GROUP OF ORDER 64 # 50

GROUP # 50

Cyclic(2) x Group(32)#30

The MAGMA library number for G is 106

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

Restriction to the Maximal Elementary Abelian Subgroup

Generated by [ G.3 * G.4 * G.6, G.3 * G.4, G.4 * 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, 0, z + y + x, z^2 + y^2, z^2 + x^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 ]

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

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

The Group H is Isomorphic to the Group of Order 32 Number 30 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.2, $.3^2 = $.4, $.4^2 = $.5, $.3^$.1 = $.3 * $.4, $.4^$.1 = $.4 * $.5 Generated by [ G.2, G.1 * G.4 ]

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

The Group H is Isomorphic to the Group of Order 32 Number 30 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.2, $.3^2 = $.4, $.4^2 = $.5, $.3^$.1 = $.3 * $.4, $.4^$.1 = $.4 * $.5 Generated by [ G.2 * G.4, G.1 * G.2 ]

The images of the generators of the cohomology of G restricted to H are
[ z, z + y, y, x, 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 # 4

The Group H is Isomorphic to the Group of Order 32 Number 30 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.2, $.3^2 = $.4, $.4^2 = $.5, $.3^$.1 = $.3 * $.4, $.4^$.1 = $.4 * $.5 Generated by [ G.2, G.1 ]

The images of the generators of the cohomology of G restricted to H are
[ z, y, 0, x, y^2 + 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 # 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.1 * G.5, G.4, G.1 * G.4 ]

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 + x, v, y^2 + w ] in the cohomology of H

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

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.2 * G.5, G.4, G.1 * G.2 * G.4 ]

of type Cyclic(2) x Group(16)#10

The images of the generators of the cohomology of G restricted to H are
[ z + y, z + y, y + x, v, y^2 + w + 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 # 7

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
[ 0, z, y + x, y^2, z*y + x^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 generated as an ideal by
[ y^2*x ]

The annihilator of the Essential Cohomology has dimension 3

CHECKS

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

THE COMPUTATION IS COMPLETE