Small group number 13 of order 625
G = M125xC5 is Direct product M125 x C_5
G has 3 minimal generators, rank 3 and exponent 25.
The centre has rank 2.
The 31 maximal subgroups are:
Ab(25,5) (5x), M125 (25x), 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 10 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, a regular element
- w in degree 3, a nilpotent element
- u in degree 5, a nilpotent element
- s in degree 7, a nilpotent element
- q in degree 9, a nilpotent element
- p in degree 10, a regular element
There are 21 minimal relations:
- y32 =
0
- y22 =
0
- y12 =
0
- y1.x1 =
0
- y1.w =
0
- x1.w =
0
- w2 =
0
- y1.u =
0
- x1.u =
0
- w.u =
0
- y1.s =
0
- x1.s =
0
- u2 =
0
- w.s =
0
- y1.q =
0
- u.s =
0
- w.q =
0
- s2 =
0
- u.q =
0
- s.q =
0
- q2 =
0
This minimal generating set constitutes a Gröbner
basis for the relations ideal.
Essential ideal:
There are 8 minimal generators:
-
y1.y2.y3
-
y1.y2.x2
-
y2.y3.w
-
y2.x2.w
-
y2.y3.u
-
y2.x2.u
-
y2.y3.s
-
y2.x2.s
Nilradical:
There are 7 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 3 and depth 2.
Here is a homogeneous system of parameters:
- h1 =
x2
in degree 2
- h2 =
p
in degree 10
- h3 =
x1
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 8 as a module over the polynomial algebra
on h1, h2.
These free generators are:
- G1 =
y1.y2.y3
in degree 3
- G2 =
y1.y2.x2
in degree 4
- G3 =
y2.y3.w
in degree 5
- G4 =
y2.x2.w
in degree 6
- G5 =
y2.y3.u
in degree 7
- G6 =
y2.x2.u
in degree 8
- G7 =
y2.y3.s
in degree 9
- G8 =
y2.x2.s
in degree 10
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
-
y3
in degree 1
-
y2
in degree 1
-
y1
in degree 1
-
y2.y3
in degree 2
-
y1.y3
in degree 2
-
y1.y2
in degree 2
-
w
in degree 3
-
y1.y2.y3
in degree 3
-
y3.w
in degree 4
-
y2.w
in degree 4
-
u
in degree 5
-
y2.y3.w
in degree 5
-
y3.u
in degree 6
-
y2.u
in degree 6
-
s
in degree 7
-
y2.y3.u
in degree 7
-
y3.s
in degree 8
-
y2.s
in degree 8
-
q
in degree 9
-
y2.y3.s
in degree 9
-
y3.q
in degree 10
-
y2.q
in degree 10
-
y2.y3.q
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
-
y1.y3
in degree 2
-
y1.y2
in degree 2
-
w
in degree 3
-
y1.y2.y3
in degree 3
-
y3.w
in degree 4
-
y2.w
in degree 4
-
u
in degree 5
-
y2.y3.w
in degree 5
-
y3.u
in degree 6
-
y2.u
in degree 6
-
s
in degree 7
-
y2.y3.u
in degree 7
-
y3.s
in degree 8
-
y2.s
in degree 8
-
y2.y3.s
in degree 9
Restriction to maximal subgroup number 1, which is V125
- y1 restricts to
0
- y2 restricts to
y2
- y3 restricts to
y1
- x1 restricts to
x2
- x2 restricts to
x1
- w restricts to
0
- u restricts to
0
- s restricts to
0
- q restricts to
y3.x24
- y2.x23.x3
- p restricts to
- x35
+ x24.x3
+ y2.y3.x24
Restriction to maximal subgroup number 2, which is 125gp2
- y1 restricts to
y1
- y2 restricts to
0
- y3 restricts to
y2
- x1 restricts to
0
- x2 restricts to
x1
- w restricts to
y1.x2
- u restricts to
- 2y1.x22
- s restricts to
y1.x23
- q restricts to
- y1.x24
- p restricts to
- x25
Restriction to maximal subgroup number 3, which is 125gp2
- y1 restricts to
- y1
- y2 restricts to
y1
- y3 restricts to
y2
- x1 restricts to
0
- x2 restricts to
x1
- w restricts to
y1.x2
- u restricts to
2y1.x22
- s restricts to
y1.x23
- q restricts to
y1.x24
- p restricts to
x25
Restriction to maximal subgroup number 4, which is 125gp2
- y1 restricts to
2y1
- y2 restricts to
y1
- y3 restricts to
y2
- x1 restricts to
0
- x2 restricts to
x1
- w restricts to
- y1.x2
- u restricts to
- y1.x22
- s restricts to
y1.x23
- q restricts to
- 2y1.x24
- p restricts to
- 2x25
Restriction to maximal subgroup number 5, which is 125gp2
- y1 restricts to
y1
- y2 restricts to
y1
- y3 restricts to
y2
- x1 restricts to
0
- x2 restricts to
x1
- w restricts to
y1.x2
- u restricts to
- 2y1.x22
- s restricts to
y1.x23
- q restricts to
- y1.x24
- p restricts to
- x25
Restriction to maximal subgroup number 6, which is 125gp2
- y1 restricts to
- 2y1
- y2 restricts to
y1
- y3 restricts to
y2
- x1 restricts to
0
- x2 restricts to
x1
- w restricts to
- y1.x2
- u restricts to
y1.x22
- s restricts to
y1.x23
- q restricts to
2y1.x24
- p restricts to
2x25
Restriction to maximal subgroup number 7, which is 125gp4
- y1 restricts to
- 2y1
- y2 restricts to
y2
- y1
- y3 restricts to
0
- x1 restricts to
x
+ y1.y2
- x2 restricts to
0
- w restricts to
- w
- u restricts to
2u
- s restricts to
s
- q restricts to
- 2q
- p restricts to
- 2p
Restriction to maximal subgroup number 8, which is 125gp4
- y1 restricts to
2y1
- y2 restricts to
y2
- 2y1
- y3 restricts to
- 2y1
- x1 restricts to
x
+ 2y1.y2
- x2 restricts to
2y1.y2
- w restricts to
- w
- u restricts to
- 2u
- s restricts to
s
- q restricts to
2q
- p restricts to
2p
Restriction to maximal subgroup number 9, which is 125gp4
- y1 restricts to
- 2y1
- y2 restricts to
y2
- y3 restricts to
- y1
- x1 restricts to
x
- x2 restricts to
y1.y2
- w restricts to
- w
- u restricts to
2u
- s restricts to
s
- q restricts to
- 2q
- p restricts to
- 2p
Restriction to maximal subgroup number 10, which is 125gp4
- y1 restricts to
- 2y1
- y2 restricts to
y2
- 2y1
- y3 restricts to
- 2y1
- x1 restricts to
x
+ 2y1.y2
- x2 restricts to
2y1.y2
- w restricts to
- w
- u restricts to
2u
- s restricts to
s
- q restricts to
- 2q
- p restricts to
- 2p
Restriction to maximal subgroup number 11, which is 125gp4
- y1 restricts to
y1
- y2 restricts to
y2
+ y1
- y3 restricts to
2y1
- x1 restricts to
x
- y1.y2
- x2 restricts to
- 2y1.y2
- w restricts to
w
- u restricts to
u
- s restricts to
s
- q restricts to
q
- p restricts to
p
Restriction to maximal subgroup number 12, which is 125gp4
- y1 restricts to
- 2y1
- y2 restricts to
y2
- 2y1
- y3 restricts to
- y2
+ 2y1
- x1 restricts to
x
+ 2y1.y2
- x2 restricts to
- x
- 2y1.y2
- w restricts to
- w
- u restricts to
2u
- s restricts to
s
- q restricts to
- 2q
- p restricts to
- 2p
Restriction to maximal subgroup number 13, which is 125gp4
- y1 restricts to
- y1
- y2 restricts to
y2
- y3 restricts to
- y2
+ y1
- x1 restricts to
x
- x2 restricts to
- x
- y1.y2
- w restricts to
w
- u restricts to
- u
- s restricts to
s
- q restricts to
- q
- p restricts to
- p
Restriction to maximal subgroup number 14, which is 125gp4
- y1 restricts to
- 2y1
- y2 restricts to
y2
- y1
- y3 restricts to
- y2
- x1 restricts to
x
+ y1.y2
- x2 restricts to
- x
- w restricts to
- w
- u restricts to
2u
- s restricts to
s
- q restricts to
- 2q
- p restricts to
- 2p
Restriction to maximal subgroup number 15, which is 125gp4
- y1 restricts to
y1
- y2 restricts to
y2
+ 2y1
- y3 restricts to
- y2
- y1
- x1 restricts to
x
- 2y1.y2
- x2 restricts to
- x
+ y1.y2
- w restricts to
w
- u restricts to
u
- s restricts to
s
- q restricts to
q
- p restricts to
p
Restriction to maximal subgroup number 16, which is 125gp4
- y1 restricts to
- 2y1
- y2 restricts to
y2
+ 2y1
- y3 restricts to
- y2
- y1
- x1 restricts to
x
- 2y1.y2
- x2 restricts to
- x
+ y1.y2
- w restricts to
- w
- u restricts to
2u
- s restricts to
s
- q restricts to
- 2q
- p restricts to
- 2p
Restriction to maximal subgroup number 17, which is 125gp4
- y1 restricts to
2y1
- y2 restricts to
y2
+ 2y1
- y3 restricts to
- 2y2
+ y1
- x1 restricts to
x
- 2y1.y2
- x2 restricts to
- 2x
- y1.y2
- w restricts to
- w
- u restricts to
- 2u
- s restricts to
s
- q restricts to
2q
- p restricts to
2p
Restriction to maximal subgroup number 18, which is 125gp4
- y1 restricts to
y1
- y2 restricts to
y2
- y1
- y3 restricts to
- 2y2
+ y1
- x1 restricts to
x
+ y1.y2
- x2 restricts to
- 2x
- y1.y2
- w restricts to
w
- u restricts to
u
- s restricts to
s
- q restricts to
q
- p restricts to
p
Restriction to maximal subgroup number 19, which is 125gp4
- y1 restricts to
y1
- y2 restricts to
y2
- y1
- y3 restricts to
- 2y2
- x1 restricts to
x
+ y1.y2
- x2 restricts to
- 2x
- w restricts to
w
- u restricts to
u
- s restricts to
s
- q restricts to
q
- p restricts to
p
Restriction to maximal subgroup number 20, which is 125gp4
- y1 restricts to
- 2y1
- y2 restricts to
y2
+ y1
- y3 restricts to
- 2y2
+ y1
- x1 restricts to
x
- y1.y2
- x2 restricts to
- 2x
- y1.y2
- w restricts to
- w
- u restricts to
2u
- s restricts to
s
- q restricts to
- 2q
- p restricts to
- 2p
Restriction to maximal subgroup number 21, which is 125gp4
- y1 restricts to
- y1
- y2 restricts to
y2
+ y1
- y3 restricts to
- 2y2
+ y1
- x1 restricts to
x
- y1.y2
- x2 restricts to
- 2x
- y1.y2
- w restricts to
w
- u restricts to
- u
- s restricts to
s
- q restricts to
- q
- p restricts to
- p
Restriction to maximal subgroup number 22, which is 125gp4
- y1 restricts to
- 2y1
- y2 restricts to
y2
+ y1
- y3 restricts to
y2
+ y1
- x1 restricts to
x
- y1.y2
- x2 restricts to
x
- y1.y2
- w restricts to
- w
- u restricts to
2u
- s restricts to
s
- q restricts to
- 2q
- p restricts to
- 2p
Restriction to maximal subgroup number 23, which is 125gp4
- y1 restricts to
2y1
- y2 restricts to
y2
+ y1
- y3 restricts to
y2
- y1
- x1 restricts to
x
- y1.y2
- x2 restricts to
x
+ y1.y2
- w restricts to
- w
- u restricts to
- 2u
- s restricts to
s
- q restricts to
2q
- p restricts to
2p
Restriction to maximal subgroup number 24, which is 125gp4
- y1 restricts to
- y1
- y2 restricts to
y2
- 2y1
- y3 restricts to
y2
- x1 restricts to
x
+ 2y1.y2
- x2 restricts to
x
- w restricts to
w
- u restricts to
- u
- s restricts to
s
- q restricts to
- q
- p restricts to
- p
Restriction to maximal subgroup number 25, which is 125gp4
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
y2
+ y1
- x1 restricts to
x
- x2 restricts to
x
- y1.y2
- w restricts to
w
- u restricts to
u
- s restricts to
s
- q restricts to
q
- p restricts to
p
Restriction to maximal subgroup number 26, which is 125gp4
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
y2
+ 2y1
- x1 restricts to
x
- x2 restricts to
x
- 2y1.y2
- w restricts to
w
- u restricts to
u
- s restricts to
s
- q restricts to
q
- p restricts to
p
Restriction to maximal subgroup number 27, which is 125gp4
- y1 restricts to
- 2y1
- y2 restricts to
y2
- 2y1
- y3 restricts to
2y2
+ y1
- x1 restricts to
x
+ 2y1.y2
- x2 restricts to
2x
- y1.y2
- w restricts to
- w
- u restricts to
2u
- s restricts to
s
- q restricts to
- 2q
- p restricts to
- 2p
Restriction to maximal subgroup number 28, which is 125gp4
- y1 restricts to
- y1
- y2 restricts to
y2
- 2y1
- y3 restricts to
2y2
+ 2y1
- x1 restricts to
x
+ 2y1.y2
- x2 restricts to
2x
- 2y1.y2
- w restricts to
w
- u restricts to
- u
- s restricts to
s
- q restricts to
- q
- p restricts to
- p
Restriction to maximal subgroup number 29, which is 125gp4
- y1 restricts to
- y1
- y2 restricts to
y2
- y3 restricts to
2y2
+ 2y1
- x1 restricts to
x
- x2 restricts to
2x
- 2y1.y2
- w restricts to
w
- u restricts to
- u
- s restricts to
s
- q restricts to
- q
- p restricts to
- p
Restriction to maximal subgroup number 30, which is 125gp4
- y1 restricts to
2y1
- y2 restricts to
y2
- y3 restricts to
2y2
+ 2y1
- x1 restricts to
x
- x2 restricts to
2x
- 2y1.y2
- w restricts to
- w
- u restricts to
- 2u
- s restricts to
s
- q restricts to
2q
- p restricts to
2p
Restriction to maximal subgroup number 31, which is 125gp4
- y1 restricts to
- 2y1
- y2 restricts to
y2
+ 2y1
- y3 restricts to
2y2
- x1 restricts to
x
- 2y1.y2
- x2 restricts to
2x
- w restricts to
- w
- u restricts to
2u
- s restricts to
s
- q restricts to
- 2q
- p restricts to
- 2p
Restriction to maximal elementary abelian number 1, which is V125
- y1 restricts to
0
- y2 restricts to
y3
- y3 restricts to
y2
- x1 restricts to
x3
- x2 restricts to
x2
- w restricts to
0
- u restricts to
0
- s restricts to
0
- q restricts to
- y3.x1.x33
+ y1.x34
- p restricts to
x1.x34
- x15
- y1.y3.x34
Restriction to the greatest central elementary abelian, which is V25
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
- y2
- x1 restricts to
0
- x2 restricts to
- x2
- w restricts to
0
- u restricts to
0
- s restricts to
0
- q restricts to
0
- p restricts to
x15
(1 + 3t + 3t2
+ t3) /
(1 - t2)2 (1 - t10)
Back to the groups of order 625