GROUP OF ORDER 16 # 7

GROUP OF ORDER 16 # 7

Cyclic(2) x Quaternion(8)

The MAGMA library number for G is 12

GrpPC : G of order 16 = 2^4 PC-Relations: G.1^2 = G.4, G.2^2 = G.4, G.2^G.1 = G.2 * G.4

The center of G is abelian of type [ 2, 2 ]
The orders of the terms of the lower central series are [ 16, 2, 1 ]
The orders of the terms of the upper central series are [ 1, 4, 16 ]
The order of the Frattini subgroup is 2
The exponent of G is 4
G has a unique maximal elementary abelian subgroup which is central in G and has order 4

The cohomology ring of G is the quotient of a polynomial ring in the variables [ z, y, x, w ] in degrees [ 1, 1, 1, 4 ] by the ideal generated by the relations
[ z^2 + z*y + y^2, y^3 ]

The Hilbert series for the cohomology ring is (t^2 + t + 1)/(t^4 - 2*t^3 + 2*t^2 - 2*t + 1)
Its numerator factors as ( t^2 + t + 1 )
Its denominator factors as ( t - 1 )^2 ( t^2 + 1 )^1

The Krull dimension of the cohomology ring is 2
The cohomology ring is Cohen-Macaulay and the longest regular sequence consists of the generators [ x^2, w ]

Restriction to the Maximal Elementary Abelian Subgroup

Generated by [ G.3, G.3 * G.4 ]

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, z + y, 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 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.2 * G.3, G.1 * G.3 ]

of type Quaternion(8)

The images of the generators of the cohomology of G restricted to H are
[ y, z, z + y, x ] in the cohomology of H

The kernel of the restriction to H of the cohomology of G is generated by
[ z + y + x ]

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.2, G.1 ]

of type Quaternion(8)

The images of the generators of the cohomology of G restricted to H are
[ y, z, 0, x ] in the cohomology of H

The kernel of the restriction to H of the cohomology of G is generated by
[ x ]

Maximal Subgroup H # 3

The group H is abelian of type [ 4, 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
[ 0, z, y, x^2 ] in the cohomology of H

The kernel of the restriction to H of the cohomology of G is generated by
[ z ]

Maximal Subgroup H # 4

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.2 * G.3, G.1 ]

of type Quaternion(8)

The images of the generators of the cohomology of G restricted to H are
[ y, z, z, x ] in the cohomology of H

The kernel of the restriction to H of the cohomology of G is generated by
[ y + x ]

Maximal Subgroup H # 5

The group H is abelian of type [ 4, 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, y, x^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 # 6

The group H is abelian of type [ 4, 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, z, y, x^2 ] 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 # 7

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.1 * G.3, G.2 ]

of type Quaternion(8)

The images of the generators of the cohomology of G restricted to H are
[ y, z, y, x ] in the cohomology of H

The kernel of the restriction to H of the cohomology of G is generated by
[ z + x ]

The essential cohomology of G is generated as an ideal by the elements
[ y^2*x^2, z*y^2*x, z*y*x^2 ]

The annihilator of the Essential Cohomology has dimension 2 no end to the new generators was found.

CHECKS

paramflag = true
qregflagflag = true
cmflag = true
essflag = true
centflag = true

THE COMPUTATION IS COMPLETE