GROUP OF ORDER 64 #90

GROUP #90

The MAGMA library number for G is 63

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

The center of G is abelian of type [ 2, 2, 2 ] .
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 4.
G has a unique maximal elementary abelian subgroup which is central in G and has order 8.

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, 2, 2 ] , by the ideal generated by the relations
z2 + zx + yx ,
zy + zx + yx + x2 ,
y2 + zx + yx .

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

The Krull dimension of the cohomology ring is 3.
The cohomology ring is Cohen-Macaulay and the longest regular sequence consists of the generators
w , v , u .

Restriction to the Maximal Elementary Abelian Subgroup

Generated by [ G.4, G.5 * G.6, G.4 * G.5 ] 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 ,
0 ,
0 ,
z2 + x2 ,
y2 ,
y2 + 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 ,
x .

This ideal is also the nilradical of the cohomology of G.

It is nilpotent of degree 4.

Restrictions to Maximal Subgroups

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

The Group H is Isomorphic to the Group of Order 32 Number12 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.5, $.3^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5 Generated by [ G.1 * G.6, G.1 * G.2 * G.3 * G.6, G.1 * G.2 * G.3 ]

of type Cyclic(2) x Group(16)#10 .

The images of the generators of the cohomology of G restricted to H are
z + y + x ,
y + x ,
y + x ,
zy + zx + w ,
y2 + zx + w + v ,
w + v
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 #2 Generated by [ G.3, G.4, G.5, G.1, G.6 ] .

The Group H is Isomorphic to the Group of Order 32 Number18 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.2^$.1 = $.2 * $.5 Generated by [ G.3, G.1 ] .

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

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

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

The Group H is Isomorphic to the Group of Order 32 Number18 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.2^$.1 = $.2 * $.5 Generated by [ G.1 * G.3 * G.6, G.1 * G.2 * G.5 ] .

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

The Group H is Isomorphic to the Group of Order 32 Number18 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.2^$.1 = $.2 * $.5 Generated by [ G.2, G.1 ] .

The images of the generators of the cohomology of G restricted to H are
z ,
y ,
0 ,
w + t ,
x + w + v ,
x + w + v + 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 #5 Generated by [ G.3, G.2, G.4, G.5, G.6 ] .

The Group H is Isomorphic to the Group of Order 32 Number18 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.2^$.1 = $.2 * $.5 Generated by [ G.3, G.2 ] .

The images of the generators of the cohomology of G restricted to H are
0 ,
z ,
y ,
w ,
x + v + u ,
x + t
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.1 * G.3 * G.6, G.2, G.4, G.5, G.6 ] .

The Group H is Isomorphic to the Group of Order 32 Number12 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.5, $.3^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5 Generated by [ G.2 * G.6, G.1 * G.2 * G.3 * G.4, G.1 * G.2 * G.3 * G.4 * G.6 ]

of type Cyclic(2) x Group(16)#10 .

The images of the generators of the cohomology of G restricted to H are
y + x ,
z + y + x ,
y + x ,
w ,
zy + y2 + v ,
zy + zx + 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 #7 Generated by [ G.3, G.4, G.5, G.1 * G.2 * G.5, G.6 ] .

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
z ,
z ,
y ,
x2 ,
zx + yx + x2 + w ,
zx + yx + v
in the cohomology of H.

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

The essential cohomology of G is generated as an ideal by
zx + yx + x2 .

It is nilpotent of degree 2.
The annihilator of the Essential Cohomology has dimension 3 .
The essential cohomology is a free module over the polynomial subring Q of the cohomology ring of G generated by w , v , u .
The essential cohomology is generated as a module over Q by the elements
[] [ zx+ yx+ x2 ] [ x3 ]

Inflations from Maximal Quotient Groups

Maximal Quotient Group Q #1.

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

The images of the generators of the cohomology of Q inflated to G are
z + x ,
z + y ,
y ,
z2 + zy + v + u ,
zw + xw + zu + xu ,
z2w + x2w + w2 + u2
in the cohomology of G.

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

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 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 .

The generators of G have images [ pcy.2, pcy.1 * pcy.2 * pcy.3 * 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 + x ,
y + x ,
x ,
w ,
z2y + zv + yv ,
z2w + x2w + zyv + v2
in the cohomology of G.

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

Maximal Quotient Group Q #3.

The kernel of the quotient is generated by G.4 * G.6 .

The Group Q 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 .

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

The images of the generators of the cohomology of Q inflated to G are
y + x ,
z + y ,
y ,
zw + xw + zv + xv + zu + yu ,
zw + yw + zv + yv + yu + xu ,
zyv + zyu + u2 ,
z2w + x2w + zyv + w2 + v2 + u2
in the cohomology of G.

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

Maximal Quotient Group Q #4.

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

The Group Q 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 .

The generators of G have images [ pcy.1, pcy.1 * pcy.2 * 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 ,
z + x ,
z ,
w ,
z2y + zw + yw + zu + yu ,
zyu + w2 + u2
in the cohomology of G.

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

Maximal Quotient Group Q #5.

The kernel of the quotient is generated by G.4 * G.5 .

The Group Q 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 .

The generators of G have images [ pcy.2, pcy.1, pcy.1 * 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
z + x ,
z + y ,
z ,
z2y + zw + yw + yv + xv + zu + yu ,
z2y + zw + xw + zv + yv + zu + xu ,
zyv + zyu + w2 + u2 ,
zyv + w2 + v2 + u2
in the cohomology of G.

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

Maximal Quotient Group Q #6.

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

The Group Q is Isomorphic to the Group of Order 32 Number12 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.5, $.3^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5

of type Cyclic(2) x Group(16)#10 .

The generators of G have images [ pcy.1, pcy.2 * pcy.4 * pcy.5, 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 + y ,
x ,
z ,
zy + w ,
zy + v + u
in the cohomology of G.

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

Maximal Quotient Group Q #7.

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

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

The images of the generators of the cohomology of Q inflated to G are
y + x ,
z + y ,
z ,
z2 + zy + v + u ,
z2y + yu + xu ,
z2w + x2w + u2
in the cohomology of G.

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

Action of Automorphisms

The groups of outer automorphisms of G has order 512, 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

    G.1 * G.4
    G.2 * G.4
    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 ,
zy + v ,
zy + 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.2
    G.3 * G.5
    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 + v ,
z2 + zy + 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.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
z ,
y ,
x ,
z2 + zy + w ,
zy + yx + v ,
zy + yx + 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.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 + v ,
z2 + zy + 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.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 + zy + v ,
z2 + u .

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

    G.1 * G.4
    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 ,
z2 + w ,
yx + v ,
yx + u .

Automorphism #7

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

    G.2 * G.6
    G.1
    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
y ,
z ,
x ,
w ,
z2 + w + v ,
z2 + w + u .

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

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

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

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

The images of the generators of the cohomology of G under the map induced by the automorphism are
x ,
z + y + x ,
z ,
w ,
z2 + u ,
z2 + zy + v .

CHECKS

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

THE COMPUTATION IS COMPLETE