GROUP OF ORDER 64 # 34

GROUP # 34

Cyclic(8) x Dihedral(8)

The MAGMA library number for G is 115

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

The center of G is abelian of type [ 2, 8 ]
The orders of the terms of the lower central series are [ 64, 2, 1 ]
The orders of the terms of the upper central series are [ 1, 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, 32 ]
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 ] in degrees [ 1, 1, 1, 2, 2 ] by the ideal generated by the relations
[ z^2, z*y + z*x + y*x + x^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, v ]

Restrictions to Maximal Elementary Abelian Subgroups

Subgroup E # 1
Generated by [ G.4, G.2 * G.3 * G.4 * G.5, 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, z, z, x^2, z*y + y^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 ]

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

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

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

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 21 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.3, $.3^2 = $.4, $.2^$.1 = $.2 * $.5 Generated by [ G.1, G.1 * G.5 * G.6 ]

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

The Group H is Isomorphic to the Group of Order 32 Number 20 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.3^2 = $.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
[ z, y, 0, v, 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 # 3

The group H is abelian of type [ 8, 2, 2 ]

The cohomology ring of H is a the quotient of a polynomial ring in the variables [ z, y, x, w ] in degrees [ 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, x, z, y^2 + w, z*y + y^2 + y*x ] 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 # 4

The Group H is Isomorphic to the Group of Order 32 Number 20 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.3^2 = $.4, $.2^$.1 = $.2 * $.5 Generated by [ G.2 * G.5 * G.6, G.1 * G.2 ]

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

The group H is abelian of type [ 8, 2, 2 ]

The cohomology ring of H is a the quotient of a polynomial ring in the variables [ z, y, x, w ] in degrees [ 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, x, x, y^2 + w, z*y + y^2 + z*x + y*x ] 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 # 6

The group H is abelian of type [ 8, 4 ]

The cohomology ring of H is a the quotient of a polynomial ring in the variables [ z, y, x, w ] in degrees [ 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
[ z, z, y, x, w ] 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 14 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4 * $.5, $.3^2 = $.4, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5 Generated by [ G.2 * G.4 * G.5, G.2, G.3 * G.4 * G.5 ]

of type Cyclic(4) x Dihedral(8)

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

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

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