GROUP OF ORDER 64 #45

GROUP #45

Abelian(2,2) x Quaternion(16)

The MAGMA library number for G is 252

GrpPC : G of order 64 = 2^6 PC-Relations: G.1^2 = G.5, G.2^2 = G.5 * G.6, G.3^2 = G.6, G.4^2 = G.5 * G.6, G.5^2 = G.6, G.3^G.1 = G.3 * G.5, G.3^G.2 = G.3 * G.5 * G.6, G.4^G.3 = G.4 * G.5, G.5^G.3 = G.5 * 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, 2, 1 ] .
The orders of the terms of the upper central series are [ 1, 8, 16, 64 ] .
The order of the Frattini subgroup is 4.
The exponent of G is 8.
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 ] in degrees [ 1, 1, 1, 1, 4 ] , by the ideal generated by the relations
z2 + y2 + zx + yx + xw + w2 ,
x3 .

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

The Krull dimension of the cohomology ring is 3.
The cohomology ring is Cohen-Macaulay and the longest regular sequence consists of the generators
z2 , y2 , v .

Restriction to the Maximal Elementary Abelian Subgroup

Generated by [ G.1 * G.2 * G.6, G.2 * G.4 * G.5 * G.6, G.1 * G.2 ] 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
z + y ,
z + y + x ,
0 ,
x ,
y4 + 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 + w ,
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.3 * G.4, G.5, G.6, G.2, G.1 * G.4 ] .

The Group H is Isomorphic to the Group of Order 32 Number25 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.4, $.3^2 = $.4 * $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4 * $.5, $.3^$.1 = $.3 * $.4, $.4^$.1 = $.4 * $.5 Generated by [ G.1 * G.2 * G.4, G.1 * G.2 * G.3 * G.6, G.2 * G.5 ]

of type Cyclic(2) x Quaternion(16) .

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

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

The Group H is Isomorphic to the Group of Order 32 Number25 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.4, $.3^2 = $.4 * $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4 * $.5, $.3^$.1 = $.3 * $.4, $.4^$.1 = $.4 * $.5 Generated by [ G.1, G.1 * G.2 * G.4, G.3 * G.4 * G.5 * G.6 ]

of type Cyclic(2) x Quaternion(16) .

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

The Group H is Isomorphic to the Group of Order 32 Number25 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.4, $.3^2 = $.4 * $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4 * $.5, $.3^$.1 = $.3 * $.4, $.4^$.1 = $.4 * $.5 Generated by [ G.1 * G.2 * G.4, G.1 * G.2 * G.3 * G.4 * G.5 * G.6, G.2 * G.5 ]

of type Cyclic(2) x Quaternion(16) .

The images of the generators of the cohomology of G restricted to H are
z + x ,
z + y + x ,
z ,
z + 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.4, G.5, G.3, G.6 ] .

The Group H is Isomorphic to the Group of Order 32 Number25 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.4, $.3^2 = $.4 * $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4 * $.5, $.3^$.1 = $.3 * $.4, $.4^$.1 = $.4 * $.5 Generated by [ G.1, G.4, G.3 ]

of type Cyclic(2) x Quaternion(16) .

The images of the generators of the cohomology of G restricted to H are
x ,
0 ,
z ,
y ,
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 #5 Generated by [ G.2 * G.4, G.5, G.3, G.6, G.1 * 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.1 * G.4 * G.5, G.2 * G.4, G.2 * G.4 * G.5 * G.6, G.3 * G.5 ]

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

The images of the generators of the cohomology of G restricted to H are
x ,
z + y ,
w ,
z + y + x ,
z4 + y4 + v
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 #6 Generated by [ G.4, G.5, G.3, G.6, G.1 * G.2 ] .

The Group H is Isomorphic to the Group of Order 32 Number25 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.4, $.3^2 = $.4 * $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4 * $.5, $.3^$.1 = $.3 * $.4, $.4^$.1 = $.4 * $.5 Generated by [ G.1 * G.2 * G.4, G.1 * G.2 * G.3 * G.4 * G.5 * G.6, G.4 * G.5 ]

of type Cyclic(2) x Quaternion(16) .

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

The Group H is Isomorphic to the Group of Order 32 Number25 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.4, $.3^2 = $.4 * $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4 * $.5, $.3^$.1 = $.3 * $.4, $.4^$.1 = $.4 * $.5 Generated by [ G.4 * G.5, G.3, G.2 ]

of type Cyclic(2) x Quaternion(16) .

The images of the generators of the cohomology of G restricted to H are
0 ,
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 .

Maximal Subgroup H #8 Generated by [ G.3 * G.4, G.2 * G.4, G.5, G.6, G.1 * 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.1 * G.4 * G.5, G.2 * G.4, G.3 * G.4 * G.5, G.2 * G.4 * G.5 * G.6 ]

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

The images of the generators of the cohomology of G restricted to H are
x ,
z + y ,
w ,
z + y + x + w ,
z4 + y4 + v
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 #9 Generated by [ G.1, G.4, G.5, G.6, G.2 ] .

The group H is abelian of type [ 8, 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
x ,
z + y + x ,
0 ,
y ,
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 #10 Generated by [ G.4, G.5, G.6, G.2, G.1 * G.3 * G.5 ] .

The Group H is Isomorphic to the Group of Order 32 Number25 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.4, $.3^2 = $.4 * $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4 * $.5, $.3^$.1 = $.3 * $.4, $.4^$.1 = $.4 * $.5 Generated by [ G.4 * G.5, G.2, G.1 * G.3 * G.6 ]

of type Cyclic(2) x Quaternion(16) .

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

Maximal Subgroup H #11 Generated by [ G.2 * G.3 * G.5 * G.6, G.1, G.4, G.5, G.6 ] .

The Group H is Isomorphic to the Group of Order 32 Number25 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.4, $.3^2 = $.4 * $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4 * $.5, $.3^$.1 = $.3 * $.4, $.4^$.1 = $.4 * $.5 Generated by [ G.1, G.2 * G.3 * G.5 * G.6, G.4 ]

of type Cyclic(2) x Quaternion(16) .

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

The Group H is Isomorphic to the Group of Order 32 Number25 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.4, $.3^2 = $.4 * $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4 * $.5, $.3^$.1 = $.3 * $.4, $.4^$.1 = $.4 * $.5 Generated by [ G.1, G.1 * G.2 * G.4, G.3 ]

of type Cyclic(2) x Quaternion(16) .

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

The Group H is Isomorphic to the Group of Order 32 Number25 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.4, $.3^2 = $.4 * $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4 * $.5, $.3^$.1 = $.3 * $.4, $.4^$.1 = $.4 * $.5 Generated by [ G.1, G.2, G.3 * G.5 * G.6 ]

of type Cyclic(2) x Quaternion(16) .

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

The Group H is Isomorphic to the Group of Order 32 Number25 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.4, $.3^2 = $.4 * $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4 * $.5, $.3^$.1 = $.3 * $.4, $.4^$.1 = $.4 * $.5 Generated by [ G.1, G.2, G.3 * G.4 * G.5 * G.6 ]

of type Cyclic(2) x Quaternion(16) .

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

The Group H is Isomorphic to the Group of Order 32 Number25 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.4, $.3^2 = $.4 * $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4 * $.5, $.3^$.1 = $.3 * $.4, $.4^$.1 = $.4 * $.5 Generated by [ G.1 * G.2 * G.4, G.4 * G.5, G.1 * G.3 * G.4 ]

of type Cyclic(2) x Quaternion(16) .

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

The essential cohomology of G is generated as an ideal by
zy2x2w + y3x2w + zyx2w2 + yx2w3 .

It is nilpotent of degree 2.
The annihilator of the Essential Cohomology has dimension 3 .

Inflations from Maximal Quotient Groups

Maximal Quotient Group Q #1.

The kernel of the quotient is generated by 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.3 * pcy.4 * pcy.5, 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 ,
y ,
z + y + w ,
x + w ,
y2 + yx + x2 + xw + w2
in the cohomology of G.

The kernel of the inflation to G of the cohomology of Q is generated by
z2 + zy + y2 + yx + x2 + zw + xw + w2 + v ,
y3 + y2w + yw2 + w3 + zv + xv .

Maximal Quotient Group Q #2.

The kernel of the quotient is generated by G.1 * G.4 * G.6 .

The Group Q is Isomorphic to the Group of Order 32 Number25 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.4, $.3^2 = $.4 * $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4 * $.5, $.3^$.1 = $.3 * $.4, $.4^$.1 = $.4 * $.5

of type Cyclic(2) x Quaternion(16) .

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

The images of the generators of the cohomology of Q inflated to G are
x ,
z + w ,
y ,
y4 + w4 + 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.4 .

The Group Q is Isomorphic to the Group of Order 32 Number25 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.4, $.3^2 = $.4 * $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4 * $.5, $.3^$.1 = $.3 * $.4, $.4^$.1 = $.4 * $.5

of type Cyclic(2) x Quaternion(16) .

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

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

The Group Q is Isomorphic to the Group of Order 32 Number25 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.4, $.3^2 = $.4 * $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4 * $.5, $.3^$.1 = $.3 * $.4, $.4^$.1 = $.4 * $.5

of type Cyclic(2) x Quaternion(16) .

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

The images of the generators of the cohomology of Q inflated to G are
x ,
y + w ,
z ,
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.2 * G.4 * G.5 * G.6 .

The Group Q is Isomorphic to the Group of Order 32 Number25 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.4, $.3^2 = $.4 * $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4 * $.5, $.3^$.1 = $.3 * $.4, $.4^$.1 = $.4 * $.5

of type Cyclic(2) x Quaternion(16) .

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

The images of the generators of the cohomology of Q inflated to G are
x ,
y + w ,
z ,
y4 + 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.1 * G.2 * G.6 .

The Group Q is Isomorphic to the Group of Order 32 Number25 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.4, $.3^2 = $.4 * $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4 * $.5, $.3^$.1 = $.3 * $.4, $.4^$.1 = $.4 * $.5

of type Cyclic(2) x Quaternion(16) .

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

The images of the generators of the cohomology of Q inflated to G are
x ,
w ,
z + y ,
y4 + w4 + 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.1 * G.2 .

The Group Q is Isomorphic to the Group of Order 32 Number25 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.4, $.3^2 = $.4 * $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4 * $.5, $.3^$.1 = $.3 * $.4, $.4^$.1 = $.4 * $.5

of type Cyclic(2) x Quaternion(16) .

The generators of G have images [ pcy.2 * pcy.4 * pcy.5, pcy.1 * pcy.4 * pcy.5, pcy.3, 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 ,
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 1536, and is generated by 10 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.6
    G.2 * G.6
    G.3
    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 ,
x ,
w ,
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

    G.1
    G.2
    G.3 * G.6
    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 ,
x ,
w ,
w4 + 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

    G.1 * G.6
    G.2
    G.3 * G.6
    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 ,
x ,
w ,
y4 + 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

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

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

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

    G.2 * G.5
    G.1 * G.5 * 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 under the map induced by the automorphism are
y + x + w ,
z + x + w ,
x ,
w ,
v .

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.2
    G.1 * 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 + x ,
y ,
x ,
x + w ,
v .

Automorphism #8

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

    G.1 * G.2 * G.4 * G.5 * G.6
    G.2
    G.1 * G.3 * G.4 * G.5 * G.6
    G.1 * 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 + x + w ,
z + y ,
x ,
z + x ,
y4 + 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

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

The images of the generators of the cohomology of G under the map induced by the automorphism are
x + w ,
y ,
x ,
z + x ,
y4 + v .

Automorphism #10

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.5
    G.3 * G.4
    G.4 * G.5
    G.5 * G.6
    G.6 .

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

CHECKS

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

THE COMPUTATION IS COMPLETE