GROUP OF ORDER 64 # 20

GROUP # 20

Cyclic(2) x Group(32)#16

The MAGMA library number for G is 195

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.2^G.1 = G.2 * G.6, G.4^G.1 = G.4 * G.6, G.4^G.2 = G.4 * G.6

The center of G is abelian of type [ 2, 2, 4 ]
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 4
G has a unique maximal elementary abelian subgroup which is normal and has order 16. Its centralizer has order 32

The cohomology ring of G is the quotient of a polynomial ring in the variables [ z, y, x, w, v, u, t ] in degrees [ 1, 1, 1, 1, 2, 3, 4 ] by the ideal generated by the relations
[ z^2 + y^2, z*y + y^2 + z*w + y*w + w^2, y*w^2 + w^3, z*u + y*u, y^4*v + u^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 [ x^2, v, t ]
A homogeneous set of parameters is the set [ x^2, v, t, z^2 ] of degrees [ 2, 2, 4, 2 ]

The hypercohomolgy spectral sequence has E2 term:

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

Restriction to the Maximal Elementary Abelian Subgroup

Generated by [ G.6, G.3 * G.5, G.1 * G.2 * G.5 * G.6, 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
[ z, z, x, 0, y^2 + x^2, z^2*y + z^2*x, z^2*y^2 + y^4 + z^2*w^2 + w^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, w ]

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 16 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4 * $.5, $.2^2 = $.4, $.3^2 = $.4, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5 Generated by [ G.1, G.2 * G.3 * G.6, G.1 * G.3 * G.4 * G.6 ]

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

The Group H is Isomorphic to the Group of Order 32 Number 16 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4 * $.5, $.2^2 = $.4, $.3^2 = $.4, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5 Generated by [ G.1 * G.4 * G.6, G.2 * G.3 * G.4 * G.5, G.2 * G.3 ]

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

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
[ z, z + y, x, z + y, v, y*x^2 + y*w, x^4 + w^2 ] 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 # 5

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

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

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

The Group H is Isomorphic to the Group of Order 32 Number 16 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4 * $.5, $.2^2 = $.4, $.3^2 = $.4, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5 Generated by [ G.2 * G.3 * G.4 * G.5, G.2 * G.3, G.1 * G.3 * G.4 * G.6 ]

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

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

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

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

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

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

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

Maximal Subgroup H # 9

The Group H is Isomorphic to the Group of Order 32 Number 16 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4 * $.5, $.2^2 = $.4, $.3^2 = $.4, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5 Generated by [ G.2, G.2 * G.4 * G.5, G.1 * G.3 * G.4 * G.6 ]

The images of the generators of the cohomology of G restricted to H are
[ z, y + x, z, z + x, x^2 + w, x^3 + z*w + y*w + v, u ] 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 16 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4 * $.5, $.2^2 = $.4, $.3^2 = $.4, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5 Generated by [ G.2, G.2 * G.3 * G.4 * G.5, G.1 * G.3 * G.4 * G.6 ]

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

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

Maximal Subgroup H # 12

The Group H is Isomorphic to the Group of Order 32 Number 16 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4 * $.5, $.2^2 = $.4, $.3^2 = $.4, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5 Generated by [ G.1 * G.4 * G.6, G.2, G.2 * G.4 * G.5 ]

The images of the generators of the cohomology of G restricted to H are
[ z, y + x, 0, z + x, x^2 + w, x^3 + z*w + y*w + v, u ] 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 16 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4 * $.5, $.2^2 = $.4, $.3^2 = $.4, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5 Generated by [ G.2 * G.3 * G.4, G.1 * G.4, G.1 * G.3 ]

The images of the generators of the cohomology of G restricted to H are
[ y + x, z, z + y, z + x, w, x*w + v, z*x*w + y*x*w + w^2 + u ] 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 # 14

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

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

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

Maximal Subgroup H # 15

The Group H is Isomorphic to the Group of Order 32 Number 16 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4 * $.5, $.2^2 = $.4, $.3^2 = $.4, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5 Generated by [ G.2 * G.4, G.2 * G.3 * G.5 * G.6, G.1 * G.3 ]

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

The annihilator of the Essential Cohomology has dimension 3

CHECKS

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

THE COMPUTATION IS COMPLETE