GROUP OF ORDER 32 # 28
GROUP # 28
The MAGMA library number for G is 10
GrpPC : G of order 32 = 2^5
PC-Relations:
G.1^2 = G.3 * G.4 * G.5,
G.2^2 = G.3,
G.4^2 = G.5,
G.2^G.1 = G.2 * G.4,
G.4^G.2 = G.4 * 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, 2, 1 ]
The orders of the terms of the upper central series are [ 1, 4, 8, 32 ]
The order of the Frattini subgroup is 8
The exponent of G is 8
G has a unique maximal elementary abelian subgroup which is central in
G and has order 4
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, 2, 2, 3, 4 ]
by the ideal generated by the relations
[
z^2 + y^2,
z*y + y^2,
y^3,
z*x + y*x,
y^2*x + z*v,
y^2*w + x^2,
y*v,
x*v,
v^2
]
The Hilbert series for the cohomology ring is 1/(t^2 - 2*t + 1)
Its denominator factors as
( t - 1
)^2
The Krull dimension of the cohomology ring is 2
The cohomology ring is Cohen-Macaulay and the longest regular sequence consists
of the generators [
w,
u
]
Restriction to the Maximal Elementary Abelian Subgroup
Generated by [ G.3, 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,
0,
0,
z^2 + y^2,
0,
z^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,
v
]
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
[
z,
0,
z*y,
y^2,
z*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 # 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.2, G.4 ]
The images of the generators of the cohomology of G
restricted to H are
[
0,
z,
y^2,
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
]
Maximal Subgroup H # 3
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.3, G.1 * G.2 * G.4, G.4 ]
of type Cyclic(2) x Quaternion(8)
The images of the generators of the cohomology of G
restricted to H are
[
y,
y,
y^2 + y*x,
y^2 + x^2,
z*y^2 + y^2*x,
w
]
in the cohomology of H
The kernel of the restriction to H of the cohomology of G
is generated by
[
z + y
]
The essential cohomology of G is
generated as an ideal by the elements
[
z*v
]
The annihilator of the Essential Cohomology has dimension 2
The essential cohomology is a free module over the polynomial subring
Q of the cohomology ring of G generated by [
w,
u
]
.
The essential cohomology is generated as a module over Q
by the elements
[]
[]
[]
[
z*v
]
CHECKS
paramflag = true
qregflagflag = true
cmflag = true
essflag = true
centflag = true
THE COMPUTATION IS COMPLETE