GROUP OF ORDER 64 # 92

GROUP # 92

The MAGMA library number for G is 68

GrpPC : G of order 64 = 2^6 PC-Relations: G.1^2 = G.3 * G.5, G.2^2 = G.3, G.4^2 = G.5, G.4^G.1 = G.4 * G.5, G.4^G.2 = 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 8
The exponent of G is 4
G has a unique maximal elementary abelian subgroup which is central in G and has order 8

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

The Hilbert series for the cohomology ring is (-t^4 - 2*t^3 - t^2 - 2*t - 1)/(t^7 - t^6 - t^5 + t^4 - t^3 + t^2 + t - 1)
Its numerator factors as ( t^4 + 2*t^3 + t^2 + 2*t + 1 )
Its denominator factors as ( t - 1 )^3 ( t + 1 )^2 ( 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 [ w, v, r ]

Restriction to the Maximal Elementary Abelian Subgroup

Generated by [ G.3 * G.5 * G.6, G.6, G.3 ] 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, 0, z^2 + x^2, z^2 + y^2, 0, 0, 0, z^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, u, 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 12 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.3, $.2^$.1 = $.2 * $.5 Generated by [ G.6, G.1 * G.3 * G.4 * G.5 * G.6, G.1 * G.6 ]

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, y^2 + v, y^2 + y*x + x^2, z^2*x, z*w + y*w + z*v + y*v, y*w + y*v, y^2*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 # 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.1 * G.2, G.2 * G.4 * G.6 ]

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

The images of the generators of the cohomology of G restricted to H are
[ 0, z, y, w, u, z*t, y*x, y*t, t^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 # 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.1 * G.4, G.2 ]

The images of the generators of the cohomology of G restricted to H are
[ y, z, y, w + t, u, z*t, y*t, y*x + y*t, t^2 ] 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 12 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.3, $.2^$.1 = $.2 * $.5 Generated by [ G.4, G.1 * 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
[ y, y, z, y^2 + x^2, w, y*w + y*v, z^2*x + y*w + y*v, z*w + z*v, y^2*x^2 + w^2 + 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 # 6

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

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

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 [ w, v, r ] .
The essential cohomology is generated as a module over Q by the elements [] [] [] [] [ x^2*s ]

CHECKS

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

THE COMPUTATION IS COMPLETE