GROUP OF ORDER 64 # 87

GROUP # 87

The MAGMA library number for G is 72

GrpPC : G of order 64 = 2^6 PC-Relations: G.1^2 = G.4 * G.5 * G.6, G.2^2 = G.4, G.3^2 = G.5, G.2^G.1 = G.2 * G.6, G.3^G.1 = 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 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 ] in degrees [ 1, 1, 1, 2, 2, 2, 3, 4 ] by the ideal generated by the relations
[ z^2 + z*x + x^2, z*y + z*x + x^2, y^2 + z*x + x^2, x^3, z*w + y*w, y*x*w + x^2*w + z*t, z*x*u + x^2*u + w^2, y*t, w*t, t^2 ]

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

The Krull dimension of the cohomology ring is 3
The cohomology ring is Cohen-Macaulay and the longest regular sequence consists of the generators [ v, u, s ]

Restriction to the Maximal Elementary Abelian Subgroup

Generated by [ 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 ]

The images of the generators of the cohomology of G restricted to E are
[ 0, 0, 0, 0, y^2, z^2 + x^2, 0, y^4 + 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, t ]

This ideal is also the nilradical of the cohomology of G.

Restrictions to Maximal Subgroups

Maximal Subgroup H # 1

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

The images of the generators of the cohomology of G restricted to H are
[ z, y, 0, x + v, x + w + u, w + t, y*x + 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
[ 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.2 * G.3, G.1 ]

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

The Group H is Isomorphic to the Group of Order 32 Number 9 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.4^$.2 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.3 * G.4 * G.6, G.1 * G.2 * G.3 * G.4 * G.5, G.3 * G.4, G.1 * G.2 * G.3 * G.5 * G.6 ]

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

The images of the generators of the cohomology of G restricted to H are
[ z + w, z + w, z + y + x + w, z*y + z*x + z*w + y*w + x*w + w^2, z*x + x^2 + z*w + x*w, y^2 + x^2 + w^2, z^2*x + x*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 # 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.2, G.1 * G.3 ]

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

Maximal Subgroup H # 6

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.1 * G.4, G.4, G.1 * G.3 * G.5 * 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, z*x + y*x, y^2 + v, y^2 + x^2 + 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 # 7

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 * G.3 ]

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

The essential cohomology of G is generated as an ideal by
[ z*t ]

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

CHECKS

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

THE COMPUTATION IS COMPLETE