GROUP OF ORDER 64 # 84

GROUP # 84

The MAGMA library number for G is 67

GrpPC : G of order 64 = 2^6 PC-Relations: G.1^2 = G.4, G.2^2 = G.5, G.2^G.1 = G.2 * G.6, G.3^G.1 = G.3 * G.6, 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 2 conjugacy classes of maximal elementary abelian p-subgroups. The orders of the maximal elementary abelian subgroups are [ 16, 16 ]
The orders of the centralizers of the maximal elementary abelian subgroups are [ 32, 16 ]
The orders of the normalizers of the maximal elementary abelian subgroups are [ 64, 64 ]

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

The Hilbert series for the cohomology ring is (t^2 + t + 1)/(t^6 - 2*t^5 - t^4 + 4*t^3 - t^2 - 2*t + 1)
Its numerator factors as ( t^2 + t + 1 )
Its denominator factors as ( t - 1 )^4 ( t + 1 )^2

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

Restrictions to Maximal Elementary Abelian Subgroups

Subgroup E # 1
Generated by [ G.2 * G.3, G.4 * G.5, G.6, G.4 ]

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

Subgroup E # 2
Generated by [ G.4 * G.5, G.6, G.3 * G.6, G.4 ]

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
[ 0, 0, z, z*y + z*w, y^2 + w^2, 0, z*x + x^2, z*w + w^2 ] in the cohomology of E

The kernel of the restriction to E (Minimal Prime) of the cohomology of G is generated by
[ z, y, u ]

The nilradical of the cohomology of G is generated by
[ z ]

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.1 * G.3 ]

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

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

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.2, G.2 * G.6, G.3 * G.5 * 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
[ 0, z + y, x, z*w + y*w + x*w, w^2, z*y + y^2, z*y + y*x, z*x + 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 # 4

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.4, G.1, G.5 ]

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

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

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

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

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

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

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

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

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

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