GROUP OF ORDER 32 # 47
GROUP # 47
The MAGMA library number for G is 7
GrpPC : G of order 32 = 2^5
PC-Relations:
G.1^2 = G.3,
G.3^2 = G.5,
G.2^G.1 = G.2 * G.4,
G.3^G.2 = G.3 * G.5,
G.4^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 2 conjugacy classes of maximal elementary abelian p-subgroups. The
orders of the maximal elementary abelian subgroups are [ 8, 8 ]
The orders of the centralizers of the maximal elementary abelian subgroups are
[ 8, 8 ]
The orders of the normalizers of the maximal elementary abelian subgroups are [
32, 32 ]
The cohomology ring of G is the quotient of a polynomial
ring in the variables [
z,
y,
x,
w,
v,
u,
t,
s
]
in degrees [ 1, 1, 2, 2, 3, 3, 4, 4 ]
by the ideal generated by the relations
[
z^2,
z*y,
z*x,
z*w,
y^2*x + x^2,
x*w + y*v,
z*v,
z*u,
y^2*v + x*v,
x*u + y*t,
z*t,
w^3 + y*w*u + y^2*s + u^2,
y*w*v + v^2,
y^2*t + x*t,
v*u + w*t,
w^2*v + y*x*s + v*t + u*t,
y*w*t + v*t,
y*u*t + t^2
]
The Hilbert series for the cohomology ring is (-t^2 + t - 1)/(t^5 - 3*t^4 +
4*t^3 - 4*t^2 + 3*t - 1)
Its numerator factors as
( t^2 - 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 [
s
]
A homogeneous set of parameters is the set [
s,
y^2,
w
]
of degrees [ 4, 2, 2 ]
The hypercohomolgy spectral sequence has E2 term:
ROW (0 0)
[
y,
z
]
[
x
]
[
v,
u,
y*x
]
[
y*u,
t
]
[
y*t
]
ROW (1 0)
[
z
]
ROW (0 1)
[
z
]
[]
[
y*x
]
ROW (1 1)
[
z
]
Restrictions to Maximal Elementary Abelian Subgroups
Subgroup E # 1
Generated by [ G.2 * G.5, G.5, G.4 ]
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
[
0,
z,
0,
z*y + y^2,
0,
z*y^2 + y^3 + z^2*x + z*x^2,
0,
z^2*y*x + z*y^2*x + z^2*x^2 + z*y*x^2 + y^2*x^2 + 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,
x,
v,
t
]
Subgroup E # 2
Generated by [ G.2 * G.3 * G.5, G.5, G.4 ]
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
[
0,
z,
z^2,
z*y + y^2,
z^2*y + z*y^2,
z^3 + z*y^2 + y^3 + z^2*x + z*x^2,
z^4 + z^2*y^2 + z*y^3 + z^3*x + z^2*x^2,
z^4 + z^3*y + z^2*y^2 + z^2*y*x + z*y^2*x + z^2*x^2 + z*y*x^2 + y^2*x^2 +
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^2 + x,
y*w + v,
y*u + t
]
The nilradical of the cohomology of G is generated by
[
z
]
Restrictions to Maximal Subgroups
Maximal Subgroup H # 1
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.1 * G.2 * G.3 ]
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,
x,
z*y^3 + y*x,
z*y^3 + w
]
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 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.3 * G.4 ]
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*y^2,
z*y^3 + y*x,
z*y^3 + w
]
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 6
GrpPC of order 16 = 2^4
PC-Relations:
$.3^2 = $.4,
$.2^$.1 = $.2 * $.4,
$.3^$.1 = $.3 * $.4,
$.3^$.2 = $.3 * $.4
Generated by [ G.2 * G.3 * G.4, G.3, G.2 ]
of type Cyclic(2) x Dihedral(8)
The images of the generators of the cohomology of G
restricted to H are
[
0,
z + y,
y^2 + x^2,
z*x + y*x + x^2,
z*y^2 + z*x^2 + y*x^2 + x^3,
z*y^2 + y^3 + z^2*x + z*x^2 + x^3 + z*w + y*w,
z^2*y^2 + y^4 + z^2*y*x + x^4 + y^2*w + x^2*w,
z^2*y^2 + z*y^3 + y^4 + z^2*y*x + z*y^2*x + x^4 + z*y*w + w^2
]
in the cohomology of H
The kernel of the restriction to H of the cohomology of G
is generated by
[
z
]
The essential cohomology of G is zero
CHECKS
paramflag = true
qregflagflag = true
cmflag = false
essflag = true
centflag = true
THE COMPUTATION IS COMPLETE