GROUP OF ORDER 64 # 69

GROUP # 69

Cyclic(2) x Group(32)#34

The MAGMA library number for G is 211

GrpPC : G of order 64 = 2^6 PC-Relations: G.1^2 = G.5 * G.6, G.2^2 = G.5, G.4^2 = G.6, G.3^G.1 = G.3 * G.5 * G.6, G.3^G.2 = G.3 * 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 4 conjugacy classes of maximal elementary abelian p-subgroups. The orders of the maximal elementary abelian subgroups are [ 16, 16, 16, 16 ]
The orders of the centralizers of the maximal elementary abelian subgroups are [ 16, 16, 16, 16 ]
The orders of the normalizers of the maximal elementary abelian subgroups are [ 64, 64, 64, 64 ]

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, 2, 2 ] by the ideal generated by the relations
[ z^2 + z*x + x*w + w^2, y^2 + y*x + x*w + w^2 ]

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

The Krull dimension of the cohomology ring is 4
The cohomology ring is Cohen-Macaulay and the longest regular sequence consists of the generators [ z^2, x^2, v, u ]

Restrictions to Maximal Elementary Abelian Subgroups

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

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

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

The nilradical of the cohomology of G is zero

Restrictions to Maximal Subgroups

Maximal Subgroup H # 1

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

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

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

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

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

The Group H is Isomorphic to the Group of Order 32 Number 34 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.3 * G.4 * G.6, G.2 * G.4 * G.5, G.1 * G.2 * G.5 * G.6 ]

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

Maximal Subgroup H # 4

The Group H is Isomorphic to the Group of Order 32 Number 34 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.3, G.2, G.1 ]

The images of the generators of the cohomology of G restricted to H are
[ z, y, x, 0, z*x + w + v, z*x + 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 # 5

The Group H is Isomorphic to the Group of Order 32 Number 34 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.2 * G.3, G.1, G.4 ]

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

The Group H is Isomorphic to the Group of Order 32 Number 34 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.3 * G.4 * G.6, G.2, G.1 ]

The images of the generators of the cohomology of G restricted to H are
[ z, y, x, x, z*x + w + v, z*x + w ] 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 # 7

The group H is abelian of type [ 4, 4, 2 ]

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

The images of the generators of the cohomology of G restricted to H are
[ x, z + x, 0, y + x, w, 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 # 8

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

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

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

The Group H is Isomorphic to the Group of Order 32 Number 34 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.1 * G.3, G.2, G.4 ]

The images of the generators of the cohomology of G restricted to H are
[ x, z, x, y, w, y^2 + 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 # 10

The Group H is Isomorphic to the Group of Order 32 Number 34 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.1 * G.2, G.1 * G.4, G.3 ]

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

The Group H is Isomorphic to the Group of Order 32 Number 34 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.3, G.2, G.4 ]

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

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

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

The images of the generators of the cohomology of G restricted to H are
[ y, y + x, x, z, y*x + x^2 + x*w + w^2, y*x + v ] 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 # 13

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.5, G.1 * G.2 * G.4 * G.6, G.1 * 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
[ y + x + w, y + x + w, x, z + w, y*x + x^2 + w^2, y*x + 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 # 14

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

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

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

The Group H is Isomorphic to the Group of Order 32 Number 34 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.3, G.1, G.4 ]

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