GROUP OF ORDER 64 # 21

GROUP # 21

Cyclic(2) x Group(32)#17

The MAGMA library number for G is 248

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

The center of G is abelian of type [ 2, 8 ]
The orders of the terms of the lower central series are [ 64, 2, 1 ]
The orders of the terms of the upper central series are [ 1, 16, 64 ]
The order of the Frattini subgroup is 4
The exponent of G is 8
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, 32, 32 ]
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 ] in degrees [ 1, 1, 1, 1, 4 ] by the ideal generated by the relations
[ z^2 + y^2, y^2x + zx^2 + y^2w + zw^2 ]

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

The Krull dimension of the cohomology ring is 3
The cohomology ring is Cohen-Macaulay and the longest regular sequence consists of the generators [ x^2, v, y^2 + z*w + w^2 ]

Restrictions to Maximal Elementary Abelian Subgroups

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

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

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

Restrictions to Maximal Subgroups

Maximal Subgroup H # 1

The Group H is Isomorphic to the Group of Order 32 Number 17 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.3^2 = $.4, $.4^2 = $.5, $.2^$.1 = $.2 * $.5 Generated by [ G.2, G.1 * G.3 * G.5 * G.6, G.1 ]

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

Maximal Subgroup H # 3

The Group H is Isomorphic to the Group of Order 32 Number 17 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.3^2 = $.4, $.4^2 = $.5, $.2^$.1 = $.2 * $.5 Generated by [ G.2, G.1 * G.4 * G.6, G.1 ]

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

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

Maximal Subgroup H # 4

The Group H is Isomorphic to the Group of Order 32 Number 17 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.3^2 = $.4, $.4^2 = $.5, $.2^$.1 = $.2 * $.5 Generated by [ G.2 * G.4, G.1 * G.4 * G.5, G.2 * G.3 * G.4 ]

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

The Group H is Isomorphic to the Group of Order 32 Number 13 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3 * $.5, $.2^2 = $.3, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.1 * G.4 * G.5 * G.6, G.3 * G.5 * G.6, G.1 ]

of type Cyclic(2) x Group(16)#11

The images of the generators of the cohomology of G restricted to H are
[ z + y, 0, x, z, 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 # 6

The Group H is Isomorphic to the Group of Order 32 Number 13 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3 * $.5, $.2^2 = $.3, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.1 * G.2 * G.4, G.1 * G.3 * G.4 * G.5, G.2 * G.4 ]

of type Cyclic(2) x Group(16)#11

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

The Group H is Isomorphic to the Group of Order 32 Number 17 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.3^2 = $.4, $.4^2 = $.5, $.2^$.1 = $.2 * $.5 Generated by [ G.2 * G.3 * G.4 * G.6, G.2 * G.3 * G.5, G.1 * G.3 * G.6 ]

The images of the generators of the cohomology of G restricted to H are
[ z, y + x, z + y + x, x, w ] 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 17 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.3^2 = $.4, $.4^2 = $.5, $.2^$.1 = $.2 * $.5 Generated by [ G.2, G.1 * G.3 * G.4 * G.5, G.1 * G.4 * G.6 ]

The images of the generators of the cohomology of G restricted to H are
[ z + y, x, z, z + y, w ] 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 17 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.3^2 = $.4, $.4^2 = $.5, $.2^$.1 = $.2 * $.5 Generated by [ G.1 * G.3 * G.5, G.1 * G.3 * G.4 * G.5 * G.6, G.2 * G.6 ]

The images of the generators of the cohomology of G restricted to H are
[ z + y, x, z + y, z, w ] 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 # 10

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
[ z + y, y, z + x, y + x, y^2*w + 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 # 11

The Group H is Isomorphic to the Group of Order 32 Number 10 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.4^$.2 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.3 * G.5 * G.6, G.1 * G.2 * G.3 * G.4 * G.5, G.1 * G.2 * G.3 * G.4, G.4 * G.5 * G.6 ]

of type Cyclic(2) x AlmostExtraSpecial(16)

The images of the generators of the cohomology of G restricted to H are
[ z + w, z + w, z + y + w, z + x + w, z^2yx + z^2yw + y^3w + y^2w^2 + yxw^2 + x^2*w^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 # 12

The Group H is Isomorphic to the Group of Order 32 Number 17 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.3^2 = $.4, $.4^2 = $.5, $.2^$.1 = $.2 * $.5 Generated by [ G.1 * G.5 * G.6, G.2 * G.3 * G.4, G.1 * G.3 * G.6 ]

The images of the generators of the cohomology of G restricted to H are
[ z + y, x, y + x, x, w ] 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 # 13

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

Maximal Subgroup H # 14

The Group H is Isomorphic to the Group of Order 32 Number 13 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3 * $.5, $.2^2 = $.3, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.2 * G.4, G.1 * G.2 * G.3 * G.4 * G.5 * G.6, G.1 * G.4 ]

of type Cyclic(2) x Group(16)#11

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

Maximal Subgroup H # 15

The Group H is Isomorphic to the Group of Order 32 Number 17 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.3^2 = $.4, $.4^2 = $.5, $.2^$.1 = $.2 * $.5 Generated by [ G.1 * G.5, G.2 * G.3 * G.4 * G.6, G.1 * G.4 * G.5 ]

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

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

The essential cohomology of G is zero

CHECKS

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

THE COMPUTATION IS COMPLETE

GROUP # 21

Cyclic(2) x Group(32)#17

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

The cohomology ring of G is the quotient of a polynomial ring in the variables [ z, y, x, w, v ] in degrees [ 1, 1, 1, 1, 4 ] by the ideal generated by the relations [ z^2 + y^2, y^2*x + z*x^2 + y^2*w + z*w^2 ]

Restrictions to Maximal Elementary Abelian Subgroups

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

The images of the generators of the cohomology of G restricted to E are [ z, z, y, z + y, 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 + w, y + x + w ]

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

The images of the generators of the cohomology of G restricted to E are [ 0, 0, y, z + y, 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 ]

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

The images of the generators of the cohomology of G restricted to E are [ z, z, y, y, 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 [ z + y ]

Restrictions to Maximal Subgroups

Maximal Subgroup H # 1

The Group H is Isomorphic to the Group of Order 32 Number 17 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.3^2 = $.4, $.4^2 = $.5, $.2^$.1 = $.2 * $.5 Generated by [ G.2, G.1 * G.3 * G.5 * G.6, G.1 ]

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

Maximal Subgroup H # 3

The Group H is Isomorphic to the Group of Order 32 Number 17 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.3^2 = $.4, $.4^2 = $.5, $.2^$.1 = $.2 * $.5 Generated by [ G.2, G.1 * G.4 * G.6, G.1 ]

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

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

Maximal Subgroup H # 4

The Group H is Isomorphic to the Group of Order 32 Number 17 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.3^2 = $.4, $.4^2 = $.5, $.2^$.1 = $.2 * $.5 Generated by [ G.2 * G.4, G.1 * G.4 * G.5, G.2 * G.3 * G.4 ]

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

The Group H is Isomorphic to the Group of Order 32 Number 13 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3 * $.5, $.2^2 = $.3, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.1 * G.4 * G.5 * G.6, G.3 * G.5 * G.6, G.1 ]

of type Cyclic(2) x Group(16)#11

The images of the generators of the cohomology of G restricted to H are [ z + y, 0, x, z, 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 # 6

The Group H is Isomorphic to the Group of Order 32 Number 13 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3 * $.5, $.2^2 = $.3, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.1 * G.2 * G.4, G.1 * G.3 * G.4 * G.5, G.2 * G.4 ]

of type Cyclic(2) x Group(16)#11

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

The Group H is Isomorphic to the Group of Order 32 Number 17 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.3^2 = $.4, $.4^2 = $.5, $.2^$.1 = $.2 * $.5 Generated by [ G.2 * G.3 * G.4 * G.6, G.2 * G.3 * G.5, G.1 * G.3 * G.6 ]

The images of the generators of the cohomology of G restricted to H are [ z, y + x, z + y + x, x, w ] 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 17 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.3^2 = $.4, $.4^2 = $.5, $.2^$.1 = $.2 * $.5 Generated by [ G.2, G.1 * G.3 * G.4 * G.5, G.1 * G.4 * G.6 ]

The images of the generators of the cohomology of G restricted to H are [ z + y, x, z, z + y, w ] 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 17 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.3^2 = $.4, $.4^2 = $.5, $.2^$.1 = $.2 * $.5 Generated by [ G.1 * G.3 * G.5, G.1 * G.3 * G.4 * G.5 * G.6, G.2 * G.6 ]

The images of the generators of the cohomology of G restricted to H are [ z + y, x, z + y, z, w ] 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 # 10

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 [ z + y, y, z + x, y + x, y^2*w + 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 # 11

of type Cyclic(2) x AlmostExtraSpecial(16)

The images of the generators of the cohomology of G restricted to H are [ z + w, z + w, z + y + w, z + x + w, z^2*y*x + z^2*y*w + y^3*w + y^2*w^2 + y*x*w^2 + x^2*w^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 # 12

The Group H is Isomorphic to the Group of Order 32 Number 17 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.3^2 = $.4, $.4^2 = $.5, $.2^$.1 = $.2 * $.5 Generated by [ G.1 * G.5 * G.6, G.2 * G.3 * G.4, G.1 * G.3 * G.6 ]

The images of the generators of the cohomology of G restricted to H are [ z + y, x, y + x, x, w ] 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 # 13

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

Maximal Subgroup H # 14

The Group H is Isomorphic to the Group of Order 32 Number 13 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3 * $.5, $.2^2 = $.3, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.2 * G.4, G.1 * G.2 * G.3 * G.4 * G.5 * G.6, G.1 * G.4 ]

of type Cyclic(2) x Group(16)#11

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

Maximal Subgroup H # 15

The Group H is Isomorphic to the Group of Order 32 Number 17 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.3^2 = $.4, $.4^2 = $.5, $.2^$.1 = $.2 * $.5 Generated by [ G.1 * G.5, G.2 * G.3 * G.4 * G.6, G.1 * G.4 * G.5 ]

The cohomology ring of G is the quotient of a polynomial ring in the variables [ z, y, x, w, v ] in degrees [ 1, 1, 1, 1, 4 ] by the ideal generated by the relations
[ z^2 + y^2, y^2x + zx^2 + y^2w + zw^2 ]

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

The images of the generators of the cohomology of G restricted to E are
[ z, z, y, z + y, 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 + w, y + x + w ]

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

The images of the generators of the cohomology of G restricted to E are
[ 0, 0, y, z + y, 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 ]

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

The images of the generators of the cohomology of G restricted to E are
[ z, z, y, y, 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
[ z + y ]

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

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

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

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

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

The cohomology ring of 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
[ z + y, y, z + x, y + x, y^2*w + 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
[ z + w, z + w, z + y + w, z + x + w, z^2yx + z^2yw + y^3w + y^2w^2 + yxw^2 + x^2*w^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
[ z + y, x, y + x, x, w ] 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 ] 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
[ 0, z, x, y, y^2x^2 + y^2w + 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
[ z ]

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

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