GROUP OF ORDER 64 # 80

GROUP # 80

The MAGMA library number for G is 210

GrpPC : G of order 64 = 2^6 PC-Relations: G.1^2 = G.6, G.2^2 = G.6, G.3^2 = G.5 * G.6, G.4^2 = G.5 * G.6, G.2^G.1 = G.2 * G.6, G.3^G.2 = G.3 * G.5, G.4^G.2 = G.4 * 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 3 conjugacy classes of maximal elementary abelian p-subgroups. The orders of the maximal elementary abelian subgroups are [ 8, 8, 8 ]
The orders of the centralizers of the maximal elementary abelian subgroups are [ 32, 16, 16 ]
The orders of the normalizers of the maximal elementary abelian subgroups are [ 64, 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, 3, 4, 4 ] by the ideal generated by the relations
[ z^2 + zy + y^2 + x^2 + yw + w^2, yx + x^2 + w^2, y^3 + x^3 + xw^2, yw^2, y^2v + x^2v + w^2v, x^3v + xw^2v + x^2u + y^2*t + v^2 ]

The Hilbert series for the cohomology ring is (-t^4 - 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^4 + 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, x^2 ] of degrees [ 4, 4, 2 ]

The hypercohomolgy spectral sequence has E2 term:

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

Restrictions to Maximal Elementary Abelian Subgroups

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

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, 0, z, z, zy^2 + zx^2, y^4 + x^4, z^2*y^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, y, x + w ]

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

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
[ z, z, z, 0, z^3 + z^2y + zy^2, z^4 + z^2y^2 + y^4 + z^2x^2 + x^4, z^4 + z^3y + z^2y^2 + 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, y + x, w ]

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

The nilradical of the cohomology of G is generated by
[ y + x + w, z*w ]

Restrictions to Maximal Subgroups

Maximal Subgroup H # 1

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

The images of the generators of the cohomology of G restricted to H are
[ z + y, x, z, z, zw + xw + v, y^2w + yxw + w^2 + u, y^2w + yxw + x^2w + w^2 + zv ] 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 38 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4 * $.5, $.2^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.1 = $.3 * $.5 Generated by [ G.2 * G.3 * G.4 * G.5 * G.6, G.3 * G.4 * G.5 * G.6, G.4 ]

The images of the generators of the cohomology of G restricted to H are
[ 0, z, z + x, z + y + x, y^3 + zw + xw + v, w^2, z^2w + x^2w + zu + yu + t ] 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 41 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4 * $.5, $.2^2 = $.4, $.3^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.2 * G.4, G.1 * G.3, G.2 ]

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

The Group H is Isomorphic to the Group of Order 32 Number 41 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4 * $.5, $.2^2 = $.4, $.3^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.1 * G.2 * G.6, G.1 * G.3, 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, y + x, z, y, z^2x + u, z^2x^2 + t, z^2*x^2 + s ] 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 # 5

The Group H is Isomorphic to the Group of Order 32 Number 38 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4 * $.5, $.2^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.1 = $.3 * $.5 Generated by [ G.1 * G.2 * G.3 * G.4 * G.5, G.3 * G.4 * G.5 * G.6, G.4 ]

The images of the generators of the cohomology of G restricted to H are
[ z, z, z + x, z + y + x, zw + xw + v, w^2, z^2w + x^2w + zu + yu + t ] 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 # 6

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

The images of the generators of the cohomology of G restricted to H are
[ z, z + y, x, y, x^3 + v, yx^3 + x^4 + u, yx^3 + x^4 + y^2w + zx*w + yxw + w^2 + zv + yv + 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 # 7

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, 0, y + x, x, yw + xw, w^2, x^2w + x^2v + v^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 # 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.1, G.2, G.3 ]

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

The kernel of the restriction to H of the cohomology of G is generated by
[ 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.2 * G.3 * G.4, G.2 * G.4 * G.5 * G.6, G.1 * G.3 * G.5 * G.6 ]

The images of the generators of the cohomology of G restricted to H are
[ y, z + x, z + y, z + x, z^3 + zyx + v, xv + t, yw + xw + yv + x*v + u + t ] 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 # 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.4 ]

of type Cyclic(4) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are
[ y, x, 0, z, xw, zyx^2 + v, zyx^2 + x^2w + 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 # 11

The Group H is Isomorphic to the Group of Order 32 Number 14 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4 * $.5, $.3^2 = $.4, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5 Generated by [ G.1 * G.2 * G.3 * G.4 * G.5 * G.6, G.1 * G.2 * G.6, G.1 * G.3 * G.6 ]

of type Cyclic(4) x Dihedral(8)

The images of the generators of the cohomology of G restricted to H are
[ z + y + x, y + x, z + x, x, zx^2 + x^3 + yw + xw + zv + yv, zy^3 + zyx^2 + x^4 + y^2v + x^2v + v^2, zy^3 + zy*x^2 + y^2w + x^2w + y^2v + x^2v + 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 + y + x + w ]

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

The images of the generators of the cohomology of G restricted to H are
[ y, z + y, x, y, v, x^4 + u, x^4 + y^2w + zx*w + yxw + w^2 + zv + yv + u ] 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 14 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4 * $.5, $.3^2 = $.4, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5 Generated by [ G.4 * G.5, G.1 * G.2 * G.3 * G.4 * G.5, G.2 * G.3 * G.4 * G.6 ]

of type Cyclic(4) x Dihedral(8)

The images of the generators of the cohomology of G restricted to H are
[ z, z + x, z + x, z + y + x, zy^2 + x^3 + zw + xw, zy^3 + z^2yx + zyx^2 + y^2v + w^2 + v^2, zy^3 + zyx^2 + y^2w + y^2v + v^2 ] 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 # 14

The Group H is Isomorphic to the Group of Order 32 Number 36 GrpPC of order 32 = 2^5 PC-Relations: $.2^2 = $.4, $.3^2 = $.4, $.2^$.1 = $.2 * $.4 * $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.2 * G.4, G.2 * G.3 * G.5, G.1 * G.2 * G.4 * G.6 ]

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

Maximal Subgroup H # 15

The Group H is Isomorphic to the Group of Order 32 Number 14 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4 * $.5, $.3^2 = $.4, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5 Generated by [ G.2, G.2 * G.3 * G.4 * G.6, G.1 * G.3 * G.5 ]

of type Cyclic(4) x Dihedral(8)

The images of the generators of the cohomology of G restricted to H are
[ z, y + x, z + x, x, zy^2 + zyx + zx^2 + x^3 + yw + xw + zv + yv, zy^3 + z^2yx + zyx^2 + x^4 + y^2v + x^2v + v^2, zy^3 + z^2yx + zyx^2 + y^2w + x^2w + y^2v + x^2v + 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 + x + w ]

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 # 80

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

The hypercohomolgy spectral sequence has E2 term:

Restrictions to Maximal Elementary Abelian Subgroups

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

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

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

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

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

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

The nilradical of the cohomology of G is generated by [ y + x + w, z*w ]

Restrictions to Maximal Subgroups

Maximal Subgroup H # 1

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

The images of the generators of the cohomology of G restricted to H are [ z + y, x, z, z, z*w + x*w + v, y^2*w + y*x*w + w^2 + u, y^2*w + y*x*w + x^2*w + w^2 + z*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 38 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4 * $.5, $.2^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.1 = $.3 * $.5 Generated by [ G.2 * G.3 * G.4 * G.5 * G.6, G.3 * G.4 * G.5 * G.6, G.4 ]

The images of the generators of the cohomology of G restricted to H are [ 0, z, z + x, z + y + x, y^3 + z*w + x*w + v, w^2, z^2*w + x^2*w + z*u + y*u + t ] 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 41 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4 * $.5, $.2^2 = $.4, $.3^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.2 * G.4, G.1 * G.3, G.2 ]

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

The Group H is Isomorphic to the Group of Order 32 Number 41 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4 * $.5, $.2^2 = $.4, $.3^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.1 * G.2 * G.6, G.1 * G.3, 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, y + x, z, y, z^2*x + u, z^2*x^2 + t, z^2*x^2 + s ] 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 # 5

The Group H is Isomorphic to the Group of Order 32 Number 38 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4 * $.5, $.2^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.1 = $.3 * $.5 Generated by [ G.1 * G.2 * G.3 * G.4 * G.5, G.3 * G.4 * G.5 * G.6, G.4 ]

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

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

The images of the generators of the cohomology of G restricted to H are [ z, z + y, x, y, x^3 + v, y*x^3 + x^4 + u, y*x^3 + x^4 + y^2*w + z*x*w + y*x*w + w^2 + z*v + y*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 # 7

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, 0, y + x, x, y*w + x*w, w^2, x^2*w + 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 [ y ]

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.1, G.2, G.3 ]

The images of the generators of the cohomology of G restricted to H are [ x, z, y, 0, y*x^2 + v, z*y*w + w^2 + u, z^2*y^2 + z*x*w + x^2*w + w^2 + z*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 # 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.2 * G.3 * G.4, G.2 * G.4 * G.5 * G.6, G.1 * G.3 * G.5 * G.6 ]

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

of type Cyclic(4) x Quaternion(8)

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

Maximal Subgroup H # 11

The Group H is Isomorphic to the Group of Order 32 Number 14 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4 * $.5, $.3^2 = $.4, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5 Generated by [ G.1 * G.2 * G.3 * G.4 * G.5 * G.6, G.1 * G.2 * G.6, G.1 * G.3 * G.6 ]

of type Cyclic(4) x Dihedral(8)

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

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

The images of the generators of the cohomology of G restricted to H are [ y, z + y, x, y, v, x^4 + u, x^4 + y^2*w + z*x*w + y*x*w + w^2 + z*v + y*v + u ] 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 14 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4 * $.5, $.3^2 = $.4, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5 Generated by [ G.4 * G.5, G.1 * G.2 * G.3 * G.4 * G.5, G.2 * G.3 * G.4 * G.6 ]

of type Cyclic(4) x Dihedral(8)

The images of the generators of the cohomology of G restricted to H are [ z, z + x, z + x, z + y + x, z*y^2 + x^3 + z*w + x*w, z*y^3 + z^2*y*x + z*y*x^2 + y^2*v + w^2 + v^2, z*y^3 + z*y*x^2 + y^2*w + y^2*v + v^2 ] 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 # 14

The Group H is Isomorphic to the Group of Order 32 Number 36 GrpPC of order 32 = 2^5 PC-Relations: $.2^2 = $.4, $.3^2 = $.4, $.2^$.1 = $.2 * $.4 * $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.2 * G.4, G.2 * G.3 * G.5, G.1 * G.2 * G.4 * G.6 ]

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

Maximal Subgroup H # 15

The Group H is Isomorphic to the Group of Order 32 Number 14 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4 * $.5, $.3^2 = $.4, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5 Generated by [ G.2, G.2 * G.3 * G.4 * G.6, G.1 * G.3 * G.5 ]

of type Cyclic(4) x Dihedral(8)

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

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

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

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

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

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

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

The nilradical of the cohomology of G is generated by
[ y + x + w, z*w ]

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

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

The images of the generators of the cohomology of G restricted to H are
[ z, y + x, z + x, x, zy^2 + zyx + zx^2 + x^3 + yw + xw + zv + yv, zy^3 + z^2yx + zyx^2 + x^4 + y^2v + x^2v + v^2, zy^3 + z^2yx + zyx^2 + y^2w + x^2w + y^2v + x^2v + 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 + x + w ]