GROUP OF ORDER 32 #8

GROUP #8

Abelian(2,2) x Dihedral(8)

The MAGMA library number for G is 46

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

The center of G is abelian of type [ 2, 2, 2 ] .
The orders of the terms of the lower central series are [ 32, 2, 1 ] .
The orders of the terms of the upper central series are [ 1, 8, 32 ] .
The order of the Frattini subgroup is 2.
The exponent of G is 4.
G has 2 conjugacy classes of maximal elementary abelian p-subgroups. The orders of the maximal elementary abelian subgroups are [ 16, 16 ] .
The orders of the centralizers of the maximal elementary abelian subgroups are [ 16, 16 ] .
The orders of the normalizers of the maximal elementary abelian subgroups are [ 32, 32 ] .

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, 1, 2 ] , by the ideal generated by the relations
zx + yx + xw .

The Hilbert series for the cohomology ring is
1 / t4 -4t3+ 6t2 -4t+ 1.
Its denominator factors as ( t-1 )4 .

The Krull dimension of the cohomology ring is 4.
The cohomology ring is Cohen-Macaulay and the longest regular sequence consists of the generators
z2 , y2 , v , x2 + w2 .

Restrictions to Maximal Elementary Abelian Subgroups

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

The cohomology ring of E is a polynomial ring in the variables z , y , x , w .

The images of the generators of the cohomology of G restricted to E are
y ,
z + x + w ,
0 ,
y + x + w ,
zy + y2 + zx + x2
in the cohomology of E.

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

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

The cohomology ring of E is a polynomial ring in the variables z , y , x , w .

The images of the generators of the cohomology of G restricted to E are
y ,
z + x + w ,
z ,
z + y + x + w ,
zy + y2 + zx + x2
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 .

The nilradical of the cohomology of G is zero.

Restrictions to Maximal Subgroups

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

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

The cohomology ring of H is a polynomial ring with variables z , y , x , w in degrees [ 1, 1, 1, 1 ] .

The images of the generators of the cohomology of G restricted to H are
x ,
z ,
0 ,
y ,
zw + yw + xw + w2
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.2, G.3, G.1, G.5 ] .

The Group H is Isomorphic to the Group of Order 16 Number6 GrpPC of order 16 = 2^4 PC-Relations: $.3^2 = $.4, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.4 Generated by [ G.1 * G.2 * G.3 * G.5, G.3 * G.5, G.2 * G.3 ]

of type Cyclic(2) x Dihedral(8) .

The images of the generators of the cohomology of G restricted to H are
y ,
y + x ,
z + y + x ,
0 ,
w
in the cohomology of H.

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

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

The Group H is Isomorphic to the Group of Order 16 Number6 GrpPC of order 16 = 2^4 PC-Relations: $.3^2 = $.4, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.4 Generated by [ G.1 * G.2 * G.3 * G.4 * G.5, G.3 * G.5, G.1 * G.3 * G.4 ]

of type Cyclic(2) x Dihedral(8) .

The images of the generators of the cohomology of G restricted to H are
y + x ,
x ,
z + y + x ,
y + x ,
w
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.2, G.1 * G.4, G.3, G.5 ] .

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

The cohomology ring of H is a polynomial ring with variables z , y , x , w in degrees [ 1, 1, 1, 1 ] .

The images of the generators of the cohomology of G restricted to H are
z + y ,
z ,
x ,
y ,
xw + w2
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 #5 Generated by [ G.2, G.1, G.5, G.3 * G.4 * G.5 ] .

The Group H is Isomorphic to the Group of Order 16 Number6 GrpPC of order 16 = 2^4 PC-Relations: $.3^2 = $.4, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.4 Generated by [ G.2, G.1, G.3 * G.4 * G.5 ]

of type Cyclic(2) x Dihedral(8) .

The images of the generators of the cohomology of G restricted to H are
y ,
z ,
x ,
x ,
w
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 #6 Generated by [ G.1 * G.3 * G.5, G.2, G.4, G.5 ] .

The Group H is Isomorphic to the Group of Order 16 Number6 GrpPC of order 16 = 2^4 PC-Relations: $.3^2 = $.4, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.4 Generated by [ G.1 * G.3 * G.5, G.2 * G.5, G.4 * G.5 ]

of type Cyclic(2) x Dihedral(8) .

The images of the generators of the cohomology of G restricted to H are
x ,
y ,
x ,
z ,
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 #7 Generated by [ G.1 * G.3 * G.5, G.1 * G.2, G.4, G.5 ] .

The Group H is Isomorphic to the Group of Order 16 Number6 GrpPC of order 16 = 2^4 PC-Relations: $.3^2 = $.4, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.4 Generated by [ G.2 * G.3 * G.5, G.2 * G.3 * G.4 * G.5, G.1 * G.3 * G.4 * G.5 ]

of type Cyclic(2) x Dihedral(8) .

The images of the generators of the cohomology of G restricted to H are
y ,
z + x ,
z + y + x ,
z + y ,
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 #8 Generated by [ G.2 * G.4, G.3, G.1, G.5 ] .

The Group H is Isomorphic to the Group of Order 16 Number6 GrpPC of order 16 = 2^4 PC-Relations: $.3^2 = $.4, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.4 Generated by [ G.2 * G.3 * G.4, G.1 * G.2 * G.3 * G.4 * G.5, G.3 * G.5 ]

of type Cyclic(2) x Dihedral(8) .

The images of the generators of the cohomology of G restricted to H are
x ,
y + x ,
z + y + x ,
y + x ,
w
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 #9 Generated by [ G.1 * G.3 * G.5, G.1 * G.2, G.1 * G.4, G.5 ] .

The group H is abelian of type [ 4, 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 + x ,
x ,
z ,
y ,
w
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 #10 Generated by [ G.1, G.5, G.2 * G.3, G.3 * G.4 * G.5 ] .

The Group H is Isomorphic to the Group of Order 16 Number6 GrpPC of order 16 = 2^4 PC-Relations: $.3^2 = $.4, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.4 Generated by [ G.1 * G.2 * G.3 * G.5, G.3 * G.4 * G.5, G.1 * G.3 * G.4 ]

of type Cyclic(2) x Dihedral(8) .

The images of the generators of the cohomology of G restricted to H are
z + y ,
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 + w .

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

The Group H is Isomorphic to the Group of Order 16 Number6 GrpPC of order 16 = 2^4 PC-Relations: $.3^2 = $.4, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.4 Generated by [ G.1 * G.3 * G.4 * G.5, G.3, G.3 * G.4 * G.5 ]

of type Cyclic(2) x Dihedral(8) .

The images of the generators of the cohomology of G restricted to H are
y ,
0 ,
z + y + x ,
y + x ,
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 #12 Generated by [ G.2, G.4, G.3, G.5 ] .

The Group H is Isomorphic to the Group of Order 16 Number6 GrpPC of order 16 = 2^4 PC-Relations: $.3^2 = $.4, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.4 Generated by [ G.2 * G.3 * G.4, G.3 * G.5, G.3 * G.4 * G.5 ]

of type Cyclic(2) x Dihedral(8) .

The images of the generators of the cohomology of G restricted to H are
0 ,
y ,
z + y + x ,
y + x ,
w
in the cohomology of H.

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

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

The Group H is Isomorphic to the Group of Order 16 Number6 GrpPC of order 16 = 2^4 PC-Relations: $.3^2 = $.4, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.4 Generated by [ G.1 * G.2 * G.3 * G.4, G.1 * G.2 * G.4 * G.5, G.4 * G.5 ]

of type Cyclic(2) x Dihedral(8) .

The images of the generators of the cohomology of G restricted to H are
y + x ,
y + x ,
x ,
z + y + x ,
w
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 #14 Generated by [ G.4, G.1, G.5, G.2 * G.3 ] .

The Group H is Isomorphic to the Group of Order 16 Number6 GrpPC of order 16 = 2^4 PC-Relations: $.3^2 = $.4, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.4 Generated by [ G.1 * G.5, G.4 * G.5, G.2 * G.3 ]

of type Cyclic(2) x Dihedral(8) .

The images of the generators of the cohomology of G restricted to H are
z ,
x ,
x ,
y ,
w
in the cohomology of H.

The kernel of the restriction to H of the cohomology of G is generated by
y + x .

Maximal Subgroup H #15 Generated by [ G.1 * G.3 * G.5, G.2, G.1 * G.4, G.5 ] .

The Group H is Isomorphic to the Group of Order 16 Number6 GrpPC of order 16 = 2^4 PC-Relations: $.3^2 = $.4, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.4 Generated by [ G.1 * G.3 * G.5, G.2, G.1 * G.2 * G.4 ]

of type Cyclic(2) x Dihedral(8) .

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

Inflations from Maximal Quotient Groups

Maximal Quotient Group Q #1

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

The group Q is abelian of type [ 2, 2, 2, 2 ] .

The cohomology ring of Q is a polynomial ring with variables z , y , x , w in degrees [ 1, 1, 1, 1 ]

The images of the generators of the cohomology of Q inflated to G are
w ,
x ,
y ,
z
in the cohomology of G.

The kernel of the inflation to G of the cohomology of Q is generated by
zy + yx + yw .

Maximal Quotient Group Q #2

The kernel of the quotient is generated by G.1 * G.2 * G.5 .

The Group Q is Isomorphic to the Group of Order 16 Number6 GrpPC of order 16 = 2^4 PC-Relations: $.3^2 = $.4, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.4

of type Cyclic(2) x Dihedral(8) .

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

The images of the generators of the cohomology of Q inflated to G are
x + w ,
z + y ,
z + y + w ,
zy + zx + zw + v
in the cohomology of G.

The kernel of the inflation is zero.

Maximal Quotient Group Q #3

The kernel of the quotient is generated by G.1 * G.2 .

The Group Q is Isomorphic to the Group of Order 16 Number6 GrpPC of order 16 = 2^4 PC-Relations: $.3^2 = $.4, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.4

of type Cyclic(2) x Dihedral(8) .

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

The images of the generators of the cohomology of Q inflated to G are
x + w ,
z + y ,
z + y + w ,
v
in the cohomology of G.

The kernel of the inflation is zero.

Maximal Quotient Group Q #4

The kernel of the quotient is generated by G.1 * G.4 .

The Group Q is Isomorphic to the Group of Order 16 Number6 GrpPC of order 16 = 2^4 PC-Relations: $.3^2 = $.4, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.4

of type Cyclic(2) x Dihedral(8) .

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

The images of the generators of the cohomology of Q inflated to G are
z + x + w ,
y ,
z + y + w ,
v
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.1 * G.4 * G.5 .

The Group Q is Isomorphic to the Group of Order 16 Number6 GrpPC of order 16 = 2^4 PC-Relations: $.3^2 = $.4, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.4

of type Cyclic(2) x Dihedral(8) .

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

The images of the generators of the cohomology of Q inflated to G are
z + x + w ,
y ,
z + y + w ,
zy + zx + zw + v
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.4 * G.5 .

The Group Q is Isomorphic to the Group of Order 16 Number6 GrpPC of order 16 = 2^4 PC-Relations: $.3^2 = $.4, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.4

of type Cyclic(2) x Dihedral(8) .

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

The images of the generators of the cohomology of Q inflated to G are
y + x + w ,
z ,
z + y + w ,
zy + zx + yw + xw + v
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.4 .

The Group Q is Isomorphic to the Group of Order 16 Number6 GrpPC of order 16 = 2^4 PC-Relations: $.3^2 = $.4, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.4

of type Cyclic(2) x Dihedral(8) .

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

The images of the generators of the cohomology of Q inflated to G are
y + x + w ,
z ,
z + y + w ,
v
in the cohomology of G.

The kernel of the inflation is zero.

Action of Automorphisms

The groups of outer automorphisms of G has order 768, and is generated by 9 elements.

Automorphism #1

The order of the class of the automorphism in the outer automorphism group is 2. The images of the generators of G are

The images of the generators of the cohomology of G under the map induced by the automorphism are
z ,
y ,
x ,
w ,
zw + yw + xw + v .

Automorphism #2

The order of the class of the automorphism in the outer automorphism group is 2. The images of the generators of G are

The images of the generators of the cohomology of G under the map induced by the automorphism are
z ,
y ,
x ,
w ,
zy + zx + yw + xw + v .

Automorphism #3

The order of the class of the automorphism in the outer automorphism group is 2. The images of the generators of G are

The images of the generators of the cohomology of G under the map induced by the automorphism are
z ,
y + x ,
x ,
x + w ,
v .

Automorphism #4

The order of the class of the automorphism in the outer automorphism group is 2. The images of the generators of G are

The images of the generators of the cohomology of G under the map induced by the automorphism are
z + x ,
y ,
x ,
x + w ,
v .

Automorphism #5

The order of the class of the automorphism in the outer automorphism group is 2. The images of the generators of G are

The images of the generators of the cohomology of G under the map induced by the automorphism are
z ,
z + w ,
x ,
z + y ,
v .

Automorphism #6

The order of the class of the automorphism in the outer automorphism group is 2. The images of the generators of G are

The images of the generators of the cohomology of G under the map induced by the automorphism are
y + x + w ,
y ,
x ,
z + y + x ,
v .

Automorphism #7

The order of the class of the automorphism in the outer automorphism group is 3. The images of the generators of G are

The images of the generators of the cohomology of G under the map induced by the automorphism are
w ,
y + w ,
x ,
z + w ,
v .

Automorphism #8

The order of the class of the automorphism in the outer automorphism group is 2. The images of the generators of G are

The images of the generators of the cohomology of G under the map induced by the automorphism are
w ,
y ,
x ,
z ,
v .

Automorphism #9

The order of the class of the automorphism in the outer automorphism group is 2. The images of the generators of G are

The images of the generators of the cohomology of G under the map induced by the automorphism are
z ,
y ,
z + y + w ,
z + y + x ,
v .

CHECKS

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

THE COMPUTATION IS COMPLETE