GROUP OF ORDER 32 # 48
GROUP # 48
The MAGMA library number for G is 8
GrpPC : G of order 32 = 2^5
PC-Relations:
G.1^2 = G.2,
G.2^2 = G.4,
G.3^2 = G.4,
G.3^G.1 = G.3 * G.5,
G.3^G.2 = G.3 * G.4,
G.5^G.1 = G.4 * G.5
The center of G is abelian of type [ 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, 2, 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 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,
v,
u,
t,
s,
r
]
in degrees [ 1, 1, 2, 2, 3, 5, 5, 6, 8 ]
by the ideal generated by the relations
[
z^2,
z*y,
y^3 + z*w,
z*x,
y^2*x + z*v,
x^2 + z*v,
x*w + y*v,
z*w^2,
y^2*v + x*v,
y^2*w^2 + y*w*v + y*u,
z*w*v + v^2 + y*u,
z*u,
z*t,
y*w^3 + w^2*v + y^2*t + w*u,
x*u,
x*t + y*s,
z*s,
y*w^2*v + y*w*u + v*t + w*s,
y^2*s + x*s,
v*u + v*t + x*s + w*s,
y^2*w*u + y^2*w*t + y*w*s + v*s,
w^5 + w*v*t + y*v*s + y^2*r + u*t + t^2,
y*w^2*u + w*v*t + w^2*s + u^2,
y*w^2*t + w^2*s + u*t,
w^3*u + y*t^2 + w*v*s + y*x*r + z*w*r + t*s,
y*w^2*s + y*u*t + u*s,
y^2*t^2 + y*t*s + s^2
]
The Hilbert series for the cohomology ring is (t^5 + t^2 + 1)/(t^8 - 2*t^7 +
2*t^6 - 2*t^5 + 2*t^4 - 2*t^3 + 2*t^2 - 2*t + 1)
Its numerator factors as
( t^5 + t^2 + 1
)
Its denominator factors as
( t - 1
)^2
( t^2 + 1
)^1
( t^4 + 1
)^1
The Krull dimension of the cohomology ring is 2
The longest regular sequence consists of the generators [
r
]
A homogeneous set of parameters is the set [
r,
w
]
of degrees [ 8, 2 ]
The hypercohomolgy spectral sequence has E2 term:
ROW (0)
[
z,
y
]
[
y^2,
x
]
[
y*x,
v
]
[
z*v
]
[
t,
u
]
[
y*t,
s
]
[
y*s
]
[
x*s
]
ROW (1)
[]
[]
[
z*w
]
[]
[]
[
v^2 + y*u
]
The spectral sequence satisfies Poincaré duality
Restriction to the Maximal Elementary Abelian Subgroup
Generated by [ G.4, 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,
0,
0,
z^5,
0,
z^8 + z^4*y^4 + y^8
]
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,
u,
s
]
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 7
GrpPC of order 16 = 2^4
PC-Relations:
$.1^2 = $.4,
$.2^2 = $.4,
$.2^$.1 = $.2 * $.4
Generated by [ G.3, G.2, G.5 ]
of type Cyclic(2) x Quaternion(8)
The images of the generators of the cohomology of G
restricted to H are
[
0,
z,
z^2 + y^2,
z*x + x^2,
y^2*x + y*x^2,
z^2*y*x^2 + z*x^4 + y*x^4,
z*x^4 + x^5 + z*w,
z*y*x^4 + y*x^5 + z*y*w,
z*y^2*x^5 + z*y*x^6 + y^2*x^6 + y*x^7 + x^8 + z*y^2*x*w + z^2*x^2*w +
y^2*x^2*w + x^4*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 # 2
The Group H is Isomorphic to the
Group of Order 16 Number 11
GrpPC of order 16 = 2^4
PC-Relations:
$.1^2 = $.3,
$.2^2 = $.3,
$.3^2 = $.4,
$.2^$.1 = $.2 * $.4
Generated by [ G.1, G.1 * G.2 * G.5 ]
The images of the generators of the cohomology of G
restricted to H are
[
z + y,
0,
z^2 + z*y,
z*y,
z*y^2 + x,
z^2*y^3 + y^2*x + z*w + y*w,
z^2*y^3,
z^2*y^4 + z*y^2*x + z*y*w + y^2*w,
z^3*y^5 + z^2*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
[
y
]
Maximal Subgroup H # 3
The Group H is Isomorphic to the
Group of Order 16 Number 11
GrpPC of order 16 = 2^4
PC-Relations:
$.1^2 = $.3,
$.2^2 = $.3,
$.3^2 = $.4,
$.2^$.1 = $.2 * $.4
Generated by [ G.1 * G.2 * G.3 * G.4 * G.5, G.1 * G.3 * G.5 ]
The images of the generators of the cohomology of G
restricted to H are
[
z + y,
z + y,
z^2 + z*y,
z*y,
z*y^2 + x,
z^2*y^3 + y^2*x + z*w + y*w,
y^2*x + z*w + y*w,
z^2*y^4 + z*y^2*x + z*y*w + y^2*w,
z^3*y^5 + z^2*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
[
z + y
]
The essential cohomology of G is
generated as an ideal by the elements
[
z*w,
z*v
]
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 [
r
]
.
The essential cohomology is generated as a module over Q
by the elements
[]
[]
[
z*w
]
[
z*v
]
[]
[
v^2 + y*u
]
CHECKS
paramflag = true
qregflagflag = true
cmflag = false
essflag = true
centflag = true
THE COMPUTATION IS COMPLETE