GROUP OF ORDER 32 #9

GROUP #9

Abelian(2,2) x Quaternion(8)

The MAGMA library number for G is 47

GrpPC : G of order 32 = 2^5 PC-Relations: G.2^2 = G.5, G.3^2 = G.5, G.4^2 = G.5, G.4^G.2 = G.4 * 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 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
y2 + x2 + yw + xw + w2 ,
w3 .

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.3, G.1 * G.2 * G.3 * G.5, G.1 * 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
z + y + x ,
z + y ,
z + y ,
0 ,
z4 + x4
in the cohomology of E.

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

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.4, G.1, G.5 ] .

The Group H is Isomorphic to the Group of Order 16 Number7 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.2^$.1 = $.2 * $.4 Generated by [ G.2, G.4, G.1 ]

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

The images of the generators of the cohomology of G restricted to H are
x ,
z ,
0 ,
y ,
w
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.4, G.3, G.1, G.5 ] .

The Group H is Isomorphic to the Group of Order 16 Number7 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.2^$.1 = $.2 * $.4 Generated by [ G.4, G.3, G.1 ]

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

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 #3 Generated by [ G.4, G.2 * G.3, G.1, 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
x ,
y ,
y ,
z ,
y4 + w2
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 #4 Generated by [ G.2, G.4, G.3, G.5 ] .

The Group H is Isomorphic to the Group of Order 16 Number7 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.2^$.1 = $.2 * $.4 Generated by [ G.2, G.4, G.2 * G.3 * G.5 ]

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

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

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

The Group H is Isomorphic to the Group of Order 16 Number7 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.2^$.1 = $.2 * $.4 Generated by [ G.2, G.2 * G.3 * G.5, G.1 * G.4 ]

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

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

Maximal Subgroup H #6 Generated by [ G.2, G.3, G.1, 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
x ,
z + y ,
y ,
0 ,
w2
in the cohomology of H.

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

Maximal Subgroup H #7 Generated by [ G.4, G.3, G.1 * G.2, G.5 ] .

The Group H is Isomorphic to the Group of Order 16 Number7 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.2^$.1 = $.2 * $.4 Generated by [ G.4, G.1 * G.2, G.1 * G.2 * G.3 * G.5 ]

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

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

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

The Group H is Isomorphic to the Group of Order 16 Number7 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.2^$.1 = $.2 * $.4 Generated by [ G.1 * G.2, G.1 * G.4, G.1 * G.2 * G.3 * G.5 ]

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

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

Maximal Subgroup H #9 Generated by [ G.1 * G.2, G.1 * G.4, G.5, G.1 * G.3 ] .

The Group H is Isomorphic to the Group of Order 16 Number7 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.2^$.1 = $.2 * $.4 Generated by [ G.2 * G.3 * G.5, G.1 * G.2, G.1 * G.4 ]

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

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

Maximal Subgroup H #10 Generated by [ G.3, G.2 * G.4 * G.5, G.1, G.5 ] .

The Group H is Isomorphic to the Group of Order 16 Number7 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.2^$.1 = $.2 * $.4 Generated by [ G.3, G.1, G.2 * G.4 * G.5 ]

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

The images of the generators of the cohomology of G restricted to H are
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 #11 Generated by [ G.3 * G.4 * G.5, G.2 * G.3, G.1, 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
x ,
y ,
z + y ,
z ,
y4 + w2
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 #12 Generated by [ G.2, G.3 * G.4 * G.5, G.1, G.5 ] .

The Group H is Isomorphic to the Group of Order 16 Number7 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.2^$.1 = $.2 * $.4 Generated by [ G.2, G.3 * G.4 * G.5, G.1 ]

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

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

The Group H is Isomorphic to the Group of Order 16 Number7 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.2^$.1 = $.2 * $.4 Generated by [ G.2, G.4, G.1 * G.2 * G.3 * G.5 ]

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

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

Maximal Subgroup H #14 Generated by [ G.2, G.1 * G.4, G.5, G.1 * G.3 ] .

The Group H is Isomorphic to the Group of Order 16 Number7 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.2^$.1 = $.2 * $.4 Generated by [ G.2, G.1 * G.4, G.1 * G.2 * G.3 * G.5 ]

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

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

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

The Group H is Isomorphic to the Group of Order 16 Number7 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.2^$.1 = $.2 * $.4 Generated by [ G.4, G.2 * G.3 * G.5, G.1 * G.2 ]

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

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

The essential cohomology of G is generated as an ideal by
z2yxw2 + z2x2w2 + zyx2w2 + zx3w2 ,
z4yx2w + z4x3w + z2yx4w + z2x5w ,
z4x2w2 + z2x4w2 .

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.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
z2 + zy + y2 + zx + x2 ,
y3 + y2x + yx2 + x3 .

Maximal Quotient Group Q #2

The kernel of the quotient is generated by G.2 * G.3 * G.5 .

The Group Q is Isomorphic to the Group of Order 16 Number7 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.2^$.1 = $.2 * $.4

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

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

The images of the generators of the cohomology of Q inflated to G are
y + x ,
w ,
z ,
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.2 * G.3 .

The Group Q is Isomorphic to the Group of Order 16 Number7 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.2^$.1 = $.2 * $.4

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

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

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

The Group Q is Isomorphic to the Group of Order 16 Number7 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.2^$.1 = $.2 * $.4

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

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

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

The Group Q is Isomorphic to the Group of Order 16 Number7 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.2^$.1 = $.2 * $.4

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

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

The images of the generators of the cohomology of Q inflated to G are
y + x ,
w ,
y ,
z4 + 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.3 * G.5 .

The Group Q is Isomorphic to the Group of Order 16 Number7 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.2^$.1 = $.2 * $.4

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

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

The images of the generators of the cohomology of Q inflated to G are
y + x ,
w ,
z + y ,
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 * G.3 .

The Group Q is Isomorphic to the Group of Order 16 Number7 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.2^$.1 = $.2 * $.4

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

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

The images of the generators of the cohomology of Q inflated to G are
y + x ,
w ,
z + y ,
z4 + 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 2304, 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.5
    G.2
    G.3
    G.4
    G.5 .

The images of the generators of the cohomology of G under the map induced by the automorphism are
z ,
y ,
x ,
w ,
z4 + 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.5
    G.3
    G.4
    G.5 .

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 #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.1 * G.4
    G.5 .

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

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

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

    G.1
    G.3
    G.2
    G.4
    G.5 .

The images of the generators of the cohomology of G under the map induced by the automorphism are
z ,
x ,
y ,
w ,
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.1
    G.1 * G.2
    G.1 * G.3
    G.1 * G.4
    G.5 .

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

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

The images of the generators of the cohomology of G under the map induced by the automorphism are
z + y ,
z ,
z + y + 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.3 * G.4
    G.1 * G.2
    G.5 .

The images of the generators of the cohomology of G under the map induced by the automorphism are
z + w ,
y + w ,
x ,
y + x ,
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.1
    G.1 * G.2
    G.3
    G.4
    G.5 .

The images of the generators of the cohomology of G under the map induced by the automorphism are
z + y ,
y ,
x ,
w ,
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.2
    G.3
    G.1 * G.2 * G.4
    G.5 .

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

CHECKS

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

THE COMPUTATION IS COMPLETE