GROUP OF ORDER 64 # 60
GROUP # 60
The MAGMA library number for G is 20
GrpPC : G of order 64 = 2^6
PC-Relations:
G.1^2 = G.3,
G.2^2 = G.4,
G.4^2 = G.6,
G.5^2 = G.6,
G.2^G.1 = G.2 * G.5,
G.3^G.2 = G.3 * G.6,
G.5^G.2 = G.5 * G.6
The center of G is abelian of type [ 2, 4 ]
The orders of the terms of the lower central series are [ 64, 4, 2, 1 ]
The orders of the terms of the upper central series are [ 1, 8, 16, 64 ]
The order of the Frattini subgroup is 16
The exponent of G is 8
G has a unique maximal elementary abelian subgroup which is normal and
has order 8. Its centralizer has order 32
The cohomology ring of G is the quotient of a polynomial
ring in the variables [
z,
y,
x,
w,
v,
u,
t,
s,
r,
q
]
in degrees [ 1, 1, 2, 2, 2, 2, 3, 3, 4, 4 ]
by the ideal generated by the relations
[
z^2,
z*y,
y^2,
z*x,
y*x + z*v,
y*w + z*v + y*u,
y*v,
x^2,
x*w + x*u + z*t,
x*v,
w*v + v*u + z*s,
v^2,
y*t,
y*s,
z*w*u + z*u^2 + v*s,
z*v*u + x*t + v*s,
v*t + z*r,
x*s + z*r,
y*r,
w^2*u + u^3 + z*w*t + s^2 + v*r,
z*u*t + v*r,
z*u*s + x*r,
t^2 + s^2,
t*s + x*r + w*r + u*r,
x*u*t + w*u*t + u^2*t + z*s^2 + s*r,
w*u*s + u^2*s + z*w*r + z*v*q + t*r,
u*s^2 + z*s*r + r^2
]
The Hilbert series for the cohomology ring is -1/(t^4 - 2*t^3 + 2*t - 1)
Its denominator factors as
( t - 1
)^3
( t + 1
)^1
The Krull dimension of the cohomology ring is 3
The longest regular sequence consists of the generators [
w,
q
]
A homogeneous set of parameters is the set [
w,
q,
u
]
of degrees [ 2, 4, 2 ]
The hypercohomolgy spectral sequence has E2 term:
ROW (0)
[
z,
y
]
[
v,
x
]
[
t,
s
]
[
r
]
[
z*r
]
ROW (1)
[]
[]
[
y*u
]
The spectral sequence satisfies Poincaré duality
Restriction to the Maximal Elementary Abelian Subgroup
Generated by [ G.4 * G.5 * G.6, G.6, G.3 * G.4 * G.5 * G.6 ]
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,
0,
0,
y^2,
0,
z^2 + y^2,
z^3 + z^2*y,
z^3 + z^2*y,
z^4 + z^2*y^2,
z^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,
x,
v,
t + s,
w*u + u^2 + r
]
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 [ 4, 4, 2 ]
The cohomology ring of H is a the quotient of a polynomial ring
in the variables [
z,
y,
x,
w,
v
]
in degrees [ 1, 1, 1, 2, 2 ]
by the ideal of relations
[
z^2,
y^2
]
The images of the generators of the cohomology of G
restricted to H are
[
z,
0,
z*x,
w,
z*y + z*x,
z*x + x^2,
x^3 + x*w + z*v,
y*x^2 + x^3 + y*w + x*w,
y*x^3 + x^4 + y*x*w + x^2*w + z*y*v + z*x*v,
z*y*w + z*x*w + x^2*v + w*v + v^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 32 Number 13
GrpPC of order 32 = 2^5
PC-Relations:
$.1^2 = $.3 * $.5,
$.2^2 = $.3,
$.3^2 = $.5,
$.4^2 = $.5,
$.2^$.1 = $.2 * $.5,
$.4^$.1 = $.4 * $.5,
$.4^$.2 = $.4 * $.5
Generated by [ G.2 * G.4 * G.5 * G.6, G.3 * G.4 * G.6, G.2 ]
of type Cyclic(2) x Group(16)#11
The images of the generators of the cohomology of G
restricted to H are
[
0,
z + y,
z*x + y*x,
x^2,
z*y + y^2,
z*y,
z^2*y + z^2*x + z*y*x + y*x^2,
z^2*y + z*y^2 + z*y*x + x^3 + w,
z^3*x + z^2*x^2 + z*y*x^2 + y*x^3 + y*w,
z^2*x^2 + z*y*x^2 + v
]
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 32 Number 16
GrpPC of order 32 = 2^5
PC-Relations:
$.1^2 = $.4 * $.5,
$.2^2 = $.4,
$.3^2 = $.4,
$.3^$.1 = $.3 * $.5,
$.3^$.2 = $.3 * $.5
Generated by [ G.1 * G.2 * G.4, G.1 * G.2 * G.4 * G.5 * G.6, G.1 * G.2 * G.6 ]
The images of the generators of the cohomology of G
restricted to H are
[
z + y + x,
z + y + x,
z^2 + x^2,
w,
z*y + z*x + x^2,
x^2 + w,
x^3 + z*w + y*w + v,
z*y*x + x^3 + x*w + v,
y*x^3 + x^4 + z*x*w + y*v,
z*y*w + y^2*w + z*x*w + w^2 + u
]
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
[
z*v
]
The annihilator of the Essential Cohomology has dimension 2
no end to the new generators was found.
CHECKS
paramflag = true
qregflagflag = true
cmflag = false
essflag = true
centflag = true
bigflag = true
restrictflag = true
THE COMPUTATION IS COMPLETE