GROUP OF ORDER 32 # 22
GROUP # 22
The MAGMA library number for G is 17
GrpPC : G of order 32 = 2^5
PC-Relations:
G.1^2 = G.2,
G.2^2 = G.4,
G.4^2 = G.5,
G.3^G.1 = G.3 * G.5
The center of G is abelian of type [ 8 ]
The orders of the terms of the lower central series are [ 32, 2, 1 ]
The orders of the terms of the upper central series are [ 1, 8, 32 ]
The order of the Frattini subgroup is 8
The exponent of G is 16
G has a unique maximal elementary abelian subgroup which is normal and
has order 4. Its centralizer has order 16
The cohomology ring of G is the quotient of a polynomial
ring in the variables [
z,
y,
x,
w
]
in degrees [ 1, 1, 3, 4 ]
by the ideal generated by the relations
[
z^2,
z*y^2,
z*x,
x^2
]
The Hilbert series for the cohomology ring is 1/(t^4 - 2*t^3 + 2*t^2 - 2*t + 1)
Its denominator factors as
( t - 1
)^2
( t^2 + 1
)^1
The Krull dimension of the cohomology ring is 2
The longest regular sequence consists of the generators [
w
]
A homogeneous set of parameters is the set [
w,
y^2
]
of degrees [ 4, 2 ]
The hypercohomolgy spectral sequence has E2 term:
ROW (0)
[
z,
y
]
[
z*y
]
[
x
]
[
y*x
]
ROW (1)
[
z
]
[
z*y
]
The spectral sequence satisfies Poincaré duality
Restriction to the Maximal Elementary Abelian Subgroup
Generated by [ G.3, G.5 ]
The cohomology ring of E is a polynomial ring in the variables [
z,
y
]
The images of the generators of the cohomology of G
restricted to E are
[
0,
z,
0,
z^2*y^2 + y^4
]
in the cohomology of E
The kernel of the restriction to E (Minimal Prime) of the
cohomology of G is generated by
[
z,
x
]
This ideal is also the nilradical of the cohomology of G.
Restrictions to Maximal Subgroups
Maximal Subgroup H # 1
The group H is abelian of type [ 8, 2 ]
The cohomology ring of H is a the quotient of a polynomial ring
in the variables [
z,
y,
x
]
in degrees [ 1, 1, 2 ]
by the ideal of relations
[
z^2
]
The images of the generators of the cohomology of G
restricted to H are
[
0,
y,
z*y^2,
y^2*x + x^2
]
in the cohomology of H
The kernel of the restriction to H of the cohomology of G
is generated by
[
z
]
Maximal Subgroup H # 2
The group H is abelian of type [ 16 ]
The cohomology ring of H is a the quotient of a polynomial ring
in the variables [
z,
y
]
in degrees [ 1, 2 ]
by the ideal of relations
[
z^2
]
The images of the generators of the cohomology of G
restricted to H are
[
z,
z,
z*y,
y^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 # 3
The group H is abelian of type [ 16 ]
The cohomology ring of H is a the quotient of a polynomial ring
in the variables [
z,
y
]
in degrees [ 1, 2 ]
by the ideal of relations
[
z^2
]
The images of the generators of the cohomology of G
restricted to H are
[
z,
0,
z*y,
y^2
]
in the cohomology of H
The kernel of the restriction to H of the cohomology of G
is generated by
[
y
]
The essential cohomology of G is
generated as an ideal by the elements
[
z*y
]
The annihilator of the Essential Cohomology has dimension 1
The essential cohomology is a free module over the polynomial subring
Q of the cohomology ring of G generated by [
w
]
.
The essential cohomology is generated as a module over Q
by the elements
[]
[
z*y
]
CHECKS
paramflag = true
qregflagflag = true
cmflag = false
essflag = true
centflag = true
THE COMPUTATION IS COMPLETE