GROUP OF ORDER 64 # 59

GROUP # 59

The MAGMA library number for G is 21

GrpPC : G of order 64 = 2^6 PC-Relations: G.1^2 = G.3, G.2^2 = G.4, G.5^2 = G.6, G.2^G.1 = G.2 * G.5, 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, 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 16
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, u, t ] in degrees [ 1, 1, 2, 2, 2, 2, 2 ] by the ideal generated by the relations
[ z^2, z*y, y^2, z*x, y*x + z*v, y*v, x^2, x*v, v^2 ]

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

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

Restriction to the Maximal Elementary Abelian Subgroup

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

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

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

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

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.1 * G.5, 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^2, v, y^2 + z*x + y*x, w, y^2 + x^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 # 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, z, z*y, y^2, z*y + z*x, x^2 + w, y^2 + x^2 ] in the cohomology of H

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

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

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 [ w, u, t ] .
The essential cohomology is generated as a module over Q by the elements [] [] [ z*v ]

CHECKS

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

THE COMPUTATION IS COMPLETE