GROUP OF ORDER 32 # 11

GROUP # 11

Cyclic(2) x Group(16)#9

The MAGMA library number for G is 22

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

The center of G is abelian of type [ 2, 2, 2 ]
The orders of the terms of the lower central series are [ 32, 2, 1 ]
The orders of the terms of the upper central series are [ 1, 8, 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 16. Its centralizer has order 16

The cohomology ring of G is the quotient of a polynomial ring in the variables [ z, y, x, w, v, u ] in degrees [ 1, 1, 1, 2, 2, 2 ] by the ideal generated by the relations
[ z^2, z*x, z*v, x^2*w + v^2 ]

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

The Krull dimension of the cohomology ring is 4
The longest regular sequence consists of the generators [ y^2, w, u ]
A homogeneous set of parameters is the set [ y^2, w, u, x^2 ] of degrees [ 2, 2, 2, 2 ]

The hypercohomolgy spectral sequence has E2 term:

ROW (0) [ z, y, x ] [ z*y, y*x, v ] [ x*v, y*v ] [ y*x*v ]
ROW (1) [ z ] [ z*y ]
The spectral sequence satisfies Poincaré duality

Restriction to the Maximal Elementary Abelian Subgroup

Generated by [ G.2 * G.3, G.2 * G.4 * G.5, G.2, G.2 * G.3 * G.5 ]

The cohomology ring of E is a polynomial ring in the variables [ z, y, x, w ]

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

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

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

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

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.4 ]

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

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

Maximal Subgroup H # 4

The group H is abelian of type [ 4, 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, w, 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 + x ]

Maximal Subgroup H # 5

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

The cohomology ring of H is a polynomial ring with variables [ z, y, x, w ] in degrees [ 1, 1, 1, 1 ]

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

The group H is abelian of type [ 4, 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, 0, w, z*y, z*y + y^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 # 7

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.2 * G.4 ]

The images of the generators of the cohomology of G restricted to H are
[ z, y, y, w, x, 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 = false
essflag = true
centflag = true

THE COMPUTATION IS COMPLETE