GROUP OF ORDER 64 # 73

GROUP # 73

Cyclic(2) x Group(32)#38

The MAGMA library number for G is 205

GrpPC : G of order 64 = 2^6 PC-Relations: G.1^2 = G.5 * G.6, G.2^2 = G.5 * G.6, G.3^2 = G.6, G.4^2 = G.5 * G.6, G.2^G.1 = G.2 * G.5, G.3^G.1 = G.3 * G.6, G.3^G.2 = G.3 * G.5, G.4^G.2 = G.4 * 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 [ 32, 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, s ] in degrees [ 1, 1, 1, 1, 2, 3, 3, 4 ] by the ideal generated by the relations
[ z^2 + y^2 + zx + x^2 + xw + w^2, zy + zx + yx + x^2 + yw + xw, y^2x + x^3, y^2w^2 + x^2w^2 + yu + zt + wt, y^2v + x^2v + yu + xu + yt + xt, zu + yu + wu + yt + xt, zx^5 + zx^4w + yx^4w + x^4w^2 + zx^3v + yx^3v + x^4v + x^3wv + zxv^2 + xwv^2 + xw^2u + yx^2t + x^3t + zw^2t + yw^2t + xw^2t + w^3t + yvu + zvt + wvt + y^2s + x^2s + u^2, yx^5 + zx^4w + x^4w^2 + zx^3v + x^3wv + zxv^2 + yxv^2 + x^2v^2 + xwv^2 + xw^2u + zx^2t + yx^2t + x^2wt + zw^2t + yw^2t + xw^2t + w^3t + zxs + x^2s + xws + u^2 + ut + t^2, x^6 + zx^4w + yx^4w + x^5w + x^4w^2 + zx^3v + yx^3v + x^4v + x^3wv + zxv^2 + xwv^2 + xw^2u + zx^2t + yx^2t + x^2wt + zw^2t + yw^2t + xw^2t + w^3t + xvu + zvt + yvt + xvt + wvt + zxs + x^2s + xws + t^2 ]

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

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

The hypercohomolgy spectral sequence has E2 term:

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

Restrictions to Maximal Elementary Abelian Subgroups

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

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, y + x + w, z^2 + zy + zx + x^2 + zw, z^3 + z^2y + zx^2 + z^2w, z^3 + z^2y + zx^2 + z^2w, z^2y^2 + y^4 + z^3*x ] 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, y + x, u + t ]

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

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 + y + x + w, z^2 + zy + x^2 + zw, 0, z^3 + z^2y + zy^2, z^4 + z^3*y + y^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, y + x, u ]

The nilradical of the cohomology of G is generated by
[ y + x ]

Restrictions to Maximal Subgroups

Maximal Subgroup H # 1

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

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

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

Maximal Subgroup H # 2

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 + x, y, z + y, x, zy + zx + yx + x^2 + zw + w^2, y^3 + zx^2 + yx^2 + zyw + y^2w + yw^2 + zv, zy^2 + y^3 + yx^2 + y^2w + zw^2 + yw^2 + zv, y^4 + y^3x + zy^2w + y^3w + zyv + y^2v + 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 ]

Maximal Subgroup H # 3

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.1 * G.2 ]

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

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

The images of the generators of the cohomology of G restricted to H are
[ y + x, z + x, y + x, z, zy + y^2 + yx + w, x^3 + xw + v + u, y^3 + yx^2 + x^3 + zw + yw + xw + u, y^4 + x^4 + xu + t ] 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 # 5

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

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

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

Maximal Subgroup H # 6

The Group H is Isomorphic to the Group of Order 32 Number 12 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.3, $.2^$.1 = $.2 * $.5 Generated by [ G.2, G.1 * G.4 * G.5 * G.6, G.4 ]

of type Cyclic(2) x Group(16)#10

The images of the generators of the cohomology of G restricted to H are
[ x, y, 0, z + x, yx + w, yx^2 + zw + yv, z^2x + yx^2 + yw + zv + yv, y^2w + y^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
[ x ]

Maximal Subgroup H # 7

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

The images of the generators of the cohomology of G restricted to H are
[ z + y, y + x, z + x, z + y + x, y^2 + w, yx^2 + xw + v + u, yx^2 + yw + xw + u, y^4 + z^2w + yxw + x^2w + w^2 + xu + t ] 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 # 8

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

The images of the generators of the cohomology of G restricted to H are
[ 0, z + x, y + x, z + y + x, zy + yx + w, xw + v + u, y^3 + zw + yw + xw + u, y^4 + x*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 # 9

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

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

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

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

Maximal Subgroup H # 11

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

The images of the generators of the cohomology of G restricted to H are
[ y, y + x, z + x, y + x, yx + x^2 + w, yx^2 + xw + v + u, yx^2 + yw + xw + u, y^4 + x^4 + z^2w + yx*w + x^2w + w^2 + xu + 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 # 12

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

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

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

Maximal Subgroup H # 13

The Group H is Isomorphic to the Group of Order 32 Number 12 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.3, $.2^$.1 = $.2 * $.5 Generated by [ G.3 * G.4 * G.6, G.1 * G.4 * G.5 * G.6, G.1 * G.6 ]

of type Cyclic(2) x Group(16)#10

The images of the generators of the cohomology of G restricted to H are
[ y + x, 0, z, z + x, y^2 + w, zx^2 + yw + zv, zx^2 + zw + yw + zv + yv, 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 ]

Maximal Subgroup H # 14

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.4, G.1 * G.2 * G.3 * G.6, G.1 * 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, y + x, z + x, z + y, y^2 + yx + x^2 + w, yx^2 + x^3 + xw + v + u, x^3 + yw + xw + u, y^4 + yx^3 + z^2w + yx*w + x^2w + w^2 + xu + t ] 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 11 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.4^$.1 = $.4 * $.5 Generated by [ G.2 * G.3, G.1 * G.2 * G.4 * G.5, G.1 * G.4 ]

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

The images of the generators of the cohomology of G restricted to H are
[ z + y, z + x, x, z + y, zy + yx + u, zw, y^2x + yx^2 + x^3 + zw + xw + xv + zu + xu, y^4 + yx^3 + 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 + 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 # 73

Cyclic(2) x Group(32)#38

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

The hypercohomolgy spectral sequence has E2 term:

Restrictions to Maximal Elementary Abelian Subgroups

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

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

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

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

The nilradical of the cohomology of G is generated by [ y + x ]

Restrictions to Maximal Subgroups

Maximal Subgroup H # 1

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

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

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

Maximal Subgroup H # 2

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

Maximal Subgroup H # 3

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.1 * G.2 ]

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

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

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

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

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

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

Maximal Subgroup H # 6

The Group H is Isomorphic to the Group of Order 32 Number 12 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.3, $.2^$.1 = $.2 * $.5 Generated by [ G.2, G.1 * G.4 * G.5 * G.6, G.4 ]

of type Cyclic(2) x Group(16)#10

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

Maximal Subgroup H # 7

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

The images of the generators of the cohomology of G restricted to H are [ z + y, y + x, z + x, z + y + x, y^2 + w, y*x^2 + x*w + v + u, y*x^2 + y*w + x*w + u, y^4 + z^2*w + y*x*w + x^2*w + w^2 + x*u + t ] 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 # 8

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

The images of the generators of the cohomology of G restricted to H are [ 0, z + x, y + x, z + y + x, z*y + y*x + w, x*w + v + u, y^3 + z*w + y*w + x*w + u, y^4 + x*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 # 9

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

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

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

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

Maximal Subgroup H # 11

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

The images of the generators of the cohomology of G restricted to H are [ y, y + x, z + x, y + x, y*x + x^2 + w, y*x^2 + x*w + v + u, y*x^2 + y*w + x*w + u, y^4 + x^4 + z^2*w + y*x*w + x^2*w + w^2 + x*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 # 12

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

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

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

Maximal Subgroup H # 13

The Group H is Isomorphic to the Group of Order 32 Number 12 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.3, $.2^$.1 = $.2 * $.5 Generated by [ G.3 * G.4 * G.6, G.1 * G.4 * G.5 * G.6, G.1 * G.6 ]

of type Cyclic(2) x Group(16)#10

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

Maximal Subgroup H # 14

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.4, G.1 * G.2 * G.3 * G.6, G.1 * 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, y + x, z + x, z + y, y^2 + y*x + x^2 + w, y*x^2 + x^3 + x*w + v + u, x^3 + y*w + x*w + u, y^4 + y*x^3 + z^2*w + y*x*w + x^2*w + w^2 + x*u + t ] 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 11 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.4^$.1 = $.4 * $.5 Generated by [ G.2 * G.3, G.1 * G.2 * G.4 * G.5, G.1 * G.4 ]

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

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

The essential cohomology of G is zero

CHECKS

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

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

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

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

The nilradical 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 + y + x, z + x, y + x, 0, zy + x^2 + w, yx^2 + x^3 + xw + v + u, y^3 + x^3 + zw + yw + xw + u, y^4 + yx^3 + xu + t ] in the cohomology of H

The kernel of the restriction to H of the cohomology of G is generated by
[ 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 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, z + x, y + x, y + x, zy + y^2 + x^2 + w, yx^2 + xw + v + u, y^3 + yx^2 + zw + yw + xw + u, y^4 + yx^3 + x^4 + x*u + t ] 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
[ y + x, z + x, y + x, z, zy + y^2 + yx + w, x^3 + xw + v + u, y^3 + yx^2 + x^3 + zw + yw + xw + u, y^4 + x^4 + xu + t ] in the cohomology of H

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

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

The images of the generators of the cohomology of G restricted to H are
[ x, y, 0, z + x, yx + w, yx^2 + zw + yv, z^2x + yx^2 + yw + zv + yv, y^2w + y^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
[ x ]

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

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

The images of the generators of the cohomology of G restricted to H are
[ y + x, 0, z, z + x, y^2 + w, zx^2 + yw + zv, zx^2 + zw + yw + zv + yv, 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 ]

The images of the generators of the cohomology of G restricted to H are
[ z + y + x, y + x, z + x, z + y, y^2 + yx + x^2 + w, yx^2 + x^3 + xw + v + u, x^3 + yw + xw + u, y^4 + yx^3 + z^2w + yx*w + x^2w + w^2 + xu + t ] 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, z + x, x, z + y, zy + yx + u, zw, y^2x + yx^2 + x^3 + zw + xw + xv + zu + xu, y^4 + yx^3 + 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 + w ]