GROUP OF ORDER 64 # 44

GROUP # 44

Abelian(2,2) x Semidihedral(16)

The MAGMA library number for G is 251

GrpPC : G of order 64 = 2^6 PC-Relations: G.1^2 = G.6, G.2^2 = G.5 * G.6, G.3^2 = G.5 * G.6, G.5^2 = G.6, G.2^G.1 = G.2 * G.5 * G.6, G.3^G.1 = G.3 * G.5 * G.6, G.4^G.1 = G.4 * G.5, G.4^G.2 = G.4 * G.5, G.4^G.3 = G.4 * G.5, G.5^G.1 = G.5 * G.6, G.5^G.4 = G.5 * 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, 2, 1 ]
The orders of the terms of the upper central series are [ 1, 8, 16, 64 ]
The order of the Frattini subgroup is 4
The exponent of G is 8
G has one conjugate class of maximal elementary abelian subgroups. Any element of the class has order 16 . Its centralizer has order 16 and its normalizer has order 32

The cohomology ring of G is the quotient of a polynomial ring in the variables [ z, y, x, w, v, u ] in degrees [ 1, 1, 1, 1, 3, 4 ] by the ideal generated by the relations
[ zy + y^2 + zx + x^2 + zw + yw + xw, z^3 + y^3 + y^2x + yx^2 + x^3 + z^2w + y^2w + x^2w + zw^2 + yw^2 + xw^2, zv + yv + xv, y^2xv + x^3v + xw^2v + y^2u + x^2u + w^2u + v^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 [ z^2, y^2, u ]
A homogeneous set of parameters is the set [ z^2, y^2, u, w^2 ] of degrees [ 2, 2, 4, 2 ]

The hypercohomolgy spectral sequence has E2 term:

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

Restriction to the Maximal Elementary Abelian Subgroup

Generated by [ G.1 * G.3 * G.4 * G.5, G.2 * G.3 * G.4, G.1 * G.2 * G.4 * G.5, G.1 * G.3 * G.4 * G.5 * G.6 ] 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 + x, z + y + w, z + y + x + w, z^3 + z^2y + zy^2 + z^2x, z^2y^2 + zy^3 + y^4 + z^3x + z^2yx + z^2x^2 + z^3w + z^2yw + zy^2w + z^2xw ] 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 ]

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

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

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

Maximal Subgroup H # 2

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

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

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

Maximal Subgroup H # 3

The Group H is Isomorphic to the Group of Order 32 Number 24 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.2 * G.4 * G.6, G.1 * G.2 * G.4 * G.6, G.1 * G.3 * G.5 * G.6 ]

of type Cyclic(2) x Semidihedral(16)

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

The Group H is Isomorphic to the Group of Order 32 Number 24 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.2 * G.3 * G.6, G.3 * G.4 * G.5, G.4 ]

of type Cyclic(2) x Semidihedral(16)

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

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

Maximal Subgroup H # 5

The Group H is Isomorphic to the Group of Order 32 Number 24 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.3 * G.4 * G.5 * G.6, G.1 * G.2 * G.4 * G.6, G.1 * G.2 * G.5 ]

of type Cyclic(2) x Semidihedral(16)

The images of the generators of the cohomology of G restricted to H are
[ y + x, y + x, z, z + x, y^3 + y^2x + w, y^4 + y^2x^2 + v ] 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 24 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.3 * G.4, G.1 * G.3 * G.5 * G.6, G.1 * G.3 * G.4 ]

of type Cyclic(2) x Semidihedral(16)

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

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

Maximal Subgroup H # 7

The Group H is Isomorphic to the Group of Order 32 Number 24 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.1 * G.2 * G.3 * G.5 * G.6, G.2 * G.3 * G.4 * G.5, G.1 * G.3 * G.4 ]

of type Cyclic(2) x Semidihedral(16)

The images of the generators of the cohomology of G restricted to H are
[ z + x, z + y, z + y + x, y + x, y^3 + z^2x + y^2x + yx^2 + w, y^2x^2 + yx^3 + x^4 + yw + x*w + v ] 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 # 8

The Group H is Isomorphic to the Group of Order 32 Number 24 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.1 * G.2 * G.6, G.1 * G.2 * G.3 * G.5, G.2 * G.3 ]

of type Cyclic(2) x Semidihedral(16)

The images of the generators of the cohomology of G restricted to H are
[ z + y, z + y + x, z + x, 0, y^3 + y^2x + w, y^4 + y^3x + x*w + 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 abelian of type [ 8, 2, 2 ]

The cohomology ring of H is a the quotient of a polynomial ring in the variables [ z, y, x, w ] in degrees [ 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
[ y + x, y, z + x, y + x, zw, zx*w + w^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 # 10

The Group H is Isomorphic to the Group of Order 32 Number 24 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.1, G.2 * G.3, G.4 ]

of type Cyclic(2) x Semidihedral(16)

The images of the generators of the cohomology of G restricted to H are
[ z, x, x, y, yx^2 + w, yx^3 + x^4 + x*w + v ] 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 # 11

The Group H is Isomorphic to the Group of Order 32 Number 24 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.1 * G.2 * G.4 * G.6, G.1 * G.3 * G.6, G.1 * G.2 * G.3 * G.5 ]

of type Cyclic(2) x Semidihedral(16)

The images of the generators of the cohomology of G restricted to H are
[ z + y + x, z + x, z + y, x, yx^2 + w, y^2x^2 + x^4 + y*w + v ] 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 24 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.1 * G.2 * G.5 * G.6, G.1 * G.2 * G.4, G.2 * G.4 ]

of type Cyclic(2) x Semidihedral(16)

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

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

Maximal Subgroup H # 13

The Group H is Isomorphic to the Group of Order 32 Number 24 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.2 * G.4 * G.5 * G.6, G.1 * G.3 * G.4 * G.6, G.1 * G.3 * G.5 ]

of type Cyclic(2) x Semidihedral(16)

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

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

Maximal Subgroup H # 14

The Group H is Isomorphic to the Group of Order 32 Number 24 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.2 * G.3 * G.4, G.1 * G.3 * G.4, G.1 * G.6 ]

of type Cyclic(2) x Semidihedral(16)

The images of the generators of the cohomology of G restricted to H are
[ z + x, y, y + x, y + x, y^3 + z^2x + w, y^3x + yw + xw + 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 # 15

The Group H is Isomorphic to the Group of Order 32 Number 24 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.3 * G.4, G.1 * G.2 * G.5, G.2 * G.3 ]

of type Cyclic(2) x Semidihedral(16)

The images of the generators of the cohomology of G restricted to H are
[ y, y + x, z + x, z, y^3 + y^2x + w, y^4 + y^3x + x*w + v ] 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 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 # 44

Abelian(2,2) x Semidihedral(16)

GrpPC : G of order 64 = 2^6 PC-Relations: G.1^2 = G.6, G.2^2 = G.5 * G.6, G.3^2 = G.5 * G.6, G.5^2 = G.6, G.2^G.1 = G.2 * G.5 * G.6, G.3^G.1 = G.3 * G.5 * G.6, G.4^G.1 = G.4 * G.5, G.4^G.2 = G.4 * G.5, G.4^G.3 = G.4 * G.5, G.5^G.1 = G.5 * G.6, G.5^G.4 = G.5 * 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 + x, z + y + w, z + y + x + w, z^3 + z^2*y + z*y^2 + z^2*x, z^2*y^2 + z*y^3 + y^4 + z^3*x + z^2*y*x + z^2*x^2 + z^3*w + z^2*y*w + z*y^2*w + z^2*x*w ] 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 ]

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

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

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

Maximal Subgroup H # 2

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

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

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

Maximal Subgroup H # 3

of type Cyclic(2) x Semidihedral(16)

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

The Group H is Isomorphic to the Group of Order 32 Number 24 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.2 * G.3 * G.6, G.3 * G.4 * G.5, G.4 ]

of type Cyclic(2) x Semidihedral(16)

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

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

Maximal Subgroup H # 5

of type Cyclic(2) x Semidihedral(16)

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

of type Cyclic(2) x Semidihedral(16)

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

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

Maximal Subgroup H # 7

of type Cyclic(2) x Semidihedral(16)

The images of the generators of the cohomology of G restricted to H are [ z + x, z + y, z + y + x, y + x, y^3 + z^2*x + y^2*x + y*x^2 + w, y^2*x^2 + y*x^3 + x^4 + y*w + x*w + v ] 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 # 8

of type Cyclic(2) x Semidihedral(16)

The images of the generators of the cohomology of G restricted to H are [ z + y, z + y + x, z + x, 0, y^3 + y^2*x + w, y^4 + y^3*x + x*w + 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 abelian of type [ 8, 2, 2 ]

The cohomology ring of H is a the quotient of a polynomial ring in the variables [ z, y, x, w ] in degrees [ 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 [ y + x, y, z + x, y + x, z*w, z*x*w + w^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 # 10

The Group H is Isomorphic to the Group of Order 32 Number 24 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.1, G.2 * G.3, G.4 ]

of type Cyclic(2) x Semidihedral(16)

The images of the generators of the cohomology of G restricted to H are [ z, x, x, y, y*x^2 + w, y*x^3 + x^4 + x*w + v ] 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 # 11

of type Cyclic(2) x Semidihedral(16)

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

of type Cyclic(2) x Semidihedral(16)

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

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

Maximal Subgroup H # 13

of type Cyclic(2) x Semidihedral(16)

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

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

Maximal Subgroup H # 14

of type Cyclic(2) x Semidihedral(16)

The images of the generators of the cohomology of G restricted to H are [ z + x, y, y + x, y + x, y^3 + z^2*x + w, y^3*x + y*w + x*w + 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 # 15

The Group H is Isomorphic to the Group of Order 32 Number 24 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.3 * G.4, G.1 * G.2 * G.5, G.2 * G.3 ]

of type Cyclic(2) x Semidihedral(16)

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

CHECKS

THE COMPUTATION IS COMPLETE

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

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
[ x + w, y + w, y + x, w, z^2x + zx^2 + xw^2 + xv, z^2y^2 + z^3x + z^2yx + y^3x + y^2x^2 + zx^3 + yxw^2 + x^2w^2 + w^4 + yxv + 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
[ z + y + x ]

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

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

The images of the generators of the cohomology of G restricted to H are
[ z + x, z + y, z + y + x, y + x, y^3 + z^2x + y^2x + yx^2 + w, y^2x^2 + yx^3 + x^4 + yw + x*w + v ] 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, z + y + x, z + x, 0, y^3 + y^2x + w, y^4 + y^3x + x*w + v ] 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 ] in degrees [ 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
[ y + x, y, z + x, y + x, zw, zx*w + w^2 ] in the cohomology of H

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

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

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

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

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

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

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