The hypercohomolgy spectral sequence has E2 term:

Restrictions to Maximal Elementary Abelian Subgroups

Subgroup E #1
Generated by [ G.1 * G.3 * G.5 * G.6, G.4, G.6, G.4 * G.5 ]

The images of the generators of the cohomology of G restricted to E are

z ,

0 ,

z ,

zy + y² ,

zy + zw ,

zx + x² ,

z² + y² + w²
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 .

Subgroup E #2
Generated by [ G.4, G.6, G.2 * G.3 * G.4 * G.5 * G.6, G.4 * G.5 ]

The images of the generators of the cohomology of G restricted to E are

0 ,

z ,

z ,

zy + y² ,

zy + zw ,

z² + zx + x² ,

y² + w²
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 .

The nilradical of the cohomology of G is generated by

z + y + x

It is nilpotent of degree 2.

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Restrictions to Maximal Subgroups

Maximal Subgroup H #1 Generated by [ G.1, G.4, G.5, G.6, G.2 * G.3 * G.6 ] .

The Group H is Isomorphic to the Group of Order 32 Number11 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^$.1 = $.2 * $.5 Generated by [ G.1 * G.2 * G.3 * G.4, G.2 * G.3 * G.4 * G.6, G.6 ]

of type Cyclic(2) x Group(16)#9 .

The images of the generators of the cohomology of G restricted to H are

z ,

z + y ,

z + y ,

w + v + u ,

zx + w ,

y² + zx + yx + x² + u ,

v
in the cohomology of H.

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

y + x .

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Maximal Subgroup H #2 Generated by [ G.3, G.2, G.4, G.5, G.6 ] .

The Group H is Isomorphic to the Group of Order 32 Number11 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^$.1 = $.2 * $.5 Generated by [ G.5, G.3 * G.5, G.2 * G.3 * G.4 * G.5 * G.6 ]

of type Cyclic(2) x Group(16)#9 .

The images of the generators of the cohomology of G restricted to H are

0 ,

y ,

z + y ,

yx + x² ,

w ,

y² + u ,

v
in the cohomology of H.

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

z .

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Maximal Subgroup H #3 Generated by [ G.3, G.1 * G.2, G.4, G.5, G.6 ] .

The group H is abelian of type [ 4, 4, 2 ]

The cohomology ring of H is a the quotient of a polynomial ring in the variables
[ z, y, x, w, v ] , in degrees [ 1, 1, 1, 2, 2 ] , by the ideal of relations

z² , y² .

The images of the generators of the cohomology of G restricted to H are

y ,

y ,

z ,

v ,

zx ,

zx + x² ,

w
in the cohomology of H.

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

z + y .

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Maximal Subgroup H #4 Generated by [ G.1 * G.2, G.4, G.5, G.6, G.2 * G.3 * G.6 ] .

The Group H is Isomorphic to the Group of Order 32 Number8 GrpPC of order 32 = 2^5 PC-Relations: $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.1 * G.3 * G.5 * G.6, G.2 * G.3 * G.4 * G.6, G.2 * G.3 * G.4, G.2 * G.3 * G.6 ]

of type Abelian(2,2) x Dihedral(8) .

The images of the generators of the cohomology of G restricted to H are

x ,

z + y + w ,

z + y + x + w ,

v ,

zw + yw + xw + w² ,

z² + zy + y² + zx + yw + xw + w² ,

x² + w²
in the cohomology of H.

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

z + y + x .

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Maximal Subgroup H #5 Generated by [ G.2, G.4, G.5, G.6, G.1 * G.3 * G.6 ] .

The Group H is Isomorphic to the Group of Order 32 Number11 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^$.1 = $.2 * $.5 Generated by [ G.6, G.1 * G.2 * G.3 * G.4 * G.5, G.1 * G.3 * G.4 * G.6 ]

of type Cyclic(2) x Group(16)#9 .

The images of the generators of the cohomology of G restricted to H are

z + y ,

z ,

z + y ,

w + v + u ,

y² + zx + w ,

zx + yx + x² + u ,

v
in the cohomology of H.

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

z + x .

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Maximal Subgroup H #6 Generated by [ G.1, G.2, G.4, G.5, G.6 ] .

The Group H is Isomorphic to the Group of Order 32 Number12 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.5, $.3^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5 Generated by [ G.1 * G.2 * G.5, G.2 * G.6, G.1 * G.2 * G.5 * G.6 ]

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

The images of the generators of the cohomology of G restricted to H are

y + x ,

z + y + x ,

0 ,

zy + v ,

zy ,

zy + y² + zx + w ,

zy + zx + w
in the cohomology of H.

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

x .

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Maximal Subgroup H #7 Generated by [ G.1, G.3, G.4, G.5, G.6 ] .

The Group H is Isomorphic to the Group of Order 32 Number11 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^$.1 = $.2 * $.5 Generated by [ G.1 * G.3 * G.4 * G.5 * G.6, G.5, G.3 * G.5 ]

of type Cyclic(2) x Group(16)#9 .

The images of the generators of the cohomology of G restricted to H are

y ,

0 ,

z + y ,

yx + x² ,

y² + w ,

u ,

v
in the cohomology of H.

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

y .

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The essential cohomology of G is zero.

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Inflations from Maximal Quotient Groups

Maximal Quotient Group Q #1.

The kernel of the quotient is generated by G.4 * G.6 .

The Group Q is Isomorphic to the Group of Order 32 Number34 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.3^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.2 = $.3 * $.5 .

The generators of G have images [ pcy.3, pcy.2, pcy.1 * pcy.2 * pcy.4 ] in the quotient group.

The images of the generators of the cohomology of Q inflated to G are

z ,

z + y ,

x ,

z² + w ,

z² + y² + v + u + t
in the cohomology of G.

The kernel of the inflation to G of the cohomology of Q is generated by

y² + x² .

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Maximal Quotient Group Q #2.

The kernel of the quotient is generated by G.5 .

The Group Q is Isomorphic to the Group of Order 32 Number11 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^$.1 = $.2 * $.5

of type Cyclic(2) x Group(16)#9 .

The generators of G have images [ pcy.1 * pcy.4, pcy.1 * pcy.2, pcy.1 * pcy.2 * pcy.4 ] in the quotient group.

The images of the generators of the cohomology of Q inflated to G are

z + y + x ,

z + y ,

z + x ,

z² + y² + yx + v ,

t ,

yx + u
in the cohomology of G.

The kernel of the inflation to G of the cohomology of Q is generated by

yx + x² .

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Maximal Quotient Group Q #3.

The kernel of the quotient is generated by G.4 * G.5 * G.6 .

The Group Q is Isomorphic to the Group of Order 32 Number39 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4 * $.5, $.2^2 = $.5, $.3^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.2 = $.3 * $.5 .

The generators of G have images [ pcy.1 * pcy.3 * pcy.4 * pcy.5, pcy.2, pcy.2 * pcy.3 ] in the quotient group.

The images of the generators of the cohomology of Q inflated to G are

x ,

z + y ,

z + x ,

v + u + t ,

y²x + xw + xu ,

z²w + y²w + zxu + yxu + w² + u²
in the cohomology of G.

The kernel of the inflation to G of the cohomology of Q is generated by

yx + x² .

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Maximal Quotient Group Q #4.

The kernel of the quotient is generated by G.5 * G.6 .

The Group Q is Isomorphic to the Group of Order 32 Number14 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4 * $.5, $.2^2 = $.4, $.2^$.1 = $.2 * $.5, $.3^$.2 = $.3 * $.5

of type Cyclic(4) x Dihedral(8) .

The generators of G have images [ pcy.2 * pcy.4 * pcy.5, pcy.2 * pcy.3 * pcy.5, pcy.1 ] in the quotient group.

The images of the generators of the cohomology of Q inflated to G are

z ,

y + x ,

y ,

z² + y² + t ,

w + v + u
in the cohomology of G.

The kernel of the inflation to G of the cohomology of Q is generated by

zx .

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Maximal Quotient Group Q #5.

The kernel of the quotient is generated by G.4 * G.5 .

The Group Q is Isomorphic to the Group of Order 32 Number36 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4 * $.5, $.3^2 = $.4, $.2^$.1 = $.2 * $.4, $.3^$.2 = $.3 * $.5 .

The generators of G have images [ pcy.3, pcy.1 * pcy.2 * pcy.4, pcy.2 * pcy.4 ] in the quotient group.

The images of the generators of the cohomology of Q inflated to G are

y ,

z + y ,

x ,

z² + y² + yx + u ,

z² + yx + w + v + t
in the cohomology of G.

The kernel of the inflation to G of the cohomology of Q is generated by

yx + x² .

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Maximal Quotient Group Q #6.

The kernel of the quotient is generated by G.6 .

The Group Q is Isomorphic to the Group of Order 32 Number14 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4 * $.5, $.2^2 = $.4, $.2^$.1 = $.2 * $.5, $.3^$.2 = $.3 * $.5

of type Cyclic(4) x Dihedral(8) .

The generators of G have images [ pcy.1 * pcy.3 * pcy.4 * pcy.5, pcy.1 * pcy.4, pcy.2 * pcy.3 * pcy.5 ] in the quotient group.

The images of the generators of the cohomology of Q inflated to G are

y + x ,

z ,

z + x ,

z² + t ,

z² + w + t
in the cohomology of G.

The kernel of the inflation to G of the cohomology of Q is generated by

zx .

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Maximal Quotient Group Q #7.

The kernel of the quotient is generated by G.4 .

The Group Q is Isomorphic to the Group of Order 32 Number36 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4 * $.5, $.3^2 = $.4, $.2^$.1 = $.2 * $.4, $.3^$.2 = $.3 * $.5 .

The generators of G have images [ pcy.1 * pcy.3 * pcy.4, pcy.2, pcy.1 * pcy.2 * pcy.4 ] in the quotient group.

The images of the generators of the cohomology of Q inflated to G are

z + x ,

z + y ,

x ,

z² + y² + yx + u ,

y² + yx + w + u
in the cohomology of G.

The kernel of the inflation to G of the cohomology of Q is generated by

yx + x² .

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The depth-essential cohomology of G

The depth-essential cohomology of G is the intersection of the restrictions to the centralizers of the maximal elementary abelian p-subgroups of rank d+1 where d is the depth of the cohomology ring. For this group the depth is 3.

There are 2 conjugacy classes of subgroups which are centralizers of elementary abelian subgroups of rank 4. They are represented by the subgroups generated by

[ G.1 * G.3 * G.5 * G.6, G.1 * G.3 * G.4 * G.5 * G.6, G.1 * G.3 * G.5, G.1 * G.3 * G.4 * G.6 ]

[ G.2 * G.3 * G.5, G.2 * G.3 * G.4 * G.6, G.2 * G.3 * G.4, G.2 * G.3 * G.4 * G.5 * G.6 ] .

The depth-essential cohomology of G is generated as an ideal by

z + y + x .

The annihilator of the depth-essential cohomology has dimension 3 .

The depth-essential cohomology is a free module over the polynomial subring Q of the cohomology ring of G generated by w , u , t .
The depth-essential cohomology is generated as a module over Q by the elements

[ z+ y+ x ] [ y²+ yx ]

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Action of Automorphisms

The groups of outer automorphisms of G has order 256, and is generated by 8 elements.

Automorphism #1

G.1 * G.4 * G.6
G.2 * G.4 * G.6
G.3
G.4
G.5
G.6 .

The images of the generators of the cohomology of G under the map induced by the automorphism are

z ,

y ,

x ,

w ,

z² + y² + v ,

u ,

z² + y² + t .

Automorphism #2

G.1
G.2
G.3 * G.5
G.4
G.5
G.6 .

The images of the generators of the cohomology of G under the map induced by the automorphism are

z ,

y ,

x ,

w ,

v ,

u ,

t .

Automorphism #3

G.1
G.2
G.3 * G.4 * G.6
G.4
G.5
G.6 .

The images of the generators of the cohomology of G under the map induced by the automorphism are

z ,

y ,

x ,

w ,

z² + y² + v ,

u ,

z² + y² + t .

Automorphism #4

G.1 * G.6
G.2
G.3
G.4
G.5
G.6 .

The images of the generators of the cohomology of G under the map induced by the automorphism are

z ,

y ,

x ,

w ,

y² + yx + v ,

y² + yx + u ,

t .

Automorphism #5

G.1 * G.5
G.2
G.3
G.4
G.5
G.6 .

The images of the generators of the cohomology of G under the map induced by the automorphism are

z ,

y ,

x ,

w ,

v ,

u ,

t .

Automorphism #6

G.1 * G.4 * G.6
G.2
G.3
G.4
G.5
G.6 .

The images of the generators of the cohomology of G under the map induced by the automorphism are

z ,

y ,

x ,

w ,

z² + y² + yx + v ,

y² + yx + u ,

z² + t .

Automorphism #7

G.2 * G.5
G.1
G.3
G.4
G.5
G.6 .

The images of the generators of the cohomology of G under the map induced by the automorphism are

y ,

z ,

x ,

w ,

z² + y² + v ,

z² + y² + u ,

t .

Automorphism #8

G.1
G.2
G.1 * G.2 * G.3 * G.4 * G.5 * G.6
G.4 * G.5
G.5
G.5 * G.6 .

The images of the generators of the cohomology of G under the map induced by the automorphism are

z + x ,

y + x ,

x ,

z² + y² + w + v + u + t ,

z² + y² + v ,

z² + y² + u ,

t .

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CHECKS

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

THE COMPUTATION IS COMPLETE