GROUP OF ORDER 64 # 66
GROUP # 66
The MAGMA library number for G is 31
GrpPC : G of order 64 = 2^6
PC-Relations:
G.1^2 = G.3 * G.5 * G.6,
G.2^2 = G.3,
G.3^2 = G.4,
G.4^2 = G.6,
G.5^2 = G.6,
G.2^G.1 = G.2 * G.5,
G.5^G.2 = G.5 * G.6
The center of G is abelian of type [ 8 ]
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 16
G has 2 conjugacy classes of maximal elementary abelian p-subgroups. The
orders of the maximal elementary abelian subgroups are [ 4, 4 ]
The orders of the centralizers of the maximal elementary abelian subgroups are
[ 32, 16 ]
The orders of the normalizers of the maximal elementary abelian subgroups are [
64, 32 ]
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 + y*w,
z*x + y*x,
y^2*x + x*w + z*v + y*v,
y^2*w + x*w + z*v,
x^2,
z*w^2 + y^2*v + x*v,
w^3 + 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
]
Restrictions to Maximal Elementary Abelian Subgroups
Subgroup E # 1
Generated by [ G.6, 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,
z^3,
z^4 + 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,
y,
x
]
Subgroup E # 2
Generated by [ G.1 * G.2 * G.3 * G.6, G.6 ]
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
[
z,
z,
0,
z^2,
z^3,
z^4 + 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 + y,
y^2 + w,
x,
y*w + v
]
The nilradical of the cohomology of G is generated by
[
z + y,
x
]
Restrictions to Maximal Subgroups
Maximal Subgroup H # 1
The Group H is Isomorphic to the
Group of Order 32 Number 22
GrpPC of order 32 = 2^5
PC-Relations:
$.1^2 = $.2,
$.2^2 = $.4,
$.4^2 = $.5,
$.3^$.1 = $.3 * $.5
Generated by [ G.4 * G.5 * G.6, G.2 ]
The images of the generators of the cohomology of G
restricted to H are
[
0,
z,
z*y,
z*y + y^2,
y^3 + x,
y^4 + w
]
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 32 Number 17
GrpPC of order 32 = 2^5
PC-Relations:
$.1^2 = $.4,
$.2^2 = $.4,
$.3^2 = $.4,
$.4^2 = $.5,
$.2^$.1 = $.2 * $.5
Generated by [ G.1 * G.2 * G.4 * G.5 * G.6, G.3 * G.6, G.1 * G.2 * G.4 * G.6 ]
The images of the generators of the cohomology of G
restricted to H are
[
z + y,
z + y,
z*x + y*x + x^2,
z*y + x^2,
z*y^2 + z*y*x + x^3,
z*y^3 + x^4 + 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 # 3
The group H is abelian of type [ 16, 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,
y^3 + z*x,
y^4 + 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
[
y
]
The essential cohomology of G is zero
CHECKS
paramflag = true
qregflagflag = true
cmflag = true
essflag = true
centflag = true
bigflag = true
restrictflag = true
THE COMPUTATION IS COMPLETE