GROUP OF ORDER 32 # 49
GROUP # 49
Dihedral(32)
The MAGMA library number for G is 18
GrpPC : G of order 32 = 2^5
PC-Relations:
G.1^2 = G.3,
G.3^2 = G.4,
G.4^2 = G.5,
G.2^G.1 = G.2 * G.3,
G.3^G.2 = G.3 * G.4 * G.5,
G.4^G.2 = 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, 8, 4, 2, 1 ]
The orders of the terms of the upper central series are [ 1, 2, 4, 8, 32 ]
The order of the Frattini subgroup is 8
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
[ 4, 4 ]
The orders of the normalizers of the maximal elementary abelian subgroups are [
8, 8 ]
The cohomology ring of G is the quotient of a polynomial
ring in the variables [
z,
y,
x
]
in degrees [ 1, 1, 2 ]
by the ideal generated by the relations
[
z^2 + z*y
]
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 [
y^2,
x
]
Restrictions to Maximal Elementary Abelian Subgroups
Subgroup E # 1
Generated by [ G.2 * G.3 * 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,
z,
z*y + y^2
]
in the cohomology of E
The kernel of the restriction to E (Minimal Prime) of the
cohomology of G is generated by
[
z
]
Subgroup E # 2
Generated by [ G.1 * G.2 * 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
[
z,
z,
z*y + y^2
]
in the cohomology of E
The kernel of the restriction to E (Minimal Prime) of the
cohomology of G is generated by
[
z + y
]
The nilradical of the cohomology of G is zero
Restrictions to Maximal Subgroups
Maximal Subgroup H # 1
The Group H is Isomorphic to the
Group of Order 16 Number 12
GrpPC of order 16 = 2^4
PC-Relations:
$.2^2 = $.3 * $.4,
$.3^2 = $.4,
$.2^$.1 = $.2 * $.3,
$.3^$.1 = $.3 * $.4
Generated by [ G.3, G.2 ]
of type Dihedral(16)
The images of the generators of the cohomology of G
restricted to H are
[
0,
z,
x
]
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 12
GrpPC of order 16 = 2^4
PC-Relations:
$.2^2 = $.3 * $.4,
$.3^2 = $.4,
$.2^$.1 = $.2 * $.3,
$.3^$.1 = $.3 * $.4
Generated by [ G.1 * G.2 * G.3, G.3 ]
of type Dihedral(16)
The images of the generators of the cohomology of G
restricted to H are
[
z,
z,
x
]
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 ]
The cohomology ring of H is a the quotient of a polynomial ring
in the variables [
z,
y
]
in degrees [ 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,
y
]
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
THE COMPUTATION IS COMPLETE