The hypercohomolgy spectral sequence has E2 term:

Restrictions to Maximal Elementary Abelian Subgroups

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

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

0 ,

z ,

z ,

z ,

zy² + zx² ,

z⁴ + z²x² + x⁴ ,

z⁴ + y⁴ + z²x²
in the cohomology of E.

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

z ,

y + w ,

x + w .

Subgroup E #2
Generated by [ G.6, G.5, G.1 * G.2 * G.4 * G.6 ]

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

z ,

z ,

0 ,

z ,

z³ + z²y + zy² ,

z²x² + x⁴ ,

z²y² + y⁴
in the cohomology of E.

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

z + w ,

y + w ,

x .

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

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

z ,

0 ,

0 ,

0 ,

z²y + zy² ,

z²x² + x⁴ ,

z²y² + y⁴
in the cohomology of E.

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

y ,

x ,

w .

The nilradical of the cohomology of G is generated by

y + w , zw + xw + w² .

It is nilpotent of degree 4.

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Restrictions to Maximal Subgroups

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

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

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

y + x ,

x ,

0 ,

z + y + x ,

x³ + zw + xw + zv + xv ,

y²w + x²w + w² ,

zx³ + zxw + zxv + w² + v²
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.2, G.6, G.4, G.5, G.1 * G.3 * G.6 ] .

The Group H is Isomorphic to the Group of Order 32 Number40 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4 * $.5, $.2^2 = $.4 * $.5, $.3^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.2 * G.4 * G.6, G.1 * G.2 * G.3 * G.4 * G.5, G.1 * G.2 * G.3 * G.5 * G.6 ] .

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

z + y ,

z + y + x ,

z + y ,

y + x ,

zyx + y²x + w ,

yw + t ,

u
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 #3 Generated by [ G.1 * G.4, G.2, G.3, G.6, G.5 ] .

The Group H is Isomorphic to the Group of Order 32 Number41 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.4 * $.5, $.2^$.1 = $.2 * $.4 * $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.1 * G.2 * G.3 * G.4 * G.6, G.1 * G.2 * G.4 * G.5 * G.6, G.2 * G.3 ] .

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

z + x ,

z + y + x ,

z + y ,

z + x ,

x³ + w + v + u ,

zw + t + s ,

zw + xv + s
in the cohomology of H.

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

z + w .

---

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

The Group H is Isomorphic to the Group of Order 32 Number38 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4 * $.5, $.2^2 = $.5, $.2^$.1 = $.2 * $.4 * $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.3 * G.4, G.1 * G.2 * G.4 * G.5 * G.6, G.1 * G.3 * G.6 ] .

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

y + x ,

x ,

z + y ,

z + x ,

y³ + zyx + x³ + zw + yw + v + u ,

zxw + yxw + w² + zv + t ,

zy²x + yxw + zv + t
in the cohomology of H.

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

z + x + w .

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

The Group H is Isomorphic to the Group of Order 32 Number38 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4 * $.5, $.2^2 = $.5, $.2^$.1 = $.2 * $.4 * $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.1 * G.2 * G.3 * G.4 * G.5 * G.6, G.1 * G.5 * G.6, G.4 * G.5 * G.6 ] .

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

y + x ,

y ,

y ,

z + y ,

y³ + zw + xw + v + u ,

zyw + zxw + t ,

zy²x + yx³ + zxw + w² + t
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.2, G.3, G.1, G.6, G.5 ] .

The Group H is Isomorphic to the Group of Order 32 Number16 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4 * $.5, $.2^2 = $.4, $.3^2 = $.5, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5 Generated by [ G.2, G.1 * G.3 * G.6, G.2 * G.3 * G.6 ] .

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

y ,

z + x ,

z + y ,

0 ,

z³ + yx² + zw + yw + xw + v ,

z²w + w² + u ,

z²w + zyw + zxw + w² + zv + xv
in the cohomology of H.

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

w .

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Maximal Subgroup H #7 Generated by [ G.6, G.4, G.5, G.1 * G.2, G.2 * G.3 * G.6 ] .

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

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

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

z ,

y + x ,

z + y + x ,

z + x ,

y³ + zyx + zx² + x³ + zw + xw + yv ,

x⁴ + y²w + x²w + v² ,

z²yx + zxw + x²w + zyv + y²v + 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 #8 Generated by [ G.2, G.3 * G.4, G.1, G.6, 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.3 * G.4, G.2 * G.5, G.1 * G.5 ]

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

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

x ,

z ,

y ,

y ,

y³ + zw + xw + zv + yv + xv ,

zy³ + z²yx + y²w + x²w + v² ,

y²w + zxw + zyv + y²v + zxv + w² + v²
in the cohomology of H.

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

x + w .

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Maximal Subgroup H #9 Generated by [ G.3, G.2 * G.4, G.1, G.6, G.5 ] .

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

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

y ,

z + y ,

y + x ,

z + y ,

y³ + zyx + zx² + yw + xw + xv ,

zy³ + y³x + z²x² + w² ,

zy³ + v²
in the cohomology of H.

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

y + w .

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Maximal Subgroup H #10 Generated by [ G.2 * G.4, G.3 * G.4, G.1, G.6, G.5 ] .

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

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

z ,

x ,

z + y ,

z + y + x ,

z³ + zyx + yw + v ,

z⁴ + w² ,

z⁴ + z²w + y²w + w² + zv + u
in the cohomology of H.

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

y + x + w .

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Maximal Subgroup H #11 Generated by [ G.3, G.6, G.4, G.5, G.1 * G.2 ] .

The Group H is Isomorphic to the Group of Order 32 Number16 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4 * $.5, $.2^2 = $.4, $.3^2 = $.5, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5 Generated by [ G.3, G.1 * G.2 * G.3, G.4 ] .

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

z ,

z ,

z + y ,

x ,

yx² + zw + yw + xw + v ,

u ,

z⁴ + yx³ + z²w + zyw + zxw + w² + zv + xv
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 #12 Generated by [ G.1 * G.4, G.3, G.2 * G.4, G.6, G.5 ] .

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 ,

x ,

z + y + x ,

y + x ,

zyx + yx² + x³ + zw + yw + xw + zv + yv + xv ,

x⁴ + x²v + v² ,

zyx² + zyw + yxw + zyv + yxv + x²v + w²
in the cohomology of H.

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

z + y + w .

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Maximal Subgroup H #13 Generated by [ G.1 * G.4, G.2 * G.4, G.3 * G.4, G.6, G.5 ] .

The Group H is Isomorphic to the Group of Order 32 Number15 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.5, $.4^2 = $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.1 * G.2 * G.6, G.2 * G.4, G.3 * G.4 ]

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

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

z ,

z + y ,

x ,

y + x ,

xw ,

x²w + w² ,

zyx² + zxw + v
in the cohomology of H.

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

z + y + x + w .

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Maximal Subgroup H #14 Generated by [ G.3, G.1, G.6, G.4, G.5 ] .

The Group H is Isomorphic to the Group of Order 32 Number41 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.4 * $.5, $.2^$.1 = $.2 * $.4 * $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.1, G.3 * G.4 * G.6, G.4 ] .

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

x ,

0 ,

z ,

z + y ,

z²x + u ,

t ,

xv + t + s
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 #15 Generated by [ G.2, G.3, G.6, G.4, G.5 ] .

The Group H is Isomorphic to the Group of Order 32 Number37 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4 * $.5, $.3^2 = $.4, $.2^$.1 = $.2 * $.4 * $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.2, G.3 * G.4 * G.5 * G.6, G.2 * G.3 ] .

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

0 ,

z + y ,

z + x ,

x ,

z²x + zyx + zw + yw + xw + v ,

w² ,

y²w + zxw + yxw + w² + u
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.5 * G.6 .

The Group Q is Isomorphic to the Group of Order 32 Number10 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.2 = $.3 * $.5, $.4^$.2 = $.4 * $.5

of type Cyclic(2) x AlmostExtraSpecial(16) .

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

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

y + w ,

x + w ,

w ,

z + y ,

zy³ + zyxw + yx²w + yv + wv + u + t
in the cohomology of G.

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

y² + x² + xw ,

x²w + xw² .

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Maximal Quotient Group Q #2.

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

The Group Q is Isomorphic to the Group of Order 32 Number10 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.2 = $.3 * $.5, $.4^$.2 = $.4 * $.5

of type Cyclic(2) x AlmostExtraSpecial(16) .

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

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

x + w ,

w ,

y + w ,

z + w ,

yx³ + zyxw + yx²w + yv + xv + wv + t
in the cohomology of G.

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

zy + y² + zx + yx + x² + zw + yw ,

yx² + x³ + x²w .

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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 Number10 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.2 = $.3 * $.5, $.4^$.2 = $.4 * $.5

of type Cyclic(2) x AlmostExtraSpecial(16) .

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

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

y + w ,

x ,

x + w ,

z + y ,

u
in the cohomology of G.

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

y² + yw + xw ,

x²w .

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

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

[ G.4 * G.5, G.1 * G.2 * G.4 * G.6, G.1 * G.2 * G.4 * G.5 ]

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

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

zx + yx + xw .

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 u , t .
The depth-essential cohomology is generated as a module over Q by the elements

[] [ zx+ yx+ xw ] [ zxw+ yxw+ xw², zyx+ zxw ] [ zyxw+ yxw²+ xw³ ]

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Action of Automorphisms

The groups of outer automorphisms of G has order 64, and is generated by 6 elements.

Automorphism #1

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

y²w + w³ + v ,

u ,

zy³ + zyw² + t .

Automorphism #2

G.1 * G.6
G.2 * 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 ,

xw² + v ,

u ,

zy³ + zyxw + yx²w + z²w² + t .

Automorphism #3

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

y²w + w³ + v ,

u ,

zy³ + zyw² + t .

Automorphism #4

G.1 * G.6
G.2
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 ,

y²w + w³ + v ,

u ,

zy³ + zyw² + t .

Automorphism #5

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

w ,

z²y + z²w + y²w + w³ + v ,

u ,

zy³ + zyw² + t .

Automorphism #6

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

z + w ,

x ,

z + y ,

z³ + v ,

u ,

z³y + y³w + yx²w + t .

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CHECKS

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

THE COMPUTATION IS COMPLETE