GROUP OF ORDER 64 #100

GROUP #100

The MAGMA library number for G is 117

GrpPC : G of order 64 = 2^6 PC-Relations: G.1^2 = G.2, G.2^2 = G.5 * G.6, G.4^2 = G.6, G.3^G.1 = G.3 * G.5, G.4^G.1 = G.4 * G.6, G.4^G.3 = G.4 * G.6

The center of G is abelian of type [ 2, 4 ] .
The orders of the terms of the lower central series are [ 64, 4, 1 ] .
The orders of the terms of the upper central series are [ 1, 8, 64 ] .
The order of the Frattini subgroup is 8.
The exponent of G is 8.
G has 2 conjugacy classes of maximal elementary abelian p-subgroups. The orders of the maximal elementary abelian subgroups are [ 8, 8 ] .
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 [ 64, 64 ] .

The cohomology ring of G is the quotient of a polynomial ring in the variables
[ z, y, x, w, v, u ] in degrees [ 1, 1, 1, 2, 3, 4 ] , by the ideal generated by the relations
z2 ,
zy + zx + yx + x2 ,
y2x + zx2 + x3 ,
zxw + yxw + x2w + zv ,
y2w2 + v2 .

The Hilbert series for the cohomology ring is
-1 / t5 -3t4+ 4t3 -4t2+ 3t -1.
Its denominator factors as ( t-1 )3 ( t2+1 ) .

The Krull dimension of the cohomology ring is 3.
The longest regular sequence consists of the generators w , u .
A homogeneous set of parameters is the set w , u , y2 of degrees [ 2, 4, 2 ] .

The hypercohomolgy spectral sequence has E2 term:

ROW (1) : [] [ z ] [ zx, yx+ x2 ] [ zx2 ]
ROW (0) : [1] [ x, z, y ] [ x2, yx, zx ] [ x3, v ] [ xv, yv ] [ x2v ]

Restrictions to Maximal Elementary Abelian Subgroups

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

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

The images of the generators of the cohomology of G restricted to E are
0 ,
z ,
z ,
zy + y2 ,
z2y + zy2 ,
z2y2 + y4 + 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 ,
xw + v .

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

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

The images of the generators of the cohomology of G restricted to E are
0 ,
z ,
0 ,
zy + y2 ,
z2y + zy2 ,
z2y2 + y4 + 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 ,
x ,
yw + v .

The nilradical of the cohomology of G is generated by
z , yw + v .

It is nilpotent of degree 2.

Restrictions to Maximal Subgroups

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

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

The images of the generators of the cohomology of G restricted to H are
z + y ,
z + x ,
z + y ,
yx ,
z2y + zy2 + zyx + yx2 + w ,
zy3 + z3x + z2yx + y2x2 + v
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 #2 Generated by [ G.3 * G.4 * G.6, G.5 * G.6, G.1 * G.3, G.2, 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.3 * G.4 * G.6, G.1 * G.3 ] .

The images of the generators of the cohomology of G restricted to H are
z ,
z + y ,
y ,
x + v ,
zw + yv ,
y2w + w2
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 #3 Generated by [ G.1, G.5 * G.6, G.2, 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.1, G.3 ] .

The images of the generators of the cohomology of G restricted to H are
z ,
y ,
0 ,
v ,
yx + zw + zv + yv ,
y2w + 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 #4 Generated by [ G.1, G.5 * G.6, G.2, G.4, 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.4 ] .

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

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

Maximal Subgroup H #5 Generated by [ G.5 * G.6, G.2, G.4, 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.2 * G.4 * G.5 * G.6, G.2 * G.3, G.3 * 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
0 ,
z + x ,
y ,
w + v ,
zyx + zx2 + zw + xw + zv + xv ,
y2w + x2w + 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 #6 Generated by [ G.5 * G.6, G.1 * G.3, G.2, G.4, G.5 ] .

The group H is abelian of type [ 8, 4 ]

The cohomology ring of H is a the quotient of a polynomial ring in the variables
[ z, y, x, w ] , in degrees [ 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
z ,
z ,
z + y ,
w ,
zx ,
x2
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 #7 Generated by [ G.1, G.3 * G.4 * G.6, G.5 * G.6, G.2, 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.1 * G.2 * G.3 * G.4 * G.5 * G.6, G.3 * G.4 * G.5, G.1 ]

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

The images of the generators of the cohomology of G restricted to H are
z + y ,
z + x ,
z + x ,
yx ,
z2y + zy2 + zyx + yx2 + w ,
zy3 + z3x + z2yx + y2x2 + v
in the cohomology of H.

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

The essential cohomology of G is generated as an ideal by
zx2 .

It is nilpotent of degree 2.
The annihilator of the Essential Cohomology has dimension 2 .
The essential cohomology is a free module over the polynomial subring Q of the cohomology ring of G generated by w , u .
The essential cohomology is generated as a module over Q by the elements
[] [] [ zx2 ]

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 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.3 * pcy.4, pcy.1 * pcy.2 * pcy.4 * 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 + x ,
x ,
zx + yx + x2 ,
w
in the cohomology of G.

The kernel of the inflation to G of the cohomology of Q is generated by
zx + w ,
yw + xw .

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

The images of the generators of the cohomology of Q inflated to G are
x ,
z + x ,
y ,
yx3 + zv + u
in the cohomology of G.

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

Maximal Quotient Group Q #3.

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

The images of the generators of the cohomology of Q inflated to G are
y + x ,
z + y + x ,
x ,
yw + v ,
y4 + yx3 + w2 + u
in the cohomology of G.

The kernel of the inflation to G of the cohomology of Q is generated by
zy + y2 + zx .

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

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

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 , u .
The depth-essential cohomology is generated as a module over Q by the elements
[ z ] [ zx, yx+ x2 ] [ zx2 ]

Action of Automorphisms

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

    G.1
    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 ,
zyx + v ,
u .

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

    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 ,
zyx + v ,
u .

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

    G.1 * G.5
    G.2
    G.3 * G.5
    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 ,
zx + yx + x2 + w ,
zyx + v ,
u .

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

    G.1 * G.2 * G.5 * G.6
    G.2 * G.5 * G.6
    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 ,
zyx + v ,
u .

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

    G.1 * G.4 * G.6
    G.2
    G.3 * G.4
    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 ,
z + y + x ,
w ,
v ,
u .

CHECKS

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

THE COMPUTATION IS COMPLETE