GROUP OF ORDER 64 # 79

GROUP # 79

The MAGMA library number for G is 214

GrpPC : G of order 64 = 2^6 PC-Relations: G.1^2 = G.5 * G.6, G.3^2 = G.6, G.4^2 = G.5, G.2^G.1 = G.2 * G.5, G.3^G.1 = G.3 * 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, 1 ]
The orders of the terms of the upper central series are [ 1, 8, 64 ]
The order of the Frattini subgroup is 4
The exponent of G is 4
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 ] in degrees [ 1, 1, 1, 1, 4, 4, 4 ] by the ideal generated by the relations
[ z^2 + zx + x^2, zy + zx + x^2 + w^2, x^3, yw^2, zv, y^4u + v^2 ]

The Hilbert series for the cohomology ring is (t^3 - t^2 - t - 1)/(t^7 - 3*t^6 + 5*t^5 - 7*t^4 + 7*t^3 - 5*t^2 + 3*t - 1)
Its numerator factors as ( t^3 - t^2 - t - 1 )
Its denominator factors as ( t - 1 )^3 ( t^2 + 1 )^2

The Krull dimension of the cohomology ring is 3
The longest regular sequence consists of the generators [ u, t ]
A homogeneous set of parameters is the set [ u, t, y^2 ] of degrees [ 4, 4, 2 ]

The hypercohomolgy spectral sequence has E2 term:

ROW (0) [ y, w, z, x ] [ w^2, x*w, y*w, z*w, z*x, y*x, x^2 ] [ x*w^2, z*x*w, y*x^2, x^2*w, y*x*w, w^3 ] [ x*w^3, v, y*x^2*w ] [ y*v, x*v, w*v ] [ y*x*v, y*w*v, x*w*v ] [ y*x*w*v ]
ROW (1) [] [ w^2, z*x + x^2 ] [ z*w^2, w^3, y*x^2, x*w^2, z*x*w + x^2*w ] [ z*w^3, x*w^3, x^2*w^2, y*x^2*w ] [ x^2*w^3 ]
The spectral sequence satisfies Poincaré duality

Restriction to the Maximal Elementary Abelian Subgroup

Generated by [ G.5, G.2, 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, z, 0, 0, z^2y^2, y^4, z^2y^2 + y^4 + 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, x, w ]

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 32 Number 35 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.5, $.3^2 = $.4 * $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.1 * G.4, G.2 * G.4, G.2 * G.3 * G.4 ]

The images of the generators of the cohomology of G restricted to H are
[ x, z + y, y, z + y + x, x^2w, x^2w + w^2 + v, w^2 ] in the cohomology of H

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

Maximal Subgroup H # 2

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
[ 0, x, z, y, x^2w, w^2, x^2v + v^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 # 3

The Group H is Isomorphic to the Group of Order 32 Number 35 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.5, $.3^2 = $.4 * $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.1, G.3, G.2 * G.4 ]

The images of the generators of the cohomology of G restricted to H are
[ x, y, z, y, x^2w, v, x^2w + w^2 + v ] in the cohomology of H

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

Maximal Subgroup H # 4

The Group H is Isomorphic to the Group of Order 32 Number 37 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^2 = $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.3, G.1 * G.2, G.2 * G.3 ]

The images of the generators of the cohomology of G restricted to H are
[ x, z + x, z + y, 0, y^2w + yx*w + zv, y^2w + yxw + w^2 + u, w^2 ] in the cohomology of H

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

Maximal Subgroup H # 5

The Group H is Isomorphic to the Group of Order 32 Number 37 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^2 = $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.1 * G.3 * G.6, G.3 * G.4, G.2 * G.3 * G.4 ]

The images of the generators of the cohomology of G restricted to H are
[ x, y, z + y + x, z + y, y^2w + yxw + zv, y^2w + yxw + w^2 + u, zv + u ] in the cohomology of H

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

Maximal Subgroup H # 6

The Group H is Isomorphic to the Group of Order 32 Number 15 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.5, $.3^2 = $.5, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5 Generated by [ G.1 * G.3 * G.6, G.1 * G.2, G.1 * G.2 * G.4 * G.5 ]

of type Cyclic(4) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are
[ z + y + x, z + y, x, y, y^2w + x^2w, w^2 + v, v ] 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 # 7

The Group H is Isomorphic to the Group of Order 32 Number 37 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^2 = $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.1 * G.4, G.3, G.2 * G.3 ]

The images of the generators of the cohomology of G restricted to H are
[ x, z, z + y, x, y^2w + yx*w + zv, y^2w + yxw + w^2 + u, w^2 ] in the cohomology of H

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

Maximal Subgroup H # 8

The Group H is Isomorphic to the Group of Order 32 Number 37 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^2 = $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.1, G.3 * G.4, G.2 * G.3 * G.4 ]

The images of the generators of the cohomology of G restricted to H are
[ x, y, z + y, z + y, y^2w + yxw + zv, y^2w + yxw + w^2 + u, zv + u ] in the cohomology of H

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

Maximal Subgroup H # 9

The Group H is Isomorphic to the Group of Order 32 Number 40 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.3^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.1 * G.4, G.3 * G.4, G.1 * G.2 * G.3 * G.4 * G.5 ]

The images of the generators of the cohomology of G restricted to H are
[ z + x, z, z + y, z + y + x, yv + xv, yv + xv + u + t, x*v + t ] in the cohomology of H

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

Maximal Subgroup H # 10

The Group H is Isomorphic to the Group of Order 32 Number 15 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.5, $.3^2 = $.5, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5 Generated by [ G.1, G.2 * G.3 * G.5, G.2 * G.3 * G.4 ]

of type Cyclic(4) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are
[ x, z + y, z + y, y, x^2*w, w^2 + v, v ] 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 # 11

The Group H is Isomorphic to the Group of Order 32 Number 15 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.5, $.3^2 = $.5, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5 Generated by [ G.3, G.1 * G.4 * G.5, G.3 * G.4 ]

of type Cyclic(4) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are
[ x, 0, z + y, z + x, x^2w, w^2 + v, x^2w + w^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 # 12

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.4, G.1, G.1 * G.2 * G.4 * G.6 ]

The images of the generators of the cohomology of G restricted to H are
[ z + y + x, x, 0, z + x, x^4 + zxw + yxw, x^4 + w^2, zxw + yxw + w^2 + u ] in the cohomology of H

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

Maximal Subgroup H # 13

The Group H is Isomorphic to the Group of Order 32 Number 40 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.3^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.2 * G.3 * G.6, G.1 * G.3 * G.4, G.1 * G.5 * G.6 ]

The images of the generators of the cohomology of G restricted to H are
[ z + x, y, z + y, z, yv + xv, yv + xv + u + t, x*v + u ] in the cohomology of H

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

Maximal Subgroup H # 14

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.3 * G.6, G.1 * G.3 * G.4 * G.6, G.1 * G.2 * G.3 * G.4 * G.5 ]

The images of the generators of the cohomology of G restricted to H are
[ z + y + x, x, z + y + x, z + x, x^4 + zxw + yxw, x^4 + w^2, zxw + yxw + w^2 + u ] in the cohomology of H

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

Maximal Subgroup H # 15

The Group H is Isomorphic to the Group of Order 32 Number 15 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.5, $.3^2 = $.5, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5 Generated by [ G.3, G.3 * G.4, G.1 * G.2 * G.5 ]

of type Cyclic(4) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are
[ x, x, z + y, z, x^2w, w^2 + v, x^2w + w^2 ] 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
[ x^2*w^3 ]

The annihilator of the Essential Cohomology has dimension 2 The essential cohomology is a free module over the polynomial subring Q of the cohomology ring of G generated by [ u, t ] .
The essential cohomology is generated as a module over Q by the elements [] [] [] [] [ x^2*w^3 ]

CHECKS

paramflag = true
qregflagflag = true
cmflag = false
essflag = true
centflag = true
bigflag = true
restrictflag = true

THE COMPUTATION IS COMPLETE

GROUP # 79

GrpPC : G of order 64 = 2^6 PC-Relations: G.1^2 = G.5 * G.6, G.3^2 = G.6, G.4^2 = G.5, G.2^G.1 = G.2 * G.5, G.3^G.1 = G.3 * G.6

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

The hypercohomolgy spectral sequence has E2 term:

Restriction to the Maximal Elementary Abelian Subgroup

The images of the generators of the cohomology of G restricted to E are [ 0, z, 0, 0, z^2*y^2, y^4, z^2*y^2 + y^4 + 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, x, w ]

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 32 Number 35 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.5, $.3^2 = $.4 * $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.1 * G.4, G.2 * G.4, G.2 * G.3 * G.4 ]

The images of the generators of the cohomology of G restricted to H are [ x, z + y, y, z + y + x, x^2*w, x^2*w + w^2 + v, w^2 ] in the cohomology of H

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

Maximal Subgroup H # 2

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 [ 0, x, z, y, x^2*w, w^2, x^2*v + v^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 # 3

The Group H is Isomorphic to the Group of Order 32 Number 35 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.5, $.3^2 = $.4 * $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.1, G.3, G.2 * G.4 ]

The images of the generators of the cohomology of G restricted to H are [ x, y, z, y, x^2*w, v, x^2*w + w^2 + v ] in the cohomology of H

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

Maximal Subgroup H # 4

The Group H is Isomorphic to the Group of Order 32 Number 37 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^2 = $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.3, G.1 * G.2, G.2 * G.3 ]

The images of the generators of the cohomology of G restricted to H are [ x, z + x, z + y, 0, y^2*w + y*x*w + z*v, y^2*w + y*x*w + w^2 + u, w^2 ] in the cohomology of H

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

Maximal Subgroup H # 5

The Group H is Isomorphic to the Group of Order 32 Number 37 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^2 = $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.1 * G.3 * G.6, G.3 * G.4, G.2 * G.3 * G.4 ]

The images of the generators of the cohomology of G restricted to H are [ x, y, z + y + x, z + y, y^2*w + y*x*w + z*v, y^2*w + y*x*w + w^2 + u, z*v + u ] in the cohomology of H

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

Maximal Subgroup H # 6

The Group H is Isomorphic to the Group of Order 32 Number 15 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.5, $.3^2 = $.5, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5 Generated by [ G.1 * G.3 * G.6, G.1 * G.2, G.1 * G.2 * G.4 * G.5 ]

of type Cyclic(4) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are [ z + y + x, z + y, x, y, y^2*w + x^2*w, w^2 + v, v ] 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 # 7

The Group H is Isomorphic to the Group of Order 32 Number 37 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^2 = $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.1 * G.4, G.3, G.2 * G.3 ]

The images of the generators of the cohomology of G restricted to H are [ x, z, z + y, x, y^2*w + y*x*w + z*v, y^2*w + y*x*w + w^2 + u, w^2 ] in the cohomology of H

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

Maximal Subgroup H # 8

The Group H is Isomorphic to the Group of Order 32 Number 37 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^2 = $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.1, G.3 * G.4, G.2 * G.3 * G.4 ]

The images of the generators of the cohomology of G restricted to H are [ x, y, z + y, z + y, y^2*w + y*x*w + z*v, y^2*w + y*x*w + w^2 + u, z*v + u ] in the cohomology of H

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

Maximal Subgroup H # 9

The Group H is Isomorphic to the Group of Order 32 Number 40 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.3^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.1 * G.4, G.3 * G.4, G.1 * G.2 * G.3 * G.4 * G.5 ]

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

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

Maximal Subgroup H # 10

The Group H is Isomorphic to the Group of Order 32 Number 15 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.5, $.3^2 = $.5, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5 Generated by [ G.1, G.2 * G.3 * G.5, G.2 * G.3 * G.4 ]

of type Cyclic(4) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are [ x, z + y, z + y, y, x^2*w, w^2 + v, v ] 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 # 11

The Group H is Isomorphic to the Group of Order 32 Number 15 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.5, $.3^2 = $.5, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5 Generated by [ G.3, G.1 * G.4 * G.5, G.3 * G.4 ]

of type Cyclic(4) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are [ x, 0, z + y, z + x, x^2*w, w^2 + v, x^2*w + w^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 # 12

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.4, G.1, G.1 * G.2 * G.4 * G.6 ]

The images of the generators of the cohomology of G restricted to H are [ z + y + x, x, 0, z + x, x^4 + z*x*w + y*x*w, x^4 + w^2, z*x*w + y*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 [ x ]

Maximal Subgroup H # 13

The Group H is Isomorphic to the Group of Order 32 Number 40 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.3^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.2 * G.3 * G.6, G.1 * G.3 * G.4, G.1 * G.5 * G.6 ]

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

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

Maximal Subgroup H # 14

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.3 * G.6, G.1 * G.3 * G.4 * G.6, G.1 * G.2 * G.3 * G.4 * G.5 ]

The images of the generators of the cohomology of G restricted to H are [ z + y + x, x, z + y + x, z + x, x^4 + z*x*w + y*x*w, x^4 + w^2, z*x*w + y*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 + x ]

Maximal Subgroup H # 15

The Group H is Isomorphic to the Group of Order 32 Number 15 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.5, $.3^2 = $.5, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5 Generated by [ G.3, G.3 * G.4, G.1 * G.2 * G.5 ]

of type Cyclic(4) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are [ x, x, z + y, z, x^2*w, w^2 + v, x^2*w + w^2 ] 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 [ x^2*w^3 ]

CHECKS

THE COMPUTATION IS COMPLETE

The cohomology ring of G is the quotient of a polynomial ring in the variables [ z, y, x, w, v, u, t ] in degrees [ 1, 1, 1, 1, 4, 4, 4 ] by the ideal generated by the relations
[ z^2 + zx + x^2, zy + zx + x^2 + w^2, x^3, yw^2, zv, y^4u + v^2 ]

The images of the generators of the cohomology of G restricted to E are
[ 0, z, 0, 0, z^2y^2, y^4, z^2y^2 + y^4 + 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, x, w ]

The images of the generators of the cohomology of G restricted to H are
[ x, z + y, y, z + y + x, x^2w, x^2w + w^2 + v, w^2 ] in the cohomology of H

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

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
[ 0, x, z, y, x^2w, w^2, x^2v + v^2 ] in the cohomology of H

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

The images of the generators of the cohomology of G restricted to H are
[ x, y, z, y, x^2w, v, x^2w + w^2 + v ] in the cohomology of H

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

The images of the generators of the cohomology of G restricted to H are
[ x, z + x, z + y, 0, y^2w + yx*w + zv, y^2w + yxw + w^2 + u, w^2 ] in the cohomology of H

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

The images of the generators of the cohomology of G restricted to H are
[ x, y, z + y + x, z + y, y^2w + yxw + zv, y^2w + yxw + w^2 + u, zv + u ] in the cohomology of H

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

The images of the generators of the cohomology of G restricted to H are
[ z + y + x, z + y, x, y, y^2w + x^2w, w^2 + v, v ] in the cohomology of H

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

The images of the generators of the cohomology of G restricted to H are
[ x, z, z + y, x, y^2w + yx*w + zv, y^2w + yxw + w^2 + u, w^2 ] in the cohomology of H

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

The images of the generators of the cohomology of G restricted to H are
[ x, y, z + y, z + y, y^2w + yxw + zv, y^2w + yxw + w^2 + u, zv + u ] in the cohomology of H

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

The images of the generators of the cohomology of G restricted to H are
[ z + x, z, z + y, z + y + x, yv + xv, yv + xv + u + t, x*v + t ] in the cohomology of H

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

The images of the generators of the cohomology of G restricted to H are
[ x, z + y, z + y, y, x^2*w, w^2 + v, v ] in the cohomology of H

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

The images of the generators of the cohomology of G restricted to H are
[ x, 0, z + y, z + x, x^2w, w^2 + v, x^2w + w^2 ] in the cohomology of H

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

The images of the generators of the cohomology of G restricted to H are
[ z + y + x, x, 0, z + x, x^4 + zxw + yxw, x^4 + w^2, zxw + yxw + w^2 + u ] in the cohomology of H

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

The images of the generators of the cohomology of G restricted to H are
[ z + x, y, z + y, z, yv + xv, yv + xv + u + t, x*v + u ] in the cohomology of H

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

The images of the generators of the cohomology of G restricted to H are
[ z + y + x, x, z + y + x, z + x, x^4 + zxw + yxw, x^4 + w^2, zxw + yxw + w^2 + u ] in the cohomology of H

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

The images of the generators of the cohomology of G restricted to H are
[ x, x, z + y, z, x^2w, w^2 + v, x^2w + w^2 ] 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
[ x^2*w^3 ]