x42.v
- 2y1.x2.x43
- y1.r =
- 2y1.x2.x43
- x3.r =
y2.q
- y2.x33.w2
- 2y2.x32.x4.w2
+ 2y2.x42.u
- 2y2.x3.x42.w2
+ 2y2.x43.w2
+ y1.x43.w2
- u2 =
0
- v.t =
0
- w2.s =
- y2.x4.s
+ 2y2.x32.x4.w2
- y2.x42.u
- y2.x3.x42.w2
- 2y2.x43.w2
- 2y1.x43.w2
- w1.s =
- y2.x42.u
- 2y1.x43.w2
- x2.r =
0
- x1.r =
0
- y1.q =
y1.x43.w2
- u.t =
0
- v.s =
- y2.x42.t
- 2y2.x43.v
- y1.x2.x44
- w2.r =
- y2.p1
- 2y2.x4.r
+ y1.x2.x44
- w1.r =
- 2y2.x43.v
+ y1.x2.x44
- x2.q =
y2.p1
+ 2y2.x4.r
+ 2y2.x42.t
- y2.x43.v
- 2y1.x2.x44
- x1.q =
y2.x43.v
+ y1.x2.x44
- y1.p1 =
2y1.x2.x44
- x3.p1 =
w2.q
+ y2.x3.q
- 2y2.x34.w2
- 2y2.x4.q
- 2y2.x33.x4.w2
- y2.x32.x42.w2
+ 2y2.x43.u
+ y2.x3.x43.w2
- 2y2.x44.w2
- 2y1.x44.w2
- t2 =
0
- u.s =
2y2.x42.s
+ 2y2.x32.x42.w2
+ y2.x43.u
+ 2y2.x3.x43.w2
+ y2.x44.w2
+ y1.x44.w2
- v.r =
0
- w1.q =
2y2.x43.u
- x2.p1 =
0
- x1.p1 =
0
- t.s =
y2.x42.r
+ y2.x43.t
- 2y2.x44.v
+ y1.x2.x45
- u.r =
- 2y2.x43.t
+ 2y1.x2.x45
- v.q =
2y2.x43.t
+ y2.x44.v
+ 2y1.x2.x45
+ y2.w2.q
- w2.p1 =
2y2.x4.p1
+ y1.x2.p2
+ y2.x42.r
- 2y2.x43.t
- y2.x44.v
+ 2y1.x2.x45
- y2.w2.q
- w1.p1 =
2y2.x44.v
- y1.x2.x45
- s2 =
0
- t.r =
0
- u.q =
2y2.x3.x4.q
+ y2.x43.s
+ y2.x44.u
- y2.x3.x44.w2
- 2y2.x45.w2
- 2y1.x45.w2
- v.p1 =
0
- s.r =
- 2y2.x42.p1
- 2y2.x43.r
- 2y2.x44.t
- y2.x4.w2.q
- t.q =
- 2y2.x43.r
- y2.x45.v
- 2y1.x2.x46
- 2y2.x3.w2.q
+ 2y2.x4.w2.q
- u.p1 =
2y2.x44.t
- 2y1.x2.x46
+ 2y2.x4.w2.q
- r2 =
0
- s.q =
x3.x4.w2.q
- 2y2.x32.x4.q
+ 2x42.w2.q
+ 2y2.x3.x42.q
- y2.x43.q
+ 2y2.x44.s
+ y2.x45.u
- 2y2.x3.x45.w2
+ y2.x46.w2
+ 2y1.x46.w2
- t.p1 =
0
- r.q =
2y2.x43.p1
- 2y2.x44.r
+ y2.x45.t
- 2y2.x46.v
- 2y1.x2.x47
- y2.x32.w2.q
- 2y2.x3.x4.w2.q
- 2y2.x42.w2.q
- s.p1 =
- 2y2.x43.p1
+ 2y1.x2.x42.p2
- y2.x44.r
- 2y2.x45.t
- 2y2.x46.v
+ y1.x2.x47
+ 2y2.x3.x4.w2.q
+ 2y2.x42.w2.q
- q2 =
0
- r.p1 =
0
- q.p1 =
- 2y2.x44.p1
+ y1.x2.x43.p2
+ 2y2.x46.t
+ y2.x47.v
- 2y2.x33.w2.q
- 2y2.x32.x4.w2.q
- y2.x3.x42.w2.q
+ y2.x43.w2.q
- p12 =
0
This minimal generating set constitutes a Gröbner
basis for the relations ideal.
Essential ideal:
There are 7 minimal generators:
-
y1.x2
-
y1.w2
-
y2.v
-
y2.u
-
y2.t
-
y2.s
- y2.x3.x4.w2
- 2y2.x42.w2
-
y2.r
Nilradical:
There are 13 minimal generators:
-
y2
-
y1
-
x2
-
x1
-
w2
-
w1
-
v
-
u
-
t
-
s
-
r
-
q
-
p1
This cohomology ring was obtained from a calculation
out to degree 20. The cohomology ring approximation
is stable from degree 20 onwards, and
Carlson's tests detect stability from degree 20
onwards.
This cohomology ring has dimension 3 and depth 2.
Here is a homogeneous system of parameters:
- h1 =
x4
in degree 2
- h2 =
p2
in degree 10
- h3 =
x3
in degree 2
The first
2 terms h1, h2 form
a regular sequence of maximum length.
The remaining
term h3 is
annihilated by the class
y1.
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 7 as a module over the polynomial algebra
on h1, h2.
These free generators are:
- G1 =
y1.x2
in degree 3
- G2 =
y1.w2
in degree 4
- G3 =
y2.v
in degree 5
- G4 =
y2.u
in degree 6
- G5 =
y2.t
in degree 7
- G6 =
y2.s
- y2.x3.x4.w2
- 2y2.x42.w2
in degree 8
- G7 =
y2.r
in degree 9
The essential ideal squares to zero.
A basis for R/(h1, h2, h3) 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
-
x2
in degree 2
-
x1
in degree 2
-
w2
in degree 3
-
w1
in degree 3
-
y1.x2
in degree 3
-
v
in degree 4
-
u
in degree 5
-
y2.v
in degree 5
-
t
in degree 6
-
s
in degree 7
-
y2.t
in degree 7
-
r
in degree 8
-
q
in degree 9
-
y2.r
in degree 9
-
p1
in degree 10
-
y2.p1
in degree 11
A basis for AnnR/(h1, h2)(h3) is as follows.
Carlson's Koszul condition stipulates that this must
be confined to degrees less than 12.
-
y1
in degree 1
-
w1
in degree 3
-
y1.x2
in degree 3
-
y1.w2
in degree 4
-
u
in degree 5
-
y2.v
in degree 5
-
y2.u
in degree 6
-
s
in degree 7
-
y2.t
in degree 7
-
y2.s
in degree 8
-
y2.r
in degree 9
Restriction to maximal subgroup number 1, which is V125
- y1 restricts to
0
- y2 restricts to
y1
- x1 restricts to
0
- x2 restricts to
y1.y2
- x3 restricts to
x1
- x4 restricts to
x3
- w1 restricts to
0
- w2 restricts to
y2.x1
- y1.x2
- v restricts to
y1.y2.x1
- u restricts to
2y1.x1.x3
- t restricts to
2y1.y2.x1.x3
- 2y1.y2.x12
- s restricts to
2y2.x1.x32
+ y2.x12.x3
- y1.x33
- 2y1.x2.x32
+ 2y1.x1.x32
- y1.x1.x2.x3
- 2y1.x12.x3
- r restricts to
y1.y3.x13
- 2y1.y2.x33
+ y1.y2.x1.x32
+ 2y1.y2.x12.x3
+ y1.y2.x13
- q restricts to
y3.x14
+ y2.x1.x33
- 2y2.x12.x32
- y2.x13.x3
+ 2y2.x14
- y1.x34
- y1.x2.x33
- 2y1.x1.x33
+ 2y1.x1.x2.x32
+ 2y1.x12.x32
+ y1.x12.x2.x3
- y1.x13.x3
- 2y1.x13.x2
- p1 restricts to
y2.y3.x14
- 2y1.y3.x13.x3
- y1.y3.x13.x2
+ y1.y3.x14
+ 2y1.y2.x34
- 2y1.y2.x1.x33
+ 2y1.y2.x12.x32
- y1.y2.x13.x3
- p2 restricts to
x25
- 2x1.x34
+ x12.x33
+ 2x13.x32
- x14.x3
- x14.x2
+ 2y2.y3.x14
+ y1.y3.x13.x3
- 2y1.y3.x13.x2
+ y1.y3.x14
- y1.y2.x1.x33
- y1.y2.x13.x3
Restriction to maximal subgroup number 2, which is 125gp2
- y1 restricts to
y1
- y2 restricts to
0
- x1 restricts to
- y1.y2
- x2 restricts to
- 2y1.y2
- x3 restricts to
0
- x4 restricts to
x2
- w1 restricts to
- y1.x1
- w2 restricts to
- y1.x1
- v restricts to
2y1.y2.x2
+ y1.y2.x1
- u restricts to
2y1.x1.x2
- 2y1.x12
- t restricts to
- y1.y2.x22
- 2y1.y2.x1.x2
+ 2y1.y2.x12
- s restricts to
2y1.x1.x22
- 2y1.x12.x2
- y1.x13
- r restricts to
2y1.y2.x23
- 2y1.y2.x1.x22
+ y1.y2.x13
- q restricts to
- 2y1.x1.x23
- 2y1.x12.x22
- y1.x14
- p1 restricts to
- 2y1.y2.x1.x23
- 2y1.y2.x12.x22
- 2y1.y2.x13.x2
+ y1.y2.x14
- p2 restricts to
x15
+ y1.y2.x24
- 2y1.y2.x1.x23
+ y1.y2.x13.x2
+ 2y1.y2.x14
Restriction to maximal subgroup number 3, which is 125gp2
- y1 restricts to
- y1
- y2 restricts to
y1
- x1 restricts to
y1.y2
- x2 restricts to
- 2y1.y2
- x3 restricts to
0
- x4 restricts to
- x2
- w1 restricts to
y1.x1
- w2 restricts to
0
- v restricts to
2y1.y2.x2
- y1.y2.x1
- u restricts to
2y1.x1.x2
+ 2y1.x12
- t restricts to
y1.y2.x22
- 2y1.y2.x1.x2
- 2y1.y2.x12
- s restricts to
y1.x23
+ y1.x1.x22
- 2y1.x12.x2
+ y1.x13
- r restricts to
- y1.y2.x23
+ 2y1.y2.x1.x22
- y1.y2.x13
- q restricts to
- y1.x24
- y1.x1.x23
+ 2y1.x12.x22
+ y1.x14
- p1 restricts to
2y1.y2.x24
- 2y1.y2.x1.x23
+ 2y1.y2.x12.x22
- 2y1.y2.x13.x2
- y1.y2.x14
- p2 restricts to
x15
- y1.y2.x24
- 2y1.y2.x1.x23
+ y1.y2.x13.x2
- 2y1.y2.x14
Restriction to maximal subgroup number 4, which is 125gp2
- y1 restricts to
2y1
- y2 restricts to
y1
- x1 restricts to
- 2y1.y2
- x2 restricts to
2y1.y2
- x3 restricts to
0
- x4 restricts to
2x2
- w1 restricts to
- 2y1.x1
- w2 restricts to
2y1.x1
- v restricts to
- 2y1.y2.x2
+ 2y1.y2.x1
- u restricts to
- 2y1.x1.x2
+ y1.x12
- t restricts to
2y1.y2.x22
+ 2y1.y2.x1.x2
- y1.y2.x12
- s restricts to
2y1.x23
- 2y1.x1.x22
+ 2y1.x12.x2
- 2y1.x13
- r restricts to
y1.y2.x23
- y1.y2.x1.x22
+ 2y1.y2.x13
- q restricts to
- y1.x24
- y1.x12.x22
- 2y1.x14
- p1 restricts to
2y1.y2.x24
- 2y1.y2.x1.x23
- y1.y2.x12.x22
+ 2y1.y2.x13.x2
+ 2y1.y2.x14
- p2 restricts to
x15
+ 2y1.y2.x24
- 2y1.y2.x1.x23
- y1.y2.x13.x2
- y1.y2.x14
Restriction to maximal subgroup number 5, which is 125gp2
- y1 restricts to
y1
- y2 restricts to
y1
- x1 restricts to
- y1.y2
- x2 restricts to
- y1.y2
- x3 restricts to
0
- x4 restricts to
x2
- w1 restricts to
- y1.x1
- w2 restricts to
- 2y1.x1
- v restricts to
2y1.y2.x2
+ y1.y2.x1
- u restricts to
2y1.x1.x2
- 2y1.x12
- t restricts to
- y1.y2.x22
- 2y1.y2.x1.x2
+ 2y1.y2.x12
- s restricts to
- y1.x23
- 2y1.x12.x2
- y1.x13
- r restricts to
- 2y1.y2.x1.x22
+ y1.y2.x13
- q restricts to
- y1.x24
+ 2y1.x1.x23
- 2y1.x12.x22
- y1.x14
- p1 restricts to
2y1.y2.x24
- 2y1.y2.x1.x23
- 2y1.y2.x12.x22
- 2y1.y2.x13.x2
+ y1.y2.x14
- p2 restricts to
x15
+ y1.y2.x24
- 2y1.y2.x1.x23
+ y1.y2.x13.x2
+ 2y1.y2.x14
Restriction to maximal subgroup number 6, which is 125gp2
- y1 restricts to
- 2y1
- y2 restricts to
y1
- x1 restricts to
2y1.y2
- x2 restricts to
0
- x3 restricts to
0
- x4 restricts to
- 2x2
- w1 restricts to
2y1.x1
- w2 restricts to
y1.x1
- v restricts to
- 2y1.y2.x2
- 2y1.y2.x1
- u restricts to
- 2y1.x1.x2
- y1.x12
- t restricts to
- 2y1.y2.x22
+ 2y1.y2.x1.x2
+ y1.y2.x12
- s restricts to
- 2y1.x23
+ y1.x1.x22
+ 2y1.x12.x2
+ 2y1.x13
- r restricts to
- 2y1.y2.x23
+ y1.y2.x1.x22
- 2y1.y2.x13
- q restricts to
- y1.x24
+ y1.x1.x23
+ y1.x12.x22
+ 2y1.x14
- p1 restricts to
2y1.y2.x24
- 2y1.y2.x1.x23
+ y1.y2.x12.x22
+ 2y1.y2.x13.x2
- 2y1.y2.x14
- p2 restricts to
x15
- 2y1.y2.x24
- 2y1.y2.x1.x23
- y1.y2.x13.x2
+ y1.y2.x14
Restriction to maximal elementary abelian number 1, which is V125
- y1 restricts to
0
- y2 restricts to
y2
- x1 restricts to
0
- x2 restricts to
y2.y3
- x3 restricts to
x2
- x4 restricts to
x3
+ x1
- w1 restricts to
0
- w2 restricts to
y3.x2
- y2.x3
- v restricts to
y2.y3.x2
- u restricts to
2y2.x2.x3
+ 2y2.x1.x2
- t restricts to
2y2.y3.x2.x3
- 2y2.y3.x22
+ 2y2.y3.x1.x2
- s restricts to
2y3.x2.x32
+ y3.x22.x3
- y3.x1.x2.x3
+ y3.x1.x22
+ 2y3.x12.x2
+ 2y2.x33
+ y2.x2.x32
- 2y2.x22.x3
- 2y2.x1.x32
- 2y2.x1.x2.x3
- 2y2.x1.x22
+ 2y2.x12.x2
- y2.x13
- r restricts to
- 2y2.y3.x33
+ y2.y3.x2.x32
+ 2y2.y3.x22.x3
+ 2y2.y3.x23
- y2.y3.x1.x32
+ 2y2.y3.x1.x2.x3
+ 2y2.y3.x1.x22
- y2.y3.x12.x3
+ y2.y3.x12.x2
- 2y2.y3.x13
- y1.y2.x23
- q restricts to
y3.x2.x33
- 2y3.x22.x32
- y3.x23.x3
- 2y3.x24
- 2y3.x1.x2.x32
+ y3.x1.x22.x3
- y3.x1.x23
- 2y3.x12.x2.x3
- 2y3.x12.x22
+ y3.x13.x2
- 2y2.x34
- 2y2.x22.x32
+ 2y2.x23.x3
- 2y2.x1.x33
- 2y2.x1.x2.x32
- y2.x1.x23
+ y2.x12.x32
+ y2.x12.x2.x3
+ 2y2.x12.x22
- 2y2.x13.x2
- y2.x14
+ y1.x24
- p1 restricts to
2y2.y3.x34
- 2y2.y3.x2.x33
+ 2y2.y3.x22.x32
+ y2.y3.x23.x3
+ y2.y3.x24
- 2y2.y3.x1.x33
- y2.y3.x1.x2.x32
- y2.y3.x1.x22.x3
+ 2y2.y3.x1.x23
+ 2y2.y3.x12.x32
- y2.y3.x12.x2.x3
+ 2y2.y3.x12.x22
- 2y2.y3.x13.x3
- 2y2.y3.x13.x2
+ 2y2.y3.x14
- y1.y3.x24
- 2y1.y2.x23.x3
- y1.y2.x24
+ 2y1.y2.x1.x23
- p2 restricts to
x35
- 2x2.x34
+ x22.x33
+ 2x23.x32
- 2x24.x3
+ 2x1.x2.x33
- 2x1.x22.x32
- x1.x23.x3
- x1.x24
- 2x12.x2.x32
- 2x12.x22.x3
+ 2x12.x23
+ 2x13.x2.x3
+ x13.x22
- 2x14.x2
- y2.y3.x2.x33
- 2y2.y3.x23.x3
+ y2.y3.x24
+ 2y2.y3.x1.x2.x32
+ 2y2.y3.x12.x2.x3
- y2.y3.x13.x2
- 2y1.y3.x24
+ y1.y2.x23.x3
- y1.y2.x24
- y1.y2.x1.x23
Restriction to the greatest central elementary abelian, which is V25
- y1 restricts to
0
- y2 restricts to
0
- x1 restricts to
0
- x2 restricts to
0
- x3 restricts to
0
- x4 restricts to
- x1
- w1 restricts to
0
- w2 restricts to
0
- v restricts to
0
- u restricts to
0
- t restricts to
0
- s restricts to
0
- r restricts to
0
- q restricts to
0
- p1 restricts to
0
- p2 restricts to
2x25
+ x15
(1 + 2t + 2t2
+ 2t3 + t4) /
(1 - t2)2 (1 - t10)
Back to the groups of order 625