GROUP OF ORDER 64 # 74

GROUP # 74

Cyclic(2) x Group(32)#39

The MAGMA library number for G is 207

GrpPC : G of order 64 = 2^6 PC-Relations: G.1^2 = G.5, G.4^2 = G.5 * G.6, G.3^G.1 = G.3 * G.5, G.4^G.3 = 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, 16 ]
The orders of the centralizers of the maximal elementary abelian subgroups are [ 16, 16 ]
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, 3, 4 ] by the ideal generated by the relations
[ z^2 + zx + w^2, xw + w^2, w^3, wu, zx^5 + zx^2u + x^2t + w^2t + u^2 ]

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

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

The hypercohomolgy spectral sequence has E2 term:

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

Restrictions to Maximal Elementary Abelian Subgroups

Subgroup E # 1
Generated by [ G.2 * G.6, G.2 * G.5, G.1 * G.3 * G.5, 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, 0, zw + w^2, z^2x + zx^2 + z^2w + zw^2, z^4 + z^3x + x^4 + z^3*w + 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, w ]

Subgroup E # 2
Generated by [ G.2 * G.3 * G.6, G.2 * G.6, G.2 * G.5, 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
[ 0, z + y + x + w, z, 0, zw + w^2, z^2x + zx^2 + z^2w + zw^2, z^2x^2 + x^4 + z^2*w^2 + 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, w ]

The nilradical of the cohomology of G is generated by
[ w ]

Restrictions to Maximal Subgroups

Maximal Subgroup H # 1

The Group H is Isomorphic to the Group of Order 32 Number 39 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.4 * $.5, $.3^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.1 = $.3 * $.5 Generated by [ G.4, G.1, G.3 * G.4 ]

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

The Group H is Isomorphic to the Group of Order 32 Number 39 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.4 * $.5, $.3^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.1 = $.3 * $.5 Generated by [ G.3 * G.4 * G.6, G.1, G.1 * G.2 * G.4 * G.6 ]

The images of the generators of the cohomology of G restricted to H are
[ y + x, y, z, z + y, zx + x^2 + w, zx^2 + yx^2 + zw + yw + v, y^2x^2 + zyw + zxw + yxw + zv + xv + 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 # 3

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
[ z, x, 0, y, x^2 + v, yx^2 + yw, zyx^2 + x^4 + zyw + w^2 ] in the cohomology of H

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

Maximal Subgroup H # 4

The Group H is Isomorphic to the Group of Order 32 Number 39 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.4 * $.5, $.3^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.1 = $.3 * $.5 Generated by [ G.1 * G.2, G.4, G.3 * G.4 ]

The images of the generators of the cohomology of G restricted to H are
[ x, x, z, z + y, zx + x^2 + w, zx^2 + yx^2 + zw + v, z^2y^2 + y^2x^2 + zyw + zxw + w^2 + x*v + u ] 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 # 5

The Group H is Isomorphic to the Group of Order 32 Number 39 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.4 * $.5, $.3^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.1 = $.3 * $.5 Generated by [ G.1 * G.2 * G.5 * G.6, 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 + x, z, z + y, w, zw + v, z^2y^2 + y^2x^2 + zyw + zxw + w^2 + xv + 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 39 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.4 * $.5, $.3^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.1 = $.3 * $.5 Generated by [ G.1 * G.2 * G.4 * G.5 * G.6, G.1 * G.2 * G.3 * G.4 * G.6, G.1 ]

The images of the generators of the cohomology of G restricted to H are
[ z + y + x, z + y, z, z + y, zx + x^2 + w, zx^2 + yx^2 + zw + yw + v, z^2y^2 + y^2x^2 + zy*w + yxw + x^2w + xv + 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 # 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.2 * G.3 * G.4 * G.5, G.2 * G.3 * G.4, G.3 * G.4, 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 + w, z + y, z + y + x + w, z + y + x + w, zy + zx + zw, zy^2 + zx^2 + z^2w + w^3, z^3w + zy^2w + zx^2w + z^2w^2 + y^2w^2 + x^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
[ x + w ]

Maximal Subgroup H # 8

The Group H is Isomorphic to the Group of Order 32 Number 39 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.4 * $.5, $.3^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.1 = $.3 * $.5 Generated by [ G.4, G.1 * G.2 * G.5, G.2 * G.3 * G.4 ]

The images of the generators of the cohomology of G restricted to H are
[ x, z + x, z, z + y, zx + x^2 + w, zx^2 + yx^2 + zw + v, z^2y^2 + y^2x^2 + zyw + zxw + w^2 + x*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 # 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.1 * G.2, G.1 * G.2 * G.6, G.1 * G.2 * G.3 * 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 + x, z + y + x + w, x, 0, y^2 + yx, z^2x + zx^2 + yx^2 + xv, z^2y^2 + z^3x + z^2yx + zy^2x + y^3x + y^2x^2 + x^4 + z^2v + y^2v + x^2v + v^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 # 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.2 * G.3 * G.5, G.1 * G.2 * G.3 * G.4 * 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, z, u, xw + xv, w^2 + v^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 # 11

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.6, G.1 * G.2 * G.4 * G.5, 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 + x, z + y + x, x, z, w + v + u, zw + xw + xv, x^4 + x^2v + w^2 ] 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 # 12

The Group H is Isomorphic to the Group of Order 32 Number 39 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.4 * $.5, $.3^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.1 = $.3 * $.5 Generated by [ G.4, G.1, G.2 * G.3 * G.4 ]

The images of the generators of the cohomology of G restricted to H are
[ x, z, z, z + y, zx + x^2 + w, zx^2 + yx^2 + zw + v, z^2y^2 + y^2x^2 + zyw + zxw + w^2 + 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 ]

Maximal Subgroup H # 13

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.5, G.2 * G.3 * G.4 * 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
[ 0, z + y + x, z + x, z, u, xw + xv + x*u, w^2 + v^2 + u^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 # 14

The Group H is Isomorphic to the Group of Order 32 Number 39 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.4 * $.5, $.3^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.1 = $.3 * $.5 Generated by [ G.1 * G.2 * G.5, G.1 * G.4 * G.6, G.1 * G.2 * G.3 * G.4 * G.5 * G.6 ]

The images of the generators of the cohomology of G restricted to H are
[ z + y + x, z + x, z, z + y, zx + x^2 + w, zx^2 + yx^2 + zw + yw + v, z^2y^2 + y^2x^2 + zy*w + yxw + x^2w + xv + 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 # 15

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.4 * G.6, G.1 * G.2 * G.3 * G.5, 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 + x, z + y + x, z + x, z, u, xw + xv + xu, x^4 + x^2v + x^2*u + w^2 + u^2 ] 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 zero

CHECKS

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

THE COMPUTATION IS COMPLETE

GROUP # 74

Cyclic(2) x Group(32)#39

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

The hypercohomolgy spectral sequence has E2 term:

Restrictions to Maximal Elementary Abelian Subgroups

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

The images of the generators of the cohomology of G restricted to E are [ z, y + x + w, z, 0, z*w + w^2, z^2*x + z*x^2 + z^2*w + z*w^2, z^4 + z^3*x + x^4 + z^3*w + 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, w ]

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

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

The nilradical of the cohomology of G is generated by [ w ]

Restrictions to Maximal Subgroups

Maximal Subgroup H # 1

The Group H is Isomorphic to the Group of Order 32 Number 39 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.4 * $.5, $.3^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.1 = $.3 * $.5 Generated by [ G.4, G.1, G.3 * G.4 ]

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

The Group H is Isomorphic to the Group of Order 32 Number 39 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.4 * $.5, $.3^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.1 = $.3 * $.5 Generated by [ G.3 * G.4 * G.6, G.1, G.1 * G.2 * G.4 * G.6 ]

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

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 [ z, x, 0, y, x^2 + v, y*x^2 + y*w, z*y*x^2 + x^4 + z*y*w + w^2 ] in the cohomology of H

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

Maximal Subgroup H # 4

The Group H is Isomorphic to the Group of Order 32 Number 39 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.4 * $.5, $.3^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.1 = $.3 * $.5 Generated by [ G.1 * G.2, G.4, G.3 * G.4 ]

The images of the generators of the cohomology of G restricted to H are [ x, x, z, z + y, z*x + x^2 + w, z*x^2 + y*x^2 + z*w + v, z^2*y^2 + y^2*x^2 + z*y*w + z*x*w + w^2 + x*v + u ] 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 # 5

The Group H is Isomorphic to the Group of Order 32 Number 39 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.4 * $.5, $.3^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.1 = $.3 * $.5 Generated by [ G.1 * G.2 * G.5 * G.6, 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 + x, z, z + y, w, z*w + v, z^2*y^2 + y^2*x^2 + z*y*w + z*x*w + w^2 + x*v + 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 39 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.4 * $.5, $.3^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.1 = $.3 * $.5 Generated by [ G.1 * G.2 * G.4 * G.5 * G.6, G.1 * G.2 * G.3 * G.4 * G.6, G.1 ]

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

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 + y + x + w, z + y + x + w, z*y + z*x + z*w, z*y^2 + z*x^2 + z^2*w + w^3, z^3*w + z*y^2*w + z*x^2*w + z^2*w^2 + 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 [ x + w ]

Maximal Subgroup H # 8

The Group H is Isomorphic to the Group of Order 32 Number 39 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.4 * $.5, $.3^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.1 = $.3 * $.5 Generated by [ G.4, G.1 * G.2 * G.5, G.2 * G.3 * G.4 ]

The images of the generators of the cohomology of G restricted to H are [ x, z + x, z, z + y, z*x + x^2 + w, z*x^2 + y*x^2 + z*w + v, z^2*y^2 + y^2*x^2 + z*y*w + z*x*w + w^2 + x*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 # 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.1 * G.2, G.1 * G.2 * G.6, G.1 * G.2 * G.3 * 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 + x, z + y + x + w, x, 0, y^2 + y*x, z^2*x + z*x^2 + y*x^2 + x*v, z^2*y^2 + z^3*x + z^2*y*x + z*y^2*x + y^3*x + y^2*x^2 + x^4 + z^2*v + y^2*v + 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 [ w ]

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.2 * G.3 * G.5, G.1 * G.2 * G.3 * G.4 * 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, z, u, x*w + x*v, w^2 + v^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 # 11

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.6, G.1 * G.2 * G.4 * G.5, 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 + x, z + y + x, x, z, w + v + u, z*w + x*w + x*v, x^4 + x^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 + x + w ]

Maximal Subgroup H # 12

The Group H is Isomorphic to the Group of Order 32 Number 39 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.4 * $.5, $.3^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.1 = $.3 * $.5 Generated by [ G.4, G.1, G.2 * G.3 * G.4 ]

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

Maximal Subgroup H # 13

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.5, G.2 * G.3 * G.4 * 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 [ 0, z + y + x, z + x, z, u, x*w + x*v + x*u, w^2 + v^2 + u^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 # 14

The Group H is Isomorphic to the Group of Order 32 Number 39 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.4 * $.5, $.3^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.1 = $.3 * $.5 Generated by [ G.1 * G.2 * G.5, G.1 * G.4 * G.6, G.1 * G.2 * G.3 * G.4 * G.5 * G.6 ]

The images of the generators of the cohomology of G restricted to H are [ z + y + x, z + x, z, z + y, z*x + x^2 + w, z*x^2 + y*x^2 + z*w + y*w + v, z^2*y^2 + y^2*x^2 + z*y*w + y*x*w + x^2*w + x*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 # 15

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.4 * G.6, G.1 * G.2 * G.3 * G.5, 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 + x, z + y + x, z + x, z, u, x*w + x*v + x*u, x^4 + x^2*v + x^2*u + w^2 + u^2 ] in the cohomology of H

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

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, 3, 4 ] by the ideal generated by the relations
[ z^2 + zx + w^2, xw + w^2, w^3, wu, zx^5 + zx^2u + x^2t + w^2t + u^2 ]

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

The images of the generators of the cohomology of G restricted to E are
[ z, y + x + w, z, 0, zw + w^2, z^2x + zx^2 + z^2w + zw^2, z^4 + z^3x + x^4 + z^3*w + 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, w ]

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

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

The nilradical of the cohomology of G is generated by
[ w ]

The images of the generators of the cohomology of G restricted to H are
[ x, 0, z, z + y, zx + x^2 + w, zx^2 + yx^2 + zw + v, z^2y^2 + y^2x^2 + zyw + zxw + w^2 + x*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
[ y + x, y, z, z + y, zx + x^2 + w, zx^2 + yx^2 + zw + yw + v, y^2x^2 + zyw + zxw + yxw + zv + xv + 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 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
[ z, x, 0, y, x^2 + v, yx^2 + yw, zyx^2 + x^4 + zyw + w^2 ] 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
[ x, x, z, z + y, zx + x^2 + w, zx^2 + yx^2 + zw + v, z^2y^2 + y^2x^2 + zyw + zxw + w^2 + x*v + u ] 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
[ x, z + y + x, z, z + y, w, zw + v, z^2y^2 + y^2x^2 + zyw + zxw + w^2 + xv + 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 + y + x, z + y, z, z + y, zx + x^2 + w, zx^2 + yx^2 + zw + yw + v, z^2y^2 + y^2x^2 + zy*w + yxw + x^2w + xv + 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 + w, z + y, z + y + x + w, z + y + x + w, zy + zx + zw, zy^2 + zx^2 + z^2w + w^3, z^3w + zy^2w + zx^2w + z^2w^2 + y^2w^2 + x^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
[ x + w ]

The images of the generators of the cohomology of G restricted to H are
[ x, z + x, z, z + y, zx + x^2 + w, zx^2 + yx^2 + zw + v, z^2y^2 + y^2x^2 + zyw + zxw + w^2 + x*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 + y + x, z + y + x + w, x, 0, y^2 + yx, z^2x + zx^2 + yx^2 + xv, z^2y^2 + z^3x + z^2yx + zy^2x + y^3x + y^2x^2 + x^4 + z^2v + y^2v + x^2v + v^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
[ z, z + y + x, z + x, z, u, xw + xv, w^2 + v^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, z + y + x, x, z, w + v + u, zw + xw + xv, x^4 + x^2v + w^2 ] 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
[ x, z, z, z + y, zx + x^2 + w, zx^2 + yx^2 + zw + v, z^2y^2 + y^2x^2 + zyw + zxw + w^2 + 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 ]

The images of the generators of the cohomology of G restricted to H are
[ 0, z + y + x, z + x, z, u, xw + xv + x*u, w^2 + v^2 + u^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 + x, z, z + y, zx + x^2 + w, zx^2 + yx^2 + zw + yw + v, z^2y^2 + y^2x^2 + zy*w + yxw + x^2w + xv + 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 + y + x, z + x, z, u, xw + xv + xu, x^4 + x^2v + x^2*u + w^2 + u^2 ] in the cohomology of H

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