GROUP OF ORDER 32 # 12

GROUP # 12

Cyclic(2) x Group(16)#10

The MAGMA library number for G is 23

GrpPC : G of order 32 = 2^5 PC-Relations: G.1^2 = G.3, G.2^2 = G.3, G.2^G.1 = G.2 * 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 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 + y^2, z*y ]

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

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

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

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.4, G.1 * G.2 * G.3 * G.4 * G.5 ]

The images of the generators of the cohomology of G restricted to H are
[ z + y, y, z + y, w, 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 # 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.1 * G.2 * G.3 * G.4 * G.5, G.1 ]

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

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

The annihilator of the Essential Cohomology has dimension 3

CHECKS

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

THE COMPUTATION IS COMPLETE