GROUP OF ORDER 32 # 37
GROUP # 37
The MAGMA library number for G is 29
GrpPC : G of order 32 = 2^5
PC-Relations:
G.1^2 = G.5,
G.2^2 = G.5,
G.3^2 = G.5,
G.3^G.1 = G.3 * G.4,
G.3^G.2 = G.3 * G.5
The center of G is abelian of type [ 2, 2 ]
The orders of the terms of the lower central series are [ 32, 4, 1 ]
The orders of the terms of the upper central series are [ 1, 4, 32 ]
The order of the Frattini subgroup is 4
The exponent of G is 4
G has a unique maximal elementary abelian subgroup which is normal and
has order 8. Its centralizer has order 16
The cohomology ring of G is the quotient of a polynomial
ring in the variables [
z,
y,
x,
w,
v,
u
]
in degrees [ 1, 1, 1, 2, 3, 4 ]
by the ideal generated by the relations
[
z^2 + y^2 + y*x + x^2,
z*x,
x^3,
x*v,
y^4*w + y^2*u + y*x*u + x^2*u + v^2
]
The Hilbert series for the cohomology ring is -1/(t^5 - 3*t^4 + 4*t^3 - 4*t^2 +
3*t - 1)
Its denominator factors as
( t - 1
)^3
( t^2 + 1
)^1
The Krull dimension of the cohomology ring is 3
The longest regular sequence consists of the generators [
w,
u
]
A homogeneous set of parameters is the set [
w,
u,
z^2
]
of degrees [ 2, 4, 2 ]
The hypercohomolgy spectral sequence has E2 term:
ROW (0)
[
y,
z,
x
]
[
y*x,
x^2,
z*y
]
[
y*x^2,
v
]
[
z*v,
y*v
]
[
z*y*v
]
ROW (1)
[
x
]
[
y*x,
x^2
]
[
y*x^2
]
The spectral sequence satisfies Poincaré duality
Restriction to the Maximal Elementary Abelian Subgroup
Generated by [ G.5, G.4, G.1 * G.2 * G.4 * G.5 ]
The cohomology ring of E is a polynomial ring in the variables [
z,
y,
x
]
The images of the generators of the cohomology of G
restricted to E are
[
z,
z,
0,
z*y + y^2,
z^3 + z^2*y + z*x^2,
z^4 + z^3*y + x^4
]
in the cohomology of E
The kernel of the restriction to E (Minimal Prime) of the
cohomology of G is generated by
[
z + y,
x
]
This ideal is also the nilradical of the cohomology of G.
Restrictions to Maximal Subgroups
Maximal Subgroup H # 1
The Group H is Isomorphic to the
Group of Order 16 Number 9
GrpPC of order 16 = 2^4
PC-Relations:
$.1^2 = $.3,
$.2^$.1 = $.2 * $.4
Generated by [ G.1 * G.2, G.3 ]
The images of the generators of the cohomology of G
restricted to H are
[
y,
y,
z,
v,
y^3 + y*w + y*v,
y^4 + y^2*v + w^2 + v^2
]
in the cohomology of H
The kernel of the restriction to H of the cohomology of G
is generated by
[
z + y
]
Maximal Subgroup H # 2
The Group H is Isomorphic to the
Group of Order 16 Number 10
GrpPC of order 16 = 2^4
PC-Relations:
$.1^2 = $.3,
$.2^2 = $.4,
$.2^$.1 = $.2 * $.4
Generated by [ G.1, G.1 * G.3 * G.4 * G.5 ]
The images of the generators of the cohomology of G
restricted to H are
[
z + y,
0,
y,
w,
z*x + y*x,
x^2
]
in the cohomology of H
The kernel of the restriction to H of the cohomology of G
is generated by
[
y
]
Maximal Subgroup H # 3
The Group H is Isomorphic to the
Group of Order 16 Number 10
GrpPC of order 16 = 2^4
PC-Relations:
$.1^2 = $.3,
$.2^2 = $.4,
$.2^$.1 = $.2 * $.4
Generated by [ G.2 * G.3, G.1 * G.2 * G.3 * G.5 ]
The images of the generators of the cohomology of G
restricted to H are
[
y,
z + y,
z + y,
w,
y*x,
x^2
]
in the cohomology of H
The kernel of the restriction to H of the cohomology of G
is generated by
[
y + x
]
Maximal Subgroup H # 4
The Group H is Isomorphic to the
Group of Order 16 Number 9
GrpPC of order 16 = 2^4
PC-Relations:
$.1^2 = $.3,
$.2^$.1 = $.2 * $.4
Generated by [ G.2 * G.3, G.1 * G.2 * G.5 ]
The images of the generators of the cohomology of G
restricted to H are
[
y,
z + y,
z,
v,
y*w + y*v,
y^2*v + w^2 + v^2
]
in the cohomology of H
The kernel of the restriction to H of the cohomology of G
is generated by
[
z + y + x
]
Maximal Subgroup H # 5
The Group H is Isomorphic to the
Group of Order 16 Number 7
GrpPC of order 16 = 2^4
PC-Relations:
$.1^2 = $.4,
$.2^2 = $.4,
$.2^$.1 = $.2 * $.4
Generated by [ G.4, G.2, G.3 ]
of type Cyclic(2) x Quaternion(8)
The images of the generators of the cohomology of G
restricted to H are
[
0,
y,
z,
z*x + x^2,
z^2*x,
w
]
in the cohomology of H
The kernel of the restriction to H of the cohomology of G
is generated by
[
z
]
Maximal Subgroup H # 6
The Group H is Isomorphic to the
Group of Order 16 Number 10
GrpPC of order 16 = 2^4
PC-Relations:
$.1^2 = $.3,
$.2^2 = $.4,
$.2^$.1 = $.2 * $.4
Generated by [ G.2, G.1 * G.3 ]
The images of the generators of the cohomology of G
restricted to H are
[
z,
y,
z,
x,
z*w,
w^2
]
in the cohomology of H
The kernel of the restriction to H of the cohomology of G
is generated by
[
z + x
]
Maximal Subgroup H # 7
The group H is abelian of type [ 4, 2, 2 ]
The cohomology ring of H is a the quotient of a polynomial ring
in the variables [
z,
y,
x,
w
]
in degrees [ 1, 1, 1, 2 ]
by the ideal of relations
[
z^2
]
The images of the generators of the cohomology of G
restricted to H are
[
y,
z + y,
0,
y*x + x^2,
y^2*x + y*w,
y^3*x + w^2
]
in the cohomology of H
The kernel of the restriction to H of the cohomology of G
is generated by
[
x
]
The essential cohomology of G is zero
CHECKS
paramflag = true
qregflagflag = true
cmflag = false
essflag = true
centflag = true
THE COMPUTATION IS COMPLETE