The hypercohomolgy spectral sequence has E2 term:

Restrictions to Maximal Elementary Abelian Subgroups

Subgroup E #1
Generated by [ G.6, G.5, G.1 * G.2 * G.3 * G.5 ]

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

z ,

z ,

0 ,

zy + y² ,

z²y + zx² ,

z²y + zx² ,

z²x² + x⁴
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 ,

v + u .

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

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

0 ,

0 ,

z ,

zy + y² ,

0 ,

0 ,

z²x² + x⁴
in the cohomology of E.

The kernel of the restriction to E (Minimal Prime) of the cohomology of G is generated by

z ,

y ,

v ,

u .

The nilradical of the cohomology of G is generated by

z + y , v + u .

It is nilpotent of degree 2.

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Restrictions to Maximal Subgroups

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

The group H is abelian of type [ 8, 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

z² .

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

z + y ,

y ,

0 ,

zx + yx + x² ,

y²x + zw + yw ,

zyx + y²x + yw ,

y²w + w²
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 #2 Generated by [ G.1 * G.2, G.4, G.6, G.3, G.5 ] .

The Group H 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 Generated by [ G.1 * G.2 * G.4, G.4 * G.5 * G.6, G.3 * G.4 * G.5 ]

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

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

y ,

y ,

z + y + x ,

w + v ,

y³ + yv ,

y³ + zx² + yv ,

y²w + x²w + v²
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 #3 Generated by [ G.1 * G.4 * G.5, G.2, G.6, G.3, G.5 ] .

The Group H is Isomorphic to the Group of Order 32 Number19 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.4^2 = $.5, $.2^$.1 = $.2 * $.5 Generated by [ G.1 * G.4 * G.5, G.1 * G.2 * G.3 * G.4 ] .

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

z + y ,

z ,

z + y ,

zy + x ,

zx + yx + zw + yw ,

zx + zw ,

zyx + zyw + x² + w²
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 #4 Generated by [ G.4, G.1, G.6, G.3, G.5 ] .

The Group H is Isomorphic to the Group of Order 32 Number20 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.2, $.2^2 = $.4, $.3^$.1 = $.3 * $.5 Generated by [ G.4, G.1 ] .

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

z ,

0 ,

y ,

v ,

zw ,

yx ,

y²w + w²
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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Maximal Subgroup H #5 Generated by [ G.2 * G.4 * G.6, G.1, G.6, G.3, G.5 ] .

The Group H is Isomorphic to the Group of Order 32 Number21 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.4 * $.5, $.3^2 = $.4, $.2^$.1 = $.2 * $.5 Generated by [ G.1 * G.2 * G.3 * G.4 * G.6, G.2 * G.4 * G.6 ] .

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

y ,

z + y ,

z + y ,

y² + x + w ,

yw ,

zw + yw ,

y²w + w²
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 #6 Generated by [ G.1 * G.2, G.2 * G.4 * G.6, G.6, G.3, G.5 ] .

The Group H is Isomorphic to the Group of Order 32 Number20 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.2, $.2^2 = $.4, $.3^$.1 = $.3 * $.5 Generated by [ G.2 * G.4 * G.6, G.1 * G.2 * G.3 ] .

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

y ,

z + y ,

z ,

v ,

y³ + yw + yv ,

y³ + zw + yw + zv + yv ,

y²w + w² + v²
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 #7 Generated by [ G.4, G.2, G.6, G.3, G.5 ] .

The Group H is Isomorphic to the Group of Order 32 Number13 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5 Generated by [ G.2 * G.3 * G.4 * G.5, G.2, G.4 * G.6 ]

of type Cyclic(2) x Group(16)#11 .

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

0 ,

z + y ,

z + x ,

zx ,

z²y + zy² + z²x + zyx ,

w ,

zy³ + zyx² + 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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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.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.2 * pcy.4 * pcy.5, pcy.1 * pcy.3 * pcy.4 * pcy.5, pcy.1 * pcy.4 ] in the quotient group.

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

z + y + x ,

x ,

y ,

yx ,

yx + w
in the cohomology of G.

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

yx + w ,

yw .

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Maximal Quotient Group Q #2.

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

The Group Q is Isomorphic to the Group of Order 32 Number13 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5

of type Cyclic(2) x Group(16)#11 .

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

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

z + x ,

y + x ,

z ,

zw + yw + v + u ,

x⁴ + z²w + yxw + w² + zv + t
in the cohomology of G.

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

zx + x² .

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Maximal Quotient Group Q #3.

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

The Group Q is Isomorphic to the Group of Order 32 Number17 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5 .

The generators of G have images [ pcy.3 * pcy.4, pcy.1 * pcy.3 * 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 ,

z ,

z + y + x ,

z²y² + t
in the cohomology of G.

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

zy + y² + yx .

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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 2.

There are 2 conjugacy classes of subgroups which are centralizers of elementary abelian subgroups of rank 3. They are represented by the subgroups generated by

[ G.1 * G.2, G.1 * G.6, G.1 * G.5 ]

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

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

yx .

The annihilator of the depth-essential cohomology has dimension 2 .

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

[] [ yx ]

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Action of Automorphisms

The groups of outer automorphisms of G has order 32, and is generated by 5 elements.

Automorphism #1

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

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 ,

zy² + v ,

y³ + u ,

t .

Automorphism #3

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

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

zy + y² + yx + w ,

zy² + y³ + v ,

zy² + y³ + u ,

t .

Automorphism #5

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

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CHECKS

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

THE COMPUTATION IS COMPLETE