GROUP OF ORDER 64 # 13

GROUP # 13

Abelian(2,2,2) x Quaternion(8)

The MAGMA library number for G is 262

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

The center of G is abelian of type [ 2, 2, 2, 2 ]
The orders of the terms of the lower central series are [ 64, 2, 1 ]
The orders of the terms of the upper central series are [ 1, 16, 64 ]
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 16

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, 1, 1, 1, 4 ] by the ideal generated by the relations
[ z^2 + zy + y^2 + zx + yx + x^2 + zv + yv + v^2, y^3 + y^2x + yx^2 + x^3 + y^2v + x^2v + yv^2 + x*v^2 + v^3 ]

The Hilbert series for the cohomology ring is (t^2 + t + 1)/(t^6 - 4*t^5 + 7*t^4 - 8*t^3 + 7*t^2 - 4*t + 1)
Its numerator factors as ( t^2 + t + 1 )
Its denominator factors as ( t - 1 )^4 ( t^2 + 1 )^1

The Krull dimension of the cohomology ring is 4
The cohomology ring is Cohen-Macaulay and the longest regular sequence consists of the generators [ z^2, x^2, w^2, u ]

Restriction to the Maximal Elementary Abelian Subgroup

Generated by [ G.1 * G.2 * G.3 * G.4, G.1 * G.2 * G.3 * G.6, G.1 * G.2 * G.3, G.3 * G.4 * G.5 * G.6 ] The cohomology ring of E is a polynomial ring in the variables [ z, y, x, w ]

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

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

of type Abelian(2,2) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are
[ z + w, y + x, y, w, y, zy^3 + y^4 + zy^2x + yx^2w + x^3w + v ] in the cohomology of H

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

Maximal Subgroup H # 2

The Group H is Isomorphic to the Group of Order 32 Number 9 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.4^$.2 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.1 * G.4 * G.5, G.2 * G.3 * G.6, G.1, G.2 ]

of type Abelian(2,2) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are
[ z + y, x + w, x, z, z, zy^3 + y^4 + zy^2x + z^3w + yx^2w + x^3w + z^2w^2 + z*w^3 + v ] in the cohomology of H

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

Maximal Subgroup H # 3

The Group H is Isomorphic to the Group of Order 32 Number 9 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.4^$.2 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.1 * G.5 * G.6, G.2 * G.3 * G.5 * G.6, G.2 * G.3, G.1 * G.3 * G.4 * G.5 * G.6 ]

of type Abelian(2,2) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are
[ z + y, x + w, z + x + w, z, z + y + x, zy^3 + zy^2x + yx^2w + x^3w + v ] 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 # 4

The Group H is Isomorphic to the Group of Order 32 Number 9 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.4^$.2 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.1 * G.3 * G.6, G.1 * G.5 * G.6, G.1 * G.2 * G.6, G.3 * G.4 * G.6 ]

of type Abelian(2,2) x Quaternion(8)

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

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

Maximal Subgroup H # 5

The Group H is Isomorphic to the Group of Order 32 Number 9 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.4^$.2 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.5, G.1 * G.4, G.1 * G.3 * G.5, G.1 * G.2 * G.6 ]

of type Abelian(2,2) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are
[ z + x + w, x, z, w, z + y, y^4 + z^2*w^2 + v ] 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 # 6

The Group H is Isomorphic to the Group of Order 32 Number 9 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.4^$.2 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.1 * G.3 * G.5 * G.6, G.1 * G.5 * G.6, G.1 * G.4, G.2 ]

of type Abelian(2,2) x Quaternion(8)

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

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

Maximal Subgroup H # 7

The Group H is Isomorphic to the Group of Order 32 Number 9 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.4^$.2 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.1 * G.3 * G.5 * G.6, G.5, G.1 * G.2 * G.3 * G.4 * G.5 * G.6, G.1 * G.3 * G.4 * G.5 * G.6 ]

of type Abelian(2,2) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are
[ z + y + x, z, z + y + x, z + y, z + y + x + w, y^2w^2 + x^2w^2 + v ] 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 # 8

The Group H is Isomorphic to the Group of Order 32 Number 9 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.4^$.2 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.1 * G.3 * G.4 * G.6, G.5, G.3, G.2 ]

of type Abelian(2,2) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are
[ z, w, z + y, z, x, z^2y^2 + z^3w + y^2w^2 + zw^3 + v ] 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 # 9

The group H is abelian of type [ 4, 2, 2, 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, 1, 2 ] by the ideal of relations
[ z^2 ]

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

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

Maximal Subgroup H # 10

The Group H is Isomorphic to the Group of Order 32 Number 9 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.4^$.2 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.1 * G.3 * G.6, G.2 * G.5 * G.6, G.1 * G.3 * G.4 * G.6, G.2 * G.3 * G.4 * G.6 ]

of type Abelian(2,2) x Quaternion(8)

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

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

Maximal Subgroup H # 11

The Group H is Isomorphic to the Group of Order 32 Number 9 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.4^$.2 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.1, G.5, G.3, G.1 * G.4 ]

of type Abelian(2,2) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are
[ z + w, 0, y, w, x, z^2y^2 + y^2w^2 + v ] 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 9 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.4^$.2 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.1 * G.3, G.1 * G.3 * G.4, G.3, G.1 * G.2 * G.3 * G.4 * G.5 * G.6 ]

of type Abelian(2,2) x Quaternion(8)

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

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

Maximal Subgroup H # 13

The Group H is Isomorphic to the Group of Order 32 Number 9 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.4^$.2 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.2 * G.5 * G.6, G.1, G.1 * G.3 * G.4, G.2 * G.4 * G.6 ]

of type Abelian(2,2) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are
[ z + y, x + w, z, z + w, x, y^4 + z^3w + y^2w^2 + x^2w^2 + zw^3 + v ] in the cohomology of H

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

Maximal Subgroup H # 14

The Group H is Isomorphic to the Group of Order 32 Number 9 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.4^$.2 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.3 * G.4, G.2 * G.5 * G.6, G.2 * G.3 * G.4 * G.6, G.3 ]

of type Abelian(2,2) x Quaternion(8)

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

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

Maximal Subgroup H # 15

The Group H is Isomorphic to the Group of Order 32 Number 9 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.4^$.2 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.3 * G.4, G.1 * G.5 * G.6, G.3, G.2 ]

of type Abelian(2,2) x Quaternion(8)

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

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

Maximal Subgroup H # 16

The Group H is Isomorphic to the Group of Order 32 Number 9 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.4^$.2 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.2 * G.5 * G.6, G.1, G.1 * G.3 * G.4, G.2 ]

of type Abelian(2,2) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are
[ z + y, x + w, z, z, x, y^4 + z^2w^2 + y^2w^2 + x^2*w^2 + v ] 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 # 17

The group H is abelian of type [ 4, 2, 2, 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, 1, 2 ] by the ideal of relations
[ z^2 ]

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

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

Maximal Subgroup H # 18

The Group H is Isomorphic to the Group of Order 32 Number 9 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.4^$.2 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.1 * G.3 * G.6, G.5, G.3, G.2 ]

of type Abelian(2,2) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are
[ z, w, z + y, 0, x, z^2y^2 + z^3w + y^2w^2 + zw^3 + v ] in the cohomology of H

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

Maximal Subgroup H # 19

The Group H is Isomorphic to the Group of Order 32 Number 9 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.4^$.2 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.1, G.1 * G.2 * G.4, G.3, G.1 * G.4 ]

of type Abelian(2,2) x Quaternion(8)

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

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

Maximal Subgroup H # 20

The Group H is Isomorphic to the Group of Order 32 Number 9 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.4^$.2 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.1, G.1 * G.2 * G.4, G.5, G.1 * G.4 ]

of type Abelian(2,2) x Quaternion(8)

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

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

Maximal Subgroup H # 21

The Group H is Isomorphic to the Group of Order 32 Number 9 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.4^$.2 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.3 * G.4, G.2 * G.3 * G.5, G.1 * G.2 * G.3 * G.5 * G.6, G.2 ]

of type Abelian(2,2) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are
[ z, z + y + x, z + y + w, w, z + y, z^2y^2 + zy^3 + y^4 + zy^2x + yx^2w + x^3w + y^2w^2 + v ] in the cohomology of H

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

Maximal Subgroup H # 22

The Group H is Isomorphic to the Group of Order 32 Number 9 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.4^$.2 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.1 * G.4 * G.5, G.5, G.1 * G.2 * G.3 * G.4 * G.5 * G.6, G.1 * G.5 ]

of type Abelian(2,2) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are
[ z + y + x, z, z, z + y, z + y + x + w, y^2w^2 + x^2w^2 + 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 # 23

The Group H is Isomorphic to the Group of Order 32 Number 9 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.4^$.2 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.1 * G.5 * G.6, G.1 * G.4, G.1 * G.3 * G.5, G.2 * G.4 ]

of type Abelian(2,2) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are
[ y + x + w, z, x, z + y, x + w, zy^3 + y^4 + zy^2x + yx^2w + x^3w + v ] in the cohomology of H

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

Maximal Subgroup H # 24

The Group H is Isomorphic to the Group of Order 32 Number 9 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.4^$.2 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.1 * G.3 * G.4 * G.6, G.2 * G.4 * G.6, G.5, G.3 ]

of type Abelian(2,2) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are
[ z, w, z + y, z + w, x, z^2y^2 + z^2w^2 + y^2*w^2 + v ] 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 # 25

The Group H is Isomorphic to the Group of Order 32 Number 9 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.4^$.2 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.1 * G.3 * G.6, G.1 * G.5 * G.6, G.1 * G.2 * G.4 * G.6, G.3 * G.4 * G.6 ]

of type Abelian(2,2) x Quaternion(8)

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

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

Maximal Subgroup H # 26

The group H is abelian of type [ 4, 2, 2, 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, 1, 2 ] by the ideal of relations
[ z^2 ]

The images of the generators of the cohomology of G restricted to H are
[ y, y, z + x, w, y + x, y^4 + w^4 + v^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 # 27

The Group H is Isomorphic to the Group of Order 32 Number 9 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.4^$.2 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.2 * G.3 * G.4 * G.5, G.1 * G.3 * G.4 * G.5, G.1 * G.5 * G.6, G.1 * G.3 * G.5 ]

of type Abelian(2,2) x Quaternion(8)

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

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

Maximal Subgroup H # 28

The Group H is Isomorphic to the Group of Order 32 Number 9 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.4^$.2 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.1 * G.3 * G.5 * G.6, G.1 * G.2 * G.4, G.5, G.1 * G.4 ]

of type Abelian(2,2) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are
[ z + y + x, z, x, z + y, x + w, z^2y^2 + zy^3 + zy^2x + yx^2w + x^3w + y^2w^2 + v ] 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 # 29

The Group H is Isomorphic to the Group of Order 32 Number 9 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.4^$.2 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.1 * G.3 * G.6, G.1 * G.3 * G.4 * G.6, G.5, G.1 * G.2 * G.6 ]

of type Abelian(2,2) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are
[ z + x + w, x, z + w, w, y, y^4 + 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 # 30

The Group H is Isomorphic to the Group of Order 32 Number 9 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.4^$.2 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.1 * G.3 * G.6, G.5, G.3, G.2 * G.4 ]

of type Abelian(2,2) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are
[ z, w, z + y, w, x, z^2y^2 + z^3w + y^2w^2 + zw^3 + 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 # 31

The Group H is Isomorphic to the Group of Order 32 Number 9 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.4^$.2 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.1 * G.3 * G.6, G.2 * G.5, G.2 * G.3 * G.5 * G.6, G.1 * G.3 * G.4 * G.5 * G.6 ]

of type Abelian(2,2) x Quaternion(8)

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

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

The essential cohomology of G is generated as an ideal by
[ zy^2x^4w^2v + y^2x^5w^2v + zx^6w^2v + x^7w^2v + zy^2x^2w^4v + y^2x^3w^4v + zx^4w^4v + x^5w^4v + zy^2x^4wv^2 + y^2x^5wv^2 + zx^6wv^2 + x^7wv^2 + y^2x^4w^2v^2 + x^6w^2v^2 + zy^2xw^4v^2 + zx^3w^4v^2 + y^2x^4wv^3 + x^6wv^3 + zx^4w^2v^3 + x^5w^2v^3 + y^2xw^4v^3 + zx^2w^4v^3 + zy^2x^2wv^4 + y^2x^3wv^4 + zy^2xw^2v^4 + y^2x^2w^2v^4 + zx^3w^2v^4 + zxw^4v^4 + y^2x^2wv^5 + y^2xw^2v^5 + x^3w^2v^5 + xw^4v^5 + zx^2wv^6 + x^3wv^6 + zxw^2v^6 + x^2w^2v^6 + x^2wv^7 + xw^2v^7, zyx^8w^4v^2 + zx^9w^4v^2 + yx^9w^4v^2 + x^10w^4v^2 + zyx^4w^8v^2 + zx^5w^8v^2 + yx^5w^8v^2 + x^6w^8v^2 + zx^8w^4v^3 + yx^8w^4v^3 + zx^4w^8v^3 + yx^4w^8v^3 + zyx^8w^2v^4 + zx^9w^2v^4 + yx^9w^2v^4 + x^10w^2v^4 + x^8w^4v^4 + zyx^2w^8v^4 + zx^3w^8v^4 + yx^3w^8v^4 + zx^8w^2v^5 + yx^8w^2v^5 + zx^2w^8v^5 + yx^2w^8v^5 + x^8w^2v^6 + x^2w^8v^6 + zyx^4w^2v^8 + zx^5w^2v^8 + yx^5w^2v^8 + x^6w^2v^8 + zyx^2w^4v^8 + zx^3w^4v^8 + yx^3w^4v^8 + x^4w^4v^8 + zx^4w^2v^9 + yx^4w^2v^9 + zx^2w^4v^9 + yx^2w^4v^9 + x^4w^2v^10 + x^2w^4v^10, y^2x^8w^4v^2 + x^10w^4v^2 + y^2x^4w^8v^2 + x^6w^8v^2 + y^2x^8w^2v^4 + x^10w^2v^4 + x^8w^4v^4 + y^2x^2w^8v^4 + x^8w^2v^6 + x^2w^8v^6 + y^2x^4w^2v^8 + x^6w^2v^8 + y^2x^2w^4v^8 + x^4w^4v^8 + x^4w^2v^10 + x^2w^4v^10 ]

The annihilator of the Essential Cohomology has dimension 4 no end to the new generators was found.

CHECKS

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

THE COMPUTATION IS COMPLETE

GROUP # 13

Abelian(2,2,2) x Quaternion(8)

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

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, 1, 1, 1, 4 ] by the ideal generated by the relations [ z^2 + z*y + y^2 + z*x + y*x + x^2 + z*v + y*v + v^2, y^3 + y^2*x + y*x^2 + x^3 + y^2*v + x^2*v + y*v^2 + x*v^2 + v^3 ]

Restriction to the Maximal Elementary Abelian Subgroup

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

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

of type Abelian(2,2) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are [ z + w, y + x, y, w, y, z*y^3 + y^4 + z*y^2*x + y*x^2*w + x^3*w + v ] in the cohomology of H

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

Maximal Subgroup H # 2

The Group H is Isomorphic to the Group of Order 32 Number 9 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.4^$.2 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.1 * G.4 * G.5, G.2 * G.3 * G.6, G.1, G.2 ]

of type Abelian(2,2) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are [ z + y, x + w, x, z, z, z*y^3 + y^4 + z*y^2*x + z^3*w + y*x^2*w + x^3*w + z^2*w^2 + z*w^3 + v ] in the cohomology of H

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

Maximal Subgroup H # 3

of type Abelian(2,2) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are [ z + y, x + w, z + x + w, z, z + y + x, z*y^3 + z*y^2*x + y*x^2*w + x^3*w + v ] 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 # 4

of type Abelian(2,2) x Quaternion(8)

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

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

Maximal Subgroup H # 5

The Group H is Isomorphic to the Group of Order 32 Number 9 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.4^$.2 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.5, G.1 * G.4, G.1 * G.3 * G.5, G.1 * G.2 * G.6 ]

of type Abelian(2,2) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are [ z + x + w, x, z, w, z + y, y^4 + z^2*w^2 + v ] 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 # 6

of type Abelian(2,2) x Quaternion(8)

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

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

Maximal Subgroup H # 7

of type Abelian(2,2) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are [ z + y + x, z, z + y + x, z + y, z + y + x + w, y^2*w^2 + x^2*w^2 + v ] 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 # 8

The Group H is Isomorphic to the Group of Order 32 Number 9 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.4^$.2 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.1 * G.3 * G.4 * G.6, G.5, G.3, G.2 ]

of type Abelian(2,2) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are [ z, w, z + y, z, x, z^2*y^2 + z^3*w + y^2*w^2 + z*w^3 + v ] 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 # 9

The group H is abelian of type [ 4, 2, 2, 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, 1, 2 ] by the ideal of relations [ z^2 ]

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

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

Maximal Subgroup H # 10

of type Abelian(2,2) x Quaternion(8)

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

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

Maximal Subgroup H # 11

The Group H is Isomorphic to the Group of Order 32 Number 9 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.4^$.2 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.1, G.5, G.3, G.1 * G.4 ]

of type Abelian(2,2) x Quaternion(8)

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

of type Abelian(2,2) x Quaternion(8)

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

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

Maximal Subgroup H # 13

of type Abelian(2,2) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are [ z + y, x + w, z, z + w, x, y^4 + z^3*w + y^2*w^2 + x^2*w^2 + z*w^3 + v ] in the cohomology of H

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

Maximal Subgroup H # 14

of type Abelian(2,2) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are [ 0, z + w, z + y + x, z + y, w, z^3*w + z*w^3 + v ] in the cohomology of H

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

Maximal Subgroup H # 15

The Group H is Isomorphic to the Group of Order 32 Number 9 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.4^$.2 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.3 * G.4, G.1 * G.5 * G.6, G.3, G.2 ]

of type Abelian(2,2) x Quaternion(8)

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

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

Maximal Subgroup H # 16

The Group H is Isomorphic to the Group of Order 32 Number 9 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.4^$.2 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.2 * G.5 * G.6, G.1, G.1 * G.3 * G.4, G.2 ]

of type Abelian(2,2) x Quaternion(8)

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

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, 1, 1, 1, 4 ] by the ideal generated by the relations
[ z^2 + zy + y^2 + zx + yx + x^2 + zv + yv + v^2, y^3 + y^2x + yx^2 + x^3 + y^2v + x^2v + yv^2 + x*v^2 + v^3 ]

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

The images of the generators of the cohomology of G restricted to H are
[ z + w, y + x, y, w, y, zy^3 + y^4 + zy^2x + yx^2w + x^3w + v ] in the cohomology of H

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

The images of the generators of the cohomology of G restricted to H are
[ z + y, x + w, x, z, z, zy^3 + y^4 + zy^2x + z^3w + yx^2w + x^3w + z^2w^2 + z*w^3 + v ] in the cohomology of H

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

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

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

The images of the generators of the cohomology of G restricted to H are
[ z + x + w, x, z, w, z + y, y^4 + z^2*w^2 + v ] 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
[ z + y + w, x, z, w, z + y, z^3w + y^2w^2 + x^2w^2 + zw^3 + v ] in the cohomology of H

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

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

The kernel of the restriction to H of the cohomology of G is generated by
[ z + 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, 1, 2 ] by the ideal of relations
[ z^2 ]

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

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

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

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

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

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

The images of the generators of the cohomology of G restricted to H are
[ z + y, x + w, z, z + w, x, y^4 + z^3w + y^2w^2 + x^2w^2 + zw^3 + v ] in the cohomology of H

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

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

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

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

The kernel of the restriction to H of the cohomology of G is generated by
[ x + 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, 1, 2 ] by the ideal of relations
[ z^2 ]

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

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

The images of the generators of the cohomology of G restricted to H are
[ z, w, z + y, 0, x, z^2y^2 + z^3w + y^2w^2 + zw^3 + v ] 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
[ z + y + x, z, w, z + y, 0, y^2w^2 + x^2w^2 + v ] in the cohomology of H

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

The images of the generators of the cohomology of G restricted to H are
[ z + y + x, z, 0, z + y, w, v ] 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, z + y + x, z + y + w, w, z + y, z^2y^2 + zy^3 + y^4 + zy^2x + yx^2w + x^3w + y^2w^2 + v ] in the cohomology of H

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

The images of the generators of the cohomology of G restricted to H are
[ z + y + x, z, z, z + y, z + y + x + w, y^2w^2 + x^2w^2 + 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
[ y + x + w, z, x, z + y, x + w, zy^3 + y^4 + zy^2x + yx^2w + x^3w + v ] in the cohomology of H

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

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

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

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

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

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, 1, 2 ] by the ideal of relations
[ z^2 ]

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

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

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

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

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

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