GROUP OF ORDER 64 # 76

GROUP # 76

Cyclic(2) x Group(32)#41

The MAGMA library number for G is 209

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.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 a unique maximal elementary abelian subgroup which is normal and has order 16. Its centralizer has order 16

The cohomology ring of G is the quotient of a polynomial ring in the variables [ z, y, x, w, v, u, t, s, r ] in degrees [ 1, 1, 1, 1, 3, 3, 3, 4, 4 ] by the ideal generated by the relations
[ z^2 + y^2 + zx + x^2 + xw + w^2, zy + yx + x^2 + yw, zx^2 + yx^2 + x^3 + x^2w, y^2x + x^3, x^4 + x^3w + xv + yu + xu + yt, yx^2w + x^3w + yv + xv + yu + xu + zt + wt, zv + wv + yu + xu, zu + wu + yt + xt, y^2t + zxt + x^2t + xwt, x^3w^3 + xw^2v + yw^2u + xw^2u + zxwt + yxwt + yw^2t + xw^2t + yxs + zxr + xwr + v^2 + u^2 + vt + t^2, x^3v + x^2wv + yxwt + x^2wt + y^2s + zxs + yxs + xws + y^2r + x^2r + v^2 + vt, x^3u + y^2s + x^2s + zxr + x^2r + xwr + v^2 + u^2 + t^2, x^2wu + zxwt + xw^2t + yxs + x^2s + yxr + v^2 + vu + u^2 + ut + t^2, x^3t + zxwt + yxwt + xw^2t + yxs + x^2s + y^2r + zxr + x^2r + xwr + vt + t^2 ]

The Hilbert series for the cohomology ring is (-t^4 + t^3 + 1)/(t^8 - 4*t^7 + 8*t^6 - 12*t^5 + 14*t^4 - 12*t^3 + 8*t^2 - 4*t + 1)
Its numerator factors as ( t^4 - t^3 - 1 )
Its denominator factors as ( t - 1 )^4 ( t^2 + 1 )^2

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

The hypercohomolgy spectral sequence has E2 term:

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

Restriction to the Maximal Elementary Abelian Subgroup

Generated by [ G.2 * G.3 * G.5 * G.6, G.1 * G.4, G.1 * G.4 * G.5 * G.6, G.1 * G.4 * G.5 ] 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
[ y + x + w, z, z, y + x + w, z^3 + z^2y + zy^2 + z^2w + zw^2, z^3 + zy^2 + zx^2 + z^2w, zy^2 + z^2x + zw^2, z^2y^2 + z^3x + z^2x^2 + x^4 + w^4, z^4 + y^4 + x^4 + z^3w ] 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, x^3 + x^2*w + v + t ]

This ideal is also the nilradical of the cohomology of G.

Restrictions to Maximal Subgroups

Maximal Subgroup H # 1

The Group H is Isomorphic to the Group of Order 32 Number 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.2 * G.4 * G.5, G.3 * G.5 ]

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

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

Maximal Subgroup H # 2

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

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

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

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

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, y, y^2x + yx^2 + x^3 + zw + xu, yx^2 + x^3 + xw + xv, zy^2 + y^2x + zu + xu, y^4 + x^2u + w^2 + v^2 + u^2, 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 # 5

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

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

The images of the generators of the cohomology of G restricted to H are
[ z + x, 0, z + y, z + y + x, yw, zw + zv + yv, zv, w^2, y^2w + 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 # 6

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.2, G.3 * G.5 ]

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

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

Maximal Subgroup H # 7

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

The images of the generators of the cohomology of G restricted to H are
[ y + x, z + y, x, x, v, z^2x + y^2x + w, w + v + u, z^2x^2 + zw + zv + s, zv + x*v + t + s ] 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 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^2 + y^2x + yx^2 + x^3 + zw + xw + xv, zy^2 + yx^2 + x^3 + zw + zu + xu, zy^2 + y^2x + zw + xw + xv + zu, y^4 + x^2v + w^2 + u^2, yx^3 + x^4 + x^2u + u^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 # 9

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

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, zw + zv + yv, z^2x + zw + yw + zv + yv, zw + yw + yv, y^2v + w^2 + v^2, 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 # 10

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

The images of the generators of the cohomology of G restricted to H are
[ x, z, z + y, z, y^2x + w, w + v, u, zw + t, z^2x^2 + yw + z*v + s ] 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 # 11

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
[ x, z + y, z, z + x, zx^2 + yx^2 + zw + yv, zx^2 + yw + zv + yv, zyx + zw + yw + zv + yv, zyw + zyv + w^2 + v^2, x^4 + zyw + zyv + 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 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.5, G.2 * G.3 * G.4 * G.6, G.1 * G.4 * G.6 ]

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, z + x, z + y + x, x^3 + zw + zu + xu, yx^2 + xw + xv + zu + xu, yx^2 + x^3 + zw + xu, yx^3 + x^4 + x^2u + w^2 + v^2, yx^3 + x^2v + x^2u + w^2 + u^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 # 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.2 * G.3 * G.4 * G.6, G.1 * G.4, G.1 * G.2 * G.5 * 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, z + y, z, z + x, zw + yw + zv + yv, z^2x + zx^2 + zw, zx^2 + yx^2 + yw + zv, y^2x^2 + x^4 + y^2v + v^2, y^2x^2 + y^2*w + 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 + w ]

Maximal Subgroup H # 14

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.4 * G.5 * 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
[ x, x, z, z + y, z^2x + y^2x + w + v, z^2x + y^2x + v, z^2x + y^2x + v + u, zv + t + s, z^2x^2 + z*v + 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 # 15

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

The images of the generators of the cohomology of G restricted to H are
[ x, z + y, x, y + x, y^2x + v, z^2x + y^2x + w, w + v + u, z^2x^2 + zw + zv + s, zv + xv + t + s ] 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 generated as an ideal by
[ yv + xv + yu + xu + zt + wt, yxwt + x^2wt, zxwt + xw^2t ]

The annihilator of the Essential Cohomology has dimension 3 no end to the new generators was found.

CHECKS

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

THE COMPUTATION IS COMPLETE

GROUP # 76

Cyclic(2) x Group(32)#41

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.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:

Restriction to the Maximal Elementary Abelian Subgroup

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

This ideal is also the nilradical of the cohomology of G.

Restrictions to Maximal Subgroups

Maximal Subgroup H # 1

The Group H is Isomorphic to the Group of Order 32 Number 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.2 * G.4 * G.5, G.3 * G.5 ]

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

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

Maximal Subgroup H # 2

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

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

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

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

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, y, y^2*x + y*x^2 + x^3 + z*w + x*u, y*x^2 + x^3 + x*w + x*v, z*y^2 + y^2*x + z*u + x*u, y^4 + x^2*u + w^2 + v^2 + u^2, 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 # 5

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

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

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

Maximal Subgroup H # 6

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.2, G.3 * G.5 ]

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

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

Maximal Subgroup H # 7

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

The images of the generators of the cohomology of G restricted to H are [ y + x, z + y, x, x, v, z^2*x + y^2*x + w, w + v + u, z^2*x^2 + z*w + z*v + s, z*v + x*v + t + s ] 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 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^2 + y^2*x + y*x^2 + x^3 + z*w + x*w + x*v, z*y^2 + y*x^2 + x^3 + z*w + z*u + x*u, z*y^2 + y^2*x + z*w + x*w + x*v + z*u, y^4 + x^2*v + w^2 + u^2, y*x^3 + x^4 + x^2*u + u^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 # 9

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

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, z*w + z*v + y*v, z^2*x + z*w + y*w + z*v + y*v, z*w + y*w + y*v, y^2*v + w^2 + v^2, 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 # 10

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

The images of the generators of the cohomology of G restricted to H are [ x, z, z + y, z, y^2*x + w, w + v, u, z*w + t, z^2*x^2 + y*w + z*v + s ] 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 # 11

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 [ x, z + y, z, z + x, z*x^2 + y*x^2 + z*w + y*v, z*x^2 + y*w + z*v + y*v, z*y*x + z*w + y*w + z*v + y*v, z*y*w + z*y*v + w^2 + v^2, x^4 + z*y*w + z*y*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 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.5, G.2 * G.3 * G.4 * G.6, G.1 * G.4 * G.6 ]

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, z + x, z + y + x, x^3 + z*w + z*u + x*u, y*x^2 + x*w + x*v + z*u + x*u, y*x^2 + x^3 + z*w + x*u, y*x^3 + x^4 + x^2*u + w^2 + v^2, y*x^3 + 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 [ y + x ]

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.2 * G.3 * G.4 * G.6, G.1 * G.4, G.1 * G.2 * G.5 * 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, z + y, z, z + x, z*w + y*w + z*v + y*v, z^2*x + z*x^2 + z*w, z*x^2 + y*x^2 + y*w + z*v, y^2*x^2 + x^4 + y^2*v + v^2, y^2*x^2 + y^2*w + 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 + w ]

Maximal Subgroup H # 14

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.4 * G.5 * 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 [ x, x, z, z + y, z^2*x + y^2*x + w + v, z^2*x + y^2*x + v, z^2*x + y^2*x + v + u, z*v + t + s, z^2*x^2 + z*v + 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 # 15

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

The images of the generators of the cohomology of G restricted to H are [ x, z + y, x, y + x, y^2*x + v, z^2*x + y^2*x + w, w + v + u, z^2*x^2 + z*w + z*v + s, z*v + x*v + t + s ] 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 generated as an ideal by [ y*v + x*v + y*u + x*u + z*t + w*t, y*x*w*t + x^2*w*t, z*x*w*t + x*w^2*t ]

CHECKS

THE COMPUTATION IS COMPLETE

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

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

The kernel of the restriction to H of the cohomology of G is generated by
[ 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
[ x, z + y, z, z + x, zx^2 + yx^2 + zw + yv, zx^2 + yw + zv + yv, zyx + zw + yw + zv + yv, zyw + zyv + w^2 + v^2, x^4 + zyw + zyv + 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
[ y + x, z + x, z + x, z + y + x, x^3 + zw + zu + xu, yx^2 + xw + xv + zu + xu, yx^2 + x^3 + zw + xu, yx^3 + x^4 + x^2u + w^2 + v^2, yx^3 + x^2v + x^2u + w^2 + u^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 + x, z + y, z, z + x, zw + yw + zv + yv, z^2x + zx^2 + zw, zx^2 + yx^2 + yw + zv, y^2x^2 + x^4 + y^2v + v^2, y^2x^2 + y^2*w + 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 + w ]

The images of the generators of the cohomology of G restricted to H are
[ x, x, z, z + y, z^2x + y^2x + w + v, z^2x + y^2x + v, z^2x + y^2x + v + u, zv + t + s, z^2x^2 + z*v + 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
[ x, z + y, x, y + x, y^2x + v, z^2x + y^2x + w, w + v + u, z^2x^2 + zw + zv + s, zv + xv + t + s ] 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 generated as an ideal by
[ yv + xv + yu + xu + zt + wt, yxwt + x^2wt, zxwt + xw^2t ]