GROUP OF ORDER 16 # 14
GROUP OF ORDER 16 # 14
Quaternion(16)
The MAGMA library number for G is 9
GrpPC : G of order 16 = 2^4
PC-Relations:
G.1^2 = G.3,
G.2^2 = G.4,
G.3^2 = G.4,
G.2^G.1 = G.2 * G.3,
G.3^G.2 = G.3 * G.4
The center of G is abelian of type [ 2 ]
The orders of the terms of the lower central series are [ 16, 4, 2, 1 ]
The orders of the terms of the upper central series are [ 1, 2, 4, 16 ]
The order of the Frattini subgroup is 4
The exponent of G is 8
G has a unique maximal elementary abelian subgroup which is central in
G and has order 2
The cohomology ring of G is the quotient of a polynomial
ring in the variables [
z,
y,
x
]
in degrees [ 1, 1, 4 ]
by the ideal generated by the relations
[
z^2 + z*y,
y^3
]
The Hilbert series for the cohomology ring is (-t^2 - t - 1)/(t^3 - t^2 + t -
1)
Its numerator factors as
( t^2 + t + 1
)
Its denominator factors as
( t - 1
)
( t^2 + 1
)
The Krull dimension of the cohomology ring is 1
The cohomology ring is Cohen-Macaulay and the longest regular sequence consists
of the generators [
x
]
Restriction to the Maximal Elementary Abelian Subgroup
Generated by [ G.4 ]
The cohomology ring of E is a polynomial ring in the variables [
z
]
The images of the generators of the cohomology of G
restricted to E are
[
0,
0,
z^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
]
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 [ 8 ]
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^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 8 Number 5
GrpPC of order 8 = 2^3
PC-Relations:
$.1^2 = $.3,
$.2^2 = $.3,
$.2^$.1 = $.2 * $.3
Generated by [ G.3, G.1 * G.2 ]
of type Quaternion(8)
The images of the generators of the cohomology of G
restricted to H are
[
y,
y,
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 Isomorphic to the
Group of Order 8 Number 5
GrpPC of order 8 = 2^3
PC-Relations:
$.1^2 = $.3,
$.2^2 = $.3,
$.2^$.1 = $.2 * $.3
Generated by [ G.3, G.2 ]
of type Quaternion(8)
The images of the generators of the cohomology of G
restricted to H are
[
0,
y,
x
]
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
generated as an ideal by the elements
[
z*y^2
]
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 [
x
]
.
As a module over Q, the essential cohomology is generated
by the elements
[]
[]
[
z*y^2
]
CHECKS
paramflag = true
qregflagflag = true
cmflag = true
essflag = true
centflag = true
THE COMPUTATION IS COMPLETE