GROUP OF ORDER 32 # 24

GROUP # 24

Cyclic(2) x Semidihedral(16)

The MAGMA library number for G is 40

GrpPC : G of order 32 = 2^5 PC-Relations: G.1^2 = G.5, G.3^2 = G.5, G.4^2 = G.5, G.2^G.1 = G.2 * G.4, G.3^G.1 = G.3 * G.5, G.3^G.2 = G.3 * G.5, G.4^G.1 = G.4 * G.5, G.4^G.2 = G.4 * 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, 2, 1 ]
The orders of the terms of the upper central series are [ 1, 4, 8, 32 ]
The order of the Frattini subgroup is 4
The exponent of G is 8
G has one conjugate class of maximal elementary abelian subgroups. Any element of the class has order 8 . Its centralizer has order 8 and its normalizer has order 16

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, 3, 4 ] by the ideal generated by the relations
[ z*y, z^3, z*w, y^2*v + w^2 ]

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

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

The hypercohomolgy spectral sequence has E2 term:

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

Restriction to the Maximal Elementary Abelian Subgroup

Generated by [ G.2 * G.3, G.3 * 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, z, z + y + x, z^2*y + z*y^2, z^2*y^2 + y^4 ] 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 13 GrpPC of order 16 = 2^4 PC-Relations: $.2^2 = $.3, $.3^2 = $.4, $.2^$.1 = $.2 * $.3, $.3^$.1 = $.3 * $.4 Generated by [ G.1 * G.2, G.2 * G.3 * G.5 ]

of type Semidihedral(16)

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

The Group H is Isomorphic to the Group of Order 16 Number 13 GrpPC of order 16 = 2^4 PC-Relations: $.2^2 = $.3, $.3^2 = $.4, $.2^$.1 = $.2 * $.3, $.3^$.1 = $.3 * $.4 Generated by [ G.1 * G.2, G.2 ]

of type Semidihedral(16)

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

The group H is abelian of type [ 8, 2 ]

The cohomology ring of H is a the quotient of a polynomial ring in the variables [ z, y, x ] in degrees [ 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, z*x, 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 Isomorphic to the Group of Order 16 Number 13 GrpPC of order 16 = 2^4 PC-Relations: $.2^2 = $.3, $.3^2 = $.4, $.2^$.1 = $.2 * $.3, $.3^$.1 = $.3 * $.4 Generated by [ G.1 * G.2 * G.3, G.2 ]

of type Semidihedral(16)

The images of the generators of the cohomology of G restricted to H are
[ y, z + y, y, x, z*x + 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 Isomorphic to the Group of Order 16 Number 7 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.2^$.1 = $.2 * $.4 Generated by [ G.1, G.3 * G.4 * G.5, G.4 ]

of type Cyclic(2) x Quaternion(8)

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

Maximal Subgroup H # 6

The Group H is Isomorphic to the Group of Order 16 Number 6 GrpPC of order 16 = 2^4 PC-Relations: $.3^2 = $.4, $.2^$.1 = $.2 * $.4, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.4 Generated by [ G.2, G.2 * G.3 * G.5, G.4 ]

of type Cyclic(2) x Dihedral(8)

The images of the generators of the cohomology of G restricted to H are
[ 0, z + y, z, z*y^2 + z^2*x + y*x^2 + x^3 + z*w + y*w, z^2*y^2 + w^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 13 GrpPC of order 16 = 2^4 PC-Relations: $.2^2 = $.3, $.3^2 = $.4, $.2^$.1 = $.2 * $.3, $.3^$.1 = $.3 * $.4 Generated by [ G.1 * G.2 * G.3, G.2 * G.3 ]

of type Semidihedral(16)

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