The hypercohomolgy spectral sequence has E2 term:

Restrictions to Maximal Elementary Abelian Subgroups

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

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

z ,

y + x + w ,

y + x + w ,

z ,

z² + zw + w² ,

z³ + z²x + zx² ,

z³y + x⁴ + z²w²
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 + x .

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

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

0 ,

y + x + w ,

z + y + x + w ,

0 ,

zw + w² ,

z²x + 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 ,

w .

The nilradical of the cohomology of G is generated by

z + w

It is nilpotent of degree 3.

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Restrictions to Maximal Subgroups

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

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

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

y + x ,

x ,

0 ,

z + x ,

yx + w ,

z³ + zyx + zw + yw + v ,

z⁴ + yx³ + z²w + zxw + xv + u
in the cohomology of H.

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

x .

---

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

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

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 + x ,

x ,

y ,

y² + w + v + u ,

y³ + zv + yu ,

y³x + y²w + w² + u²
in the cohomology of H.

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

z + y + x + w .

---

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

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

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

z + y + x ,

x ,

z ,

x ,

z² + yx + w ,

z³ + zyx + yw + v ,

z⁴ + yx³ + z²w + y²w + w² + zv + xv + u
in the cohomology of H.

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

y + w .

---

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

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

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

x ,

z ,

z + y + x ,

z + y + x ,

z² + yx + w ,

yx² + zw + yw + v ,

zxw + zv + xv + u
in the cohomology of H.

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

x + w .

---

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

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

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

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

z ,

z + x ,

y + x ,

0 ,

w + v + u ,

zv + yu ,

u²
in the cohomology of H.

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

w .

---

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

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

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

y + x ,

0 ,

x ,

z + x ,

x² + w ,

z³ + yx² + zw + yw + v ,

z⁴ + yx³ + z²w + zxw + xv + u
in the cohomology of H.

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

y .

---

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

The Group H is Isomorphic to the Group of Order 32 Number9 GrpPC of order 32 = 2^5 PC-Relations: $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.4^$.2 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.3 * G.4 * G.6, G.1 * G.5, G.1, G.2 * G.3 ]

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

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

y + x ,

z ,

z + w ,

w ,

yx + yw + w² ,

y³ + y²x + y²w + yw² + xw² ,

y²x² + y³w + y²xw + yxw² + v
in the cohomology of H.

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

y + x + w .

---

Maximal Subgroup H #8 Generated by [ G.2, G.3, G.6, G.4, G.5 ] .

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

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

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

0 ,

z + x ,

y + x ,

z ,

u ,

yw + yv + zu ,

w² + v²
in the cohomology of H.

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

z .

---

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

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

y + x ,

y + x ,

y ,

z + x ,

x² + w ,

z³ + yx² + zw + yw + v ,

z²w + zxw + xv + u
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 #10 Generated by [ G.6, G.4, G.5, G.2 * G.3, G.1 ] .

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

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

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

y ,

z + y + x ,

z + y + x ,

z + y ,

y² + w + v + u ,

y³ + yw + yv + zu ,

y⁴ + y³x + y²u + w² + v²
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 #11 Generated by [ G.1 * G.4 * G.6, G.2, G.3, G.6, G.5 ] .

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

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

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

x ,

w ,

z + y ,

x ,

zy + zx + x² + zw ,

z²y + zy² + zyx + yx² + x³ + z²w + zxw + x²w + zw² + zv + yv + xv + wv ,

z²y² + zyx² + zx³ + yx³ + zx²w + z²w² + x²v + v²
in the cohomology of H.

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

z + w .

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Maximal Subgroup H #12 Generated by [ G.1 * G.4 * G.6, G.1 * G.2 * G.5, G.3, 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.1 * G.3 * G.4, G.2 * G.4, G.2 * G.3 * G.4 * G.5 ] .

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

x ,

z + y ,

z + x ,

z + y + x ,

z² + x² + w ,

zyx + zw + yw + v ,

yx³ + zxw + zv + xv + u
in the cohomology of H.

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

z + y + w .

---

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

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

z + x ,

y ,

z + x ,

y + x ,

yx + w ,

yx² + yw + v ,

yx³ + z²w + y²w + w² + xv + u
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 #14 Generated by [ G.1 * G.2 * G.5, G.6, G.4, G.5, G.1 * G.3 * 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 ,

y + x ,

x ,

z ,

w + v ,

zv + yv ,

zyv + 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 #15 Generated by [ G.1 * G.4 * G.6, G.2, G.6, G.5, G.1 * G.3 * 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.3 * G.4 * G.6, G.1 * G.2 * G.3, G.1 * G.2 * G.4 * G.5 * G.6 ] .

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

y + x ,

z + y + x ,

z + y ,

z + x ,

yx + w ,

z³ + zyx + zw + yw + v ,

z⁴ + yx³ + z²w + zxw + xv + u
in the cohomology of H.

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

z + x + w .

---

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 .

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

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

z ,

w ,

x ,

z + y + w ,

z⁴ + z³y + z²v + zyv + zxv + v² + wu + t
in the cohomology of G.

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

zy + y² + zx + zw .

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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.3, pcy.4, pcy.2 * pcy.3 * pcy.4 ] in the quotient group.

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

w ,

z ,

z + x ,

z + y ,

z³y + z²v + wu + t
in the cohomology of G.

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

zy + y² + zx + zw .

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Maximal Quotient Group Q #3.

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

The Group Q 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

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

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

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

x ,

w ,

z + w ,

y ,

z² + v
in the cohomology of G.

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

zy + yx + x² + yw ,

x³ .

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Maximal Quotient Group Q #4.

The kernel of the quotient is generated by G.2 * G.3 * G.5 .

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

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

y + x + w ,

z + y + x ,

z ,

zy + y² + yx + x² + v ,

z³ + zy² + y²x + zx² + yx² + zxw + yxw + zv + wv + u ,

z⁴ + y³x + y²x² + yx³ + z²v + yu + xu + t
in the cohomology of G.

The kernel of the inflation is zero.

Maximal Quotient Group Q #5.

The kernel of the quotient is generated by G.2 * G.3 .

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

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

y + x + w ,

z + y + x ,

y + x ,

y² + x² + v ,

z³ + zv + wv + u ,

z⁴ + z³y + z²v + yu + xu + t
in the cohomology of G.

The kernel of the inflation is zero.

Maximal Quotient Group Q #6.

The kernel of the quotient is generated by G.2 * G.3 * 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.1 * pcy.2 * pcy.5, pcy.1 * pcy.2, pcy.2 * pcy.3 * pcy.4 * pcy.5 ] in the quotient group.

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

y + x + w ,

z + y + x ,

z ,

y² + x² + v ,

z³ + z²y + zy² + y²x + zx² + yx² + zxw + yxw + xw² + zv + wv + u ,

z⁴ + y³x + y²x² + yx³ + z²v + yu + xu + t
in the cohomology of G.

The kernel of the inflation is zero.

Maximal Quotient Group Q #7.

The kernel of the quotient is generated by G.2 * G.3 * 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.1 * pcy.2 * pcy.3 * pcy.4 * pcy.5, pcy.1 * pcy.2 * pcy.3 * pcy.5, pcy.2 ] in the quotient group.

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

y + x + w ,

z + y + x ,

y + x ,

zy + y² + yx + x² + v ,

z³ + z²y + xw² + zv + wv + u ,

z⁴ + z³y + z²v + yu + xu + t
in the cohomology of G.

The kernel of the inflation is zero.

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

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

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

z + w .

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 y² , v , t .
The depth-essential cohomology is generated as a module over Q by the elements

[ z+ w ] [ zx+ yw, zx+ xw, zw+ w² ] [ zw²+ w³, zxw+ xw², yxw ] [ zxw²+ xw³ ]

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Action of Automorphisms

The groups of outer automorphisms of G has order 1024, and is generated by 10 elements.

Automorphism #1

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

y ,

x ,

w ,

z² + w² + v ,

zy² + zx² + 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 ,

z² + w² + v ,

zy² + zx² + u ,

t .

Automorphism #3

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 ,

z² + w² + v ,

zy² + zx² + u ,

t .

Automorphism #4

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 ,

z² + zy + yx + w² + v ,

z²y + zy² + zxw + yxw + u ,

zyx² + zx³ + zx²w + zxw² + t .

Automorphism #5

G.1 * G.5
G.2 * G.5
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 + zy + yx + w² + v ,

y²x + zx² + yx² + xw² + u ,

z³y + zyx² + y²x² + zx³ + y³w + y²xw + zx²w + zxw² + t .

Automorphism #6

G.1 * G.2 * G.3
G.3
G.2
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 ,

z + x ,

z + y ,

w ,

v ,

u ,

z⁴ + t .

Automorphism #7

G.1 * G.2 * G.3
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 ,

z + y ,

z + x ,

w ,

v ,

u ,

z⁴ + t .

Automorphism #8

G.1 * G.2 * G.3
G.2
G.3
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 ,

z + y + w ,

z + x + w ,

w ,

v ,

u ,

t .

Automorphism #9

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

z + x ,

x ,

z + y + x + w ,

z² + y² + x² + v ,

z³ + y³ + y²x + yx² + x³ + u ,

z³y + y³x + y²x² + yx³ + x⁴ + z²v + zyv + y²v + zxv + x²v + yu + xu + wu + t .

Automorphism #10

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

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

w ,

z + y + w ,

x ,

z ,

z² + w² + v ,

z³ + yv + xv + wv + u ,

zwv + v² + t .

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CHECKS

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

THE COMPUTATION IS COMPLETE