GROUP OF ORDER 64 #87

GROUP #87

The MAGMA library number for G is 72

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.6, G.3^G.1 = G.3 * G.5

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, t, s ] in degrees [ 1, 1, 1, 2, 2, 2, 3, 4 ] , by the ideal generated by the relations
z2 + zx + x2 ,
zy + zx + x2 ,
y2 + zx + x2 ,
x3 ,
zw + yw ,
yxw + x2w + zt ,
zxu + x2u + w2 ,
yt ,
wt ,
t2 .

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
v , u , s .

Restriction to the Maximal Elementary Abelian Subgroup

Generated by [ G.4 * G.5, G.6, G.4 ] 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 ,
0 ,
y2 ,
z2 + x2 ,
0 ,
z4
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 ,
w ,
t .

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

It is nilpotent of degree 5.

Restrictions to Maximal Subgroups

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

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.2 * G.3 ] .

The images of the generators of the cohomology of G restricted to H are
y ,
z ,
z ,
x + v ,
v + u + t ,
w + t ,
zw + yw + zu + yu + zt + yt ,
w2 + u2 + t2
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.5, G.2 * G.3, G.6, G.1 * G.3, G.4 ] .

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.3, G.1 * G.3 ] .

The images of the generators of the cohomology of G restricted to H are
y ,
z ,
z + y ,
x + v ,
v + u + t ,
w + t ,
zw + yw + zu + yu + zt + yt ,
w2 + u2 + t2
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.3, G.4 ] .

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.3 * G.4, 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 ,
0 ,
y + x ,
zy ,
w ,
y2 + w ,
zw + zv ,
w2 + v2
in the cohomology of H.

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

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

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.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
w ,
w ,
y + x ,
yw + w2 ,
z2 + zw + w2 ,
y2 + w2 ,
zw2 + yw2 ,
v
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 #5 Generated by [ G.2, G.1, G.5, G.6, G.4 ] .

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
y ,
z ,
0 ,
x + v ,
v + u + t ,
w + t ,
yx + zt + yt ,
t2
in the cohomology of H.

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

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

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 * G.3 ] .

The images of the generators of the cohomology of G restricted to H are
y ,
z ,
y ,
x + v ,
v + u + t ,
w + t ,
yx + zt + yt ,
t2
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.2, G.5, G.6, G.3, G.4 ] .

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
0 ,
z ,
y ,
zx ,
zx + x2 ,
x2 + w ,
zv ,
v2
in the cohomology of H.

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

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

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 v , u , s .
The essential cohomology is generated as a module over Q by the elements
[] [] [] [ zt ] [ x2t ]

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

The images of the generators of the cohomology of Q inflated to G are
z + y ,
z ,
z + y + x ,
y2 + w + v + u ,
zv + yv + t ,
zyv + v2 + s
in the cohomology of G.

The kernel of the inflation to G of the cohomology of Q is generated by
y2 + yx + x2 .

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 Number11 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^$.1 = $.2 * $.5

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

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

The images of the generators of the cohomology of Q inflated to G are
z + y ,
z ,
y + x ,
w ,
u ,
y2 + v
in the cohomology of G.

The kernel of the inflation to G of the cohomology of Q is generated by
y2 + yx + x2 ,
x3 .

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 Number35 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4 * $.5, $.3^2 = $.4 * $.5, $.2^$.1 = $.2 * $.4, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 .

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

The images of the generators of the cohomology of Q inflated to G are
x ,
y + x ,
z + y + x ,
w + v + u ,
zxw + yxv + y2u + yxu + v2 + u2 + s
in the cohomology of G.

The kernel of the inflation to G of the cohomology of Q is generated by
zy + zx + yx + x2 .

Maximal Quotient Group Q #4.

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

The Group Q 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

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

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

The images of the generators of the cohomology of Q inflated to G are
z + y ,
y + x ,
z ,
u ,
zyw + zyv + v2 + u2 + s
in the cohomology of G.

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

Maximal Quotient Group Q #5.

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

The Group Q 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 .

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

The images of the generators of the cohomology of Q inflated to G are
z + x ,
z + y ,
z + y + x ,
y2 + v ,
zw + t ,
s
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.6 .

The Group Q 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

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

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

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

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

Maximal Quotient Group Q #7.

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

The Group Q 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 .

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

The images of the generators of the cohomology of Q inflated to G are
z + y + x ,
z + y ,
z + x ,
y2 + v ,
zw + zv + yv + zu + yu + t ,
zyw + zyv + v2 + u2 + s
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 256, and is generated by 8 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.5 * 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 ,
y2 + w ,
v ,
y2 + u ,
t ,
s .

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.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 ,
v ,
u ,
t ,
s .

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.4 * 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 ,
y2 + w ,
v ,
y2 + u ,
t ,
s .

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.3 * 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 ,
y2 + yx + x2 + w ,
yx + x2 + v ,
u ,
zyx + t ,
s .

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.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 ,
v ,
u ,
t ,
s .

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.2
    G.3 * G.4 * 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 ,
yx + w ,
yx + x2 + v ,
x2 + u ,
t ,
s .

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.1 * 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 ,
y ,
z + x ,
w ,
v ,
u ,
t ,
s .

Automorphism #8

The order of the class of the automorphism in the outer automorphism group is 4. The images of the generators of G are

    G.1
    G.2 * G.3
    G.3
    G.4 * G.5
    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 ,
y + x ,
w ,
v ,
u ,
zv + yv + zu + yu + t ,
zyw + zyv + v2 + u2 + s .

CHECKS

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

THE COMPUTATION IS COMPLETE