GROUP OF ORDER 64 # 51

GROUP # 51

Cyclic(2) x Group(32)#31

The MAGMA library number for G is 101

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

The center of G is abelian of type [ 2, 4 ]
The orders of the terms of the lower central series are [ 64, 4, 2, 1 ]
The orders of the terms of the upper central series are [ 1, 8, 16, 64 ]
The order of the Frattini subgroup is 8
The exponent of G is 8
G has 2 conjugacy classes of maximal elementary abelian p-subgroups. The orders of the maximal elementary abelian subgroups are [ 8, 8 ]
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, 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, 2, 2, 3, 4 ] by the ideal generated by the relations
[ z^2, z*y + z*x, y*x^2 + x^3 + y*w + x*w, z*v, y^3*x + x^4 + y^2*w + x^2*w + y^2*v + x^2*v + y*u + x*u, z*x*w + x^2*v + w*v + z*u, v^2, y^2*x*v + x^3*v + x*w*v + v*u, y^2*w^2 + y*x*w*v + x^2*w*v + y^2*x*u + x^3*u + y*v*u + x*v*u + u^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 [ y^2, w, t ]

Restrictions to Maximal Elementary Abelian Subgroups

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

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, x, x, z^2 + x^2, 0, z^2*x + x^3, z^2*y^2 + y^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, v, x*w + u ]

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

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, x, z + x, z^2 + x^2, 0, z*x^2 + x^3, z^2*y^2 + y^4 + z^2*x^2 + z*x^3 ] in the cohomology of E

The kernel of the restriction to E (Minimal Prime) of the cohomology of G is generated by
[ z, x^2 + w, v, y^2*x + u ]

The nilradical of the cohomology of G is generated by
[ z, v, y^2*x + x^3 + x*w + u ]

Restrictions to Maximal Subgroups

Maximal Subgroup H # 1

The Group H is Isomorphic to the Group of Order 32 Number 31 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.2, $.3^2 = $.5, $.4^2 = $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.3, G.1 * G.2 * G.3 * G.5 ]

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

The Group H is Isomorphic to the Group of Order 32 Number 31 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.2, $.3^2 = $.5, $.4^2 = $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.3, G.1 ]

The images of the generators of the cohomology of G restricted to H are
[ z, 0, 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
[ y ]

Maximal Subgroup H # 3

The Group H is Isomorphic to the Group of Order 32 Number 31 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.2, $.3^2 = $.5, $.4^2 = $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.1 * G.4 * G.5, G.2 * G.4 * G.5 ]

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

The Group H is Isomorphic to the Group of Order 32 Number 10 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.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.4, G.5, G.2 * G.4 * G.5 ]

of type Cyclic(2) x AlmostExtraSpecial(16)

The images of the generators of the cohomology of G restricted to H are
[ 0, z, w, z*y + y^2 + z*x + x^2 + y*w + x*w + w^2, z*y + y^2 + z*w + y*w, z*y^2 + y*x^2 + y^2*w + z*w^2 + y*w^2, z*y^3 + y*x^2*w + x^2*w^2 + 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 # 5

The Group H is Isomorphic to the Group of Order 32 Number 31 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.2, $.3^2 = $.5, $.4^2 = $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.1 * G.2 * G.3 * G.4 * G.6, G.2 * G.4 * G.5 * G.6 ]

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

The Group H is Isomorphic to the Group of Order 32 Number 13 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3 * $.5, $.2^2 = $.3, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.5 * G.6, G.1 * G.2 * G.5 * G.6, G.1 * G.3 * G.5 * G.6 ]

of type Cyclic(2) x Group(16)#11

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

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