GROUP OF ORDER 64 # 89

GROUP # 89

The MAGMA library number for G is 62

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

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 8
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, s, r, q, p ] in degrees [ 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4 ] by the ideal generated by the relations
[ z^2 + y^2, z*y + y^2 + x^2, z*x + y*x, y^2*x + y*w + x*t, y*x^2, z*w + y*w, x*w + z*t + y*t, y^2*v + x^2*u + x^2*t + t^2, y*x*v + w*t + x*s, x^2*v + w^2, y^2*t + w^2 + y*s + x*r, y*x*t + x*s, z*s + y*s, z*r + y*r, x*t^2 + y*x*s + w*s, y*w*u + x*u*t + y*x*s + w*s + t*r + y*p, z*v*t + y*v*t + y*t^2 + t*s + x*p, y*x*r + w*r + x*p, z*p + y*p, y^4*u + w^2*v + y*w*r + x^2*q + y*x*p + r^2, w^2*t + x*v*r + y*x*p + w*p, x*v*s + x*u*s + x*t*s + y*v*r + y^2*p + s*r + t*p, x*t*s + y^2*p + s*r, y*t*s + x*v*r + s^2 + w*p, y*v*t^2 + y^2*u*s + v*t*s + z*t*q + y*t*q + y*w*p + x*v*p + r*p, w*t*s + x*s^2 + t^2*r + s*p, v*r^2 + p^2 ]

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

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

The hypercohomolgy spectral sequence has E2 term:

ROW (0) [ y, z, x ] [ y*x, w, x^2, t ] [ z*t, x*t, y*t, s, r ] [ x*r, y*r, p, t^2 ] [ x*p, y*p ] [ w*p ]
ROW (1) [ z + y ] [ x^2 ] [ z*t + y*t ] [ t^2 ]
The spectral sequence satisfies Poincaré duality

Restriction to the Maximal Elementary Abelian Subgroup

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

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

The images of the generators of the cohomology of G restricted to H are
[ 0, y, z, v, t, u, x, y*w, z*w + z*t, w^2, w*v + v*t ] 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 18 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.2^$.1 = $.2 * $.5 Generated by [ G.2 * G.3, G.1 ]

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

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

The Group H is Isomorphic to the Group of Order 32 Number 18 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.2^$.1 = $.2 * $.5 Generated by [ G.2, G.1 * G.3 ]

The images of the generators of the cohomology of G restricted to H are
[ y, z, y, x + v, w + t, v + u + t, x, z*u + y*u, y*x + y*w + y*u + y*t, u^2, x*w + w*v + x*u + v*u + x*t + v*t ] 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 # 5

The Group H is Isomorphic to the Group of Order 32 Number 18 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.2^$.1 = $.2 * $.5 Generated by [ G.3, G.1 ]

The images of the generators of the cohomology of G restricted to H are
[ y, 0, z, v, t, u, x, y*w + y*t, z*w, w^2 + t^2, 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 # 6

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

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

The images of the generators of the cohomology of G restricted to H are
[ x, z + x, z, z*y, x^2 + w, y^2 + x^2 + w, z*y + x^2 + v, x^3 + z*w + x*v + z*u, y*x^2 + x^3 + x*v + z*u, x^4 + w^2 + u^2, y*x^3 + x^4 + y*x*v + z*y*u + 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 # 7

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

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

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

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

The essential cohomology of G is generated as an ideal by
[ x^2 ]

The annihilator of the Essential Cohomology has dimension 3 The essential cohomology is a free module over the polynomial subring Q of the cohomology ring of G generated by [ v, u, q ] .
The essential cohomology is generated as a module over Q by the elements [] [ x^2 ] [] [ t^2 ]

CHECKS

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

THE COMPUTATION IS COMPLETE