GROUP OF ORDER 32 # 41

GROUP # 41

The MAGMA library number for G is 33

GrpPC : G of order 32 = 2^5 PC-Relations: G.1^2 = G.4 * G.5, G.2^2 = G.4, G.3^2 = G.5, G.2^G.1 = G.2 * G.4, G.3^G.1 = G.3 * G.4, G.3^G.2 = G.3 * G.5

The center of G is abelian of type [ 2, 2 ]
The orders of the terms of the lower central series are [ 32, 4, 1 ]
The orders of the terms of the upper central series are [ 1, 4, 32 ]
The order of the Frattini subgroup is 4
The exponent of G is 4
G has a unique maximal elementary abelian subgroup which is normal and has order 8. Its centralizer 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, s ] in degrees [ 1, 1, 1, 3, 3, 3, 4, 4 ] by the ideal generated by the relations
[ z^2 + y*x + x^2, z*y + y^2 + z*x + y*x + x^2, z*x^2 + x^3, y^2*x + y*x^2, z*w + x*u, y*w + z*v + x*v + z*u + y*u + x*u, x*w + z*v + x*v + y*u + x*u, y*v + z*u + x*u, y*x*u, x^6 + x^2*t + y^2*s + z*x*s + x^2*s + w*v + v^2 + v*u, x^3*u + x^2*t + y*x*s + x^2*s + w^2, y^2*t + y^2*s + z*x*s + y*x*s + x^2*s + w^2 + w*v + v*u + u^2, z*x*t + x^2*t + y^2*s + w^2 + w*u, y*x*t + y^2*s + y*x*s + w^2 + u^2 ]

The Hilbert series for the cohomology ring is (t^4 - t^3 - 1)/(t^7 - 3*t^6 + 5*t^5 - 7*t^4 + 7*t^3 - 5*t^2 + 3*t - 1)
Its numerator factors as ( t^4 - t^3 - 1 )
Its denominator factors as ( t - 1 )^3 ( t^2 + 1 )^2

The Krull dimension of the cohomology ring is 3
The longest regular sequence consists of the generators [ t, s ]
A homogeneous set of parameters is the set [ t, s, z^2 ] of degrees [ 4, 4, 2 ]

The hypercohomolgy spectral sequence has E2 term:

ROW (0) [ x, z, y ] [ y^2, x^2, z*x ] [ u, w, v ] [ x*u, x*v, y*u, z*u, z*v ] [ y^2*u ] [ v*u ] [ x*v*u ]
ROW (1) [ y ] [ y^2, z*x + x^2, y*x ] [ y*x^2 ] [ z*v + x*v, z*u + x*u, y*u ] [ z*x*u + x^2*u, y^2*u ]
The spectral sequence satisfies Poincaré duality

Restriction to the Maximal Elementary Abelian Subgroup

Generated by [ G.5, G.4, G.1 * G.3 * G.5 ]

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
[ z, 0, z, z^3 + z^2*y + z*y^2 + z^2*x + z*x^2, z^3 + z^2*x + z*x^2, z^3 + z^2*y + z*y^2 + z^2*x + z*x^2, z^2*x^2 + x^4, z^3*y + y^4 + z^3*x + 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 + x, y, w + u ]

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 16 Number 10 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.2^$.1 = $.2 * $.4 Generated by [ G.1, G.2 ]

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

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

Maximal Subgroup H # 2

The Group H is Isomorphic to the Group of Order 16 Number 9 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.3, $.2^$.1 = $.2 * $.4 Generated by [ G.1 * G.3 * G.5, G.2 * G.3 * G.5 ]

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

The group H is abelian of type [ 4, 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, z + y, z*x + y*x + z*w, y*x + z*w + y*w, z*x, x^2, w^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 # 4

The Group H is Isomorphic to the Group of Order 16 Number 9 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.3, $.2^$.1 = $.2 * $.4 Generated by [ G.2, G.1 * G.3 * G.4 ]

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

The Group H is Isomorphic to the Group of Order 16 Number 9 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.3, $.2^$.1 = $.2 * $.4 Generated by [ G.3, G.1 * G.3 * G.5 ]

The images of the generators of the cohomology of G restricted to H are
[ y, 0, z + y, y^3 + y*x + z*w + y*w + y*v, y^3 + y*x + y*w, y^3 + y*x + y*w + z*v + y*v, y^2*w + w^2, y^2*x + y^2*w + y^2*v + v^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 # 6

The Group H is Isomorphic to the Group of Order 16 Number 10 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.2^$.1 = $.2 * $.4 Generated by [ G.2, G.3 ]

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

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

Maximal Subgroup H # 7

The Group H is Isomorphic to the Group of Order 16 Number 10 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.2^$.1 = $.2 * $.4 Generated by [ G.1 * G.2, G.3 ]

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

The annihilator of the Essential Cohomology has dimension 2 The essential cohomology is a free module over the polynomial subring Q of the cohomology ring of G generated by [ t, s ] .
The essential cohomology is generated as a module over Q by the elements [] [] [ y*x^2 ] [] [ z*x*u + x^2*u, y^2*u ]

CHECKS

paramflag = true
qregflagflag = true
cmflag = false
essflag = true
centflag = true

THE COMPUTATION IS COMPLETE