GROUP OF ORDER 64 # 68

GROUP # 68

Cyclic(2) x Group(32)#33

The MAGMA library number for G is 202

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

The center of G is abelian of type [ 2, 2, 2 ]
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 2 conjugacy classes of maximal elementary abelian p-subgroups. The orders of the maximal elementary abelian subgroups are [ 16, 32 ]
The orders of the centralizers of the maximal elementary abelian subgroups are [ 16, 32 ]
The orders of the normalizers of the maximal elementary abelian subgroups are [ 64, 64 ]

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, 2, 2, 2 ] by the ideal generated by the relations
[ z^2 + zw, zx + zw, zw^2 + zv, xw*v + zwu + w^2u + x^2t + zwt + v^2 ]

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

The Krull dimension of the cohomology ring is 5
The longest regular sequence consists of the generators [ y^2, x^2, u, t ]
A homogeneous set of parameters is the set [ y^2, x^2, u, t, w^2 ] of degrees [ 2, 2, 2, 2, 2 ]

The hypercohomolgy spectral sequence has E2 term:

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

Restrictions to Maximal Elementary Abelian Subgroups

Subgroup E # 1
Generated by [ G.1 * G.3 * G.4, G.2 * G.5, G.2 * G.6, G.2 ]

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 + x + w, z, z, z^2, z^2 + zw + w^2, z^2 + zx + x^2 ] in the cohomology of E

The kernel of the restriction to E (Minimal Prime) of the cohomology of G is generated by
[ z + w, x + w, w^2 + v ]

Subgroup E # 2
Generated by [ G.2 * G.4 * G.6, G.2 * G.5, G.2 * G.6, G.2 * G.3 * G.4 * G.6, G.2 ]

The cohomology ring of E is a polynomial ring in the variables [ z, y, x, w, v ]

The images of the generators of the cohomology of G restricted to E are
[ 0, z + y + x + w + v, x, z + x, zx + x^2 + xw + zv + xv, xv + v^2, zw + x*w + w^2 ] in the cohomology of E

The kernel of the restriction to E (Minimal Prime) of the cohomology of G is generated by
[ z ]

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 32 Number 8 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5 Generated by [ G.1, G.1 * G.5, G.3 * G.4 * G.5, G.2 ]

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

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

The Group H is Isomorphic to the Group of Order 32 Number 33 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.3^2 = $.4 * $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.2 * G.4, G.1, G.1 * G.3 ]

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

The Group H is Isomorphic to the Group of Order 32 Number 33 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.3^2 = $.4 * $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.3 * G.4, G.1 * G.2, G.1 * G.3 ]

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

The Group H is Isomorphic to the Group of Order 32 Number 33 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.3^2 = $.4 * $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.1, G.4, G.1 * G.2 * G.3 ]

The images of the generators of the cohomology of G restricted to H are
[ z + x, z, z, y, w, z^2 + x^2 + u, y*x + w + v + u ] 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 Isomorphic to the Group of Order 32 Number 33 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.3^2 = $.4 * $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.3, G.1 * G.2, G.1 * G.4 * G.6 ]

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

The Group H is Isomorphic to the Group of Order 32 Number 33 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.3^2 = $.4 * $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.1 * G.2, G.4, G.1 * G.3 ]

The images of the generators of the cohomology of G restricted to H are
[ z + x, x, z, y, w, z^2 + x^2 + u, y*x + w + v + u ] 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 11 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.4^$.1 = $.4 * $.5 Generated by [ G.2 * G.4 * G.5, G.1 * G.2 * G.4, G.2 ]

of type Cyclic(2) x Group(16)#9

The images of the generators of the cohomology of G restricted to H are
[ z, z + y + x, 0, z + x, x^2 + v, x^2 + w, 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 # 8

The Group H is Isomorphic to the Group of Order 32 Number 11 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.4^$.1 = $.4 * $.5 Generated by [ G.2 * G.3 * G.4 * G.5, G.1 * G.2 * G.4, G.2 ]

of type Cyclic(2) x Group(16)#9

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

The Group H is Isomorphic to the Group of Order 32 Number 8 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5 Generated by [ G.4 * G.6, G.1 * G.3 * G.5, G.2, G.1 * G.3 ]

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

The images of the generators of the cohomology of G restricted to H are
[ z + y, w, z + y, x, zy + y^2 + zx, zy + zx + yx, zx + 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 # 10

The Group H is Isomorphic to the Group of Order 32 Number 11 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.4^$.1 = $.4 * $.5 Generated by [ G.1 * G.2 * G.3, G.2 * G.3 * G.6, G.2 ]

of type Cyclic(2) x Group(16)#9

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

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

Maximal Subgroup H # 11

The group H is abelian of type [ 2, 2, 2, 2, 2 ]

The cohomology ring of H is a polynomial ring with variables [ z, y, x, w, v ] in degrees [ 1, 1, 1, 1, 1 ]

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

The Group H is Isomorphic to the Group of Order 32 Number 8 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5 Generated by [ G.1 * G.3 * G.4 * G.5 * G.6, G.1 * G.2 * G.4 * G.6, G.1 * G.2 * G.4, G.2 ]

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

The images of the generators of the cohomology of G restricted to H are
[ z + y + x, z + y + w, x, z + y + x, zy + y^2 + zx + yx + x^2, zx + x^2 + v, zy + zx + x^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 # 13

The Group H is Isomorphic to the Group of Order 32 Number 33 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.3^2 = $.4 * $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.3 * G.4, G.1, G.1 * G.2 * G.3 ]

The images of the generators of the cohomology of G restricted to H are
[ z + x, x, y + x, y, w, z^2 + yx + x^2 + v, yx + w + 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 33 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.3^2 = $.4 * $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.1, G.4, G.1 * G.3 ]

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

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

Maximal Subgroup H # 15

The Group H is Isomorphic to the Group of Order 32 Number 33 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.3^2 = $.4 * $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.3, G.1 * G.2, G.1 * G.2 * G.4 ]

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

CHECKS

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

THE COMPUTATION IS COMPLETE

GROUP # 68

Cyclic(2) x Group(32)#33

GrpPC : G of order 64 = 2^6 PC-Relations: G.1^2 = G.5 * G.6, G.3^G.1 = G.3 * G.5, G.4^G.1 = G.4 * 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, 2, 2, 2 ] by the ideal generated by the relations [ z^2 + z*w, z*x + z*w, z*w^2 + z*v, x*w*v + z*w*u + w^2*u + x^2*t + z*w*t + v^2 ]

The hypercohomolgy spectral sequence has E2 term:

Restrictions to Maximal Elementary Abelian Subgroups

Subgroup E # 1 Generated by [ G.1 * G.3 * G.4, G.2 * G.5, G.2 * G.6, G.2 ]

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

The kernel of the restriction to E (Minimal Prime) of the cohomology of G is generated by [ z + w, x + w, w^2 + v ]

Subgroup E # 2 Generated by [ G.2 * G.4 * G.6, G.2 * G.5, G.2 * G.6, G.2 * G.3 * G.4 * G.6, G.2 ]

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

The kernel of the restriction to E (Minimal Prime) of the cohomology of G is generated by [ z ]

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 32 Number 8 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5 Generated by [ G.1, G.1 * G.5, G.3 * G.4 * G.5, G.2 ]

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

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

The Group H is Isomorphic to the Group of Order 32 Number 33 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.3^2 = $.4 * $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.2 * G.4, G.1, G.1 * G.3 ]

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

The Group H is Isomorphic to the Group of Order 32 Number 33 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.3^2 = $.4 * $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.3 * G.4, G.1 * G.2, G.1 * G.3 ]

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

The Group H is Isomorphic to the Group of Order 32 Number 33 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.3^2 = $.4 * $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.1, G.4, G.1 * G.2 * G.3 ]

The images of the generators of the cohomology of G restricted to H are [ z + x, z, z, y, w, z^2 + x^2 + u, y*x + w + v + u ] 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 Isomorphic to the Group of Order 32 Number 33 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.3^2 = $.4 * $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.3, G.1 * G.2, G.1 * G.4 * G.6 ]

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

The Group H is Isomorphic to the Group of Order 32 Number 33 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.3^2 = $.4 * $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.1 * G.2, G.4, G.1 * G.3 ]

The images of the generators of the cohomology of G restricted to H are [ z + x, x, z, y, w, z^2 + x^2 + u, y*x + w + v + u ] 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 11 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.4^$.1 = $.4 * $.5 Generated by [ G.2 * G.4 * G.5, G.1 * G.2 * G.4, G.2 ]

of type Cyclic(2) x Group(16)#9

The images of the generators of the cohomology of G restricted to H are [ z, z + y + x, 0, z + x, x^2 + v, x^2 + w, 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 # 8

The Group H is Isomorphic to the Group of Order 32 Number 11 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.4^$.1 = $.4 * $.5 Generated by [ G.2 * G.3 * G.4 * G.5, G.1 * G.2 * G.4, G.2 ]

of type Cyclic(2) x Group(16)#9

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

The Group H is Isomorphic to the Group of Order 32 Number 8 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5 Generated by [ G.4 * G.6, G.1 * G.3 * G.5, G.2, G.1 * G.3 ]

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

The images of the generators of the cohomology of G restricted to H are [ z + y, w, z + y, x, z*y + y^2 + z*x, z*y + z*x + y*x, z*x + 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 # 10

The Group H is Isomorphic to the Group of Order 32 Number 11 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.4^$.1 = $.4 * $.5 Generated by [ G.1 * G.2 * G.3, G.2 * G.3 * G.6, G.2 ]

of type Cyclic(2) x Group(16)#9

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

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

Maximal Subgroup H # 11

The group H is abelian of type [ 2, 2, 2, 2, 2 ]

The cohomology ring of H is a polynomial ring with variables [ z, y, x, w, v ] in degrees [ 1, 1, 1, 1, 1 ]

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

The Group H is Isomorphic to the Group of Order 32 Number 8 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5 Generated by [ G.1 * G.3 * G.4 * G.5 * G.6, G.1 * G.2 * G.4 * G.6, G.1 * G.2 * G.4, G.2 ]

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

The images of the generators of the cohomology of G restricted to H are [ z + y + x, z + y + w, x, z + y + x, z*y + y^2 + z*x + y*x + x^2, z*x + x^2 + v, z*y + z*x + x^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 # 13

The Group H is Isomorphic to the Group of Order 32 Number 33 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.3^2 = $.4 * $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.3 * G.4, G.1, G.1 * G.2 * G.3 ]

The images of the generators of the cohomology of G restricted to H are [ z + x, x, y + x, y, w, z^2 + y*x + x^2 + v, y*x + w + 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 33 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.3^2 = $.4 * $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.1, G.4, G.1 * G.3 ]

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

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

Maximal Subgroup H # 15

The Group H is Isomorphic to the Group of Order 32 Number 33 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.3^2 = $.4 * $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.3, G.1 * G.2, G.1 * G.2 * G.4 ]

The images of the generators of the cohomology of G restricted to H are [ z + x, z + x, y, z, w, y*x + w + v + u, z^2 + x^2 + u ] 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, t ] in degrees [ 1, 1, 1, 1, 2, 2, 2 ] by the ideal generated by the relations
[ z^2 + zw, zx + zw, zw^2 + zv, xw*v + zwu + w^2u + x^2t + zwt + v^2 ]

Subgroup E # 1
Generated by [ G.1 * G.3 * G.4, G.2 * G.5, G.2 * G.6, G.2 ]

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

The kernel of the restriction to E (Minimal Prime) of the cohomology of G is generated by
[ z + w, x + w, w^2 + v ]

Subgroup E # 2
Generated by [ G.2 * G.4 * G.6, G.2 * G.5, G.2 * G.6, G.2 * G.3 * G.4 * G.6, G.2 ]

The images of the generators of the cohomology of G restricted to E are
[ 0, z + y + x + w + v, x, z + x, zx + x^2 + xw + zv + xv, xv + v^2, zw + x*w + w^2 ] in the cohomology of E

The kernel of the restriction to E (Minimal Prime) of the cohomology of G is generated by
[ z ]

The images of the generators of the cohomology of G restricted to H are
[ z + y, w, x, x, zy + y^2 + yx + x^2, y^2 + zx + yx + v, z*x + v ] 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, y, z, y, w, z^2 + x^2 + u, y*x + w + v + u ] 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
[ z + x, z, y + x, y, w, z^2 + yx + x^2 + v, yx + w + v + u ] 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 + x, z, z, y, w, z^2 + x^2 + u, y*x + w + v + u ] 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
[ z + x, x, y, z, yx + w, yx + w + v + u, z^2 + x^2 + u ] 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
[ z + x, x, z, y, w, z^2 + x^2 + u, y*x + w + v + u ] 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
[ z, z + y + x, 0, z + x, x^2 + v, x^2 + w, 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, z + y + x, x, z + x, x^2 + v, w + v + u, 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, w, z + y, x, zy + y^2 + zx, zy + zx + yx, zx + 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, z + y + x, z + x, 0, x^2 + v, u, x^2 + w ] 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
[ 0, z, x, y, yw + xv, xw + w^2, yv + 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
[ z + y + x, z + y + w, x, z + y + x, zy + y^2 + zx + yx + x^2, zx + x^2 + v, zy + zx + x^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
[ z + x, x, y + x, y, w, z^2 + yx + x^2 + v, yx + w + 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 + x, 0, z, y, w, z^2 + x^2 + u, y*x + w + v + u ] 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 + x, z + x, y, z, w, y*x + w + v + u, z^2 + x^2 + u ] in the cohomology of H

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