GROUP OF ORDER 32 #41

GROUP #41

The MAGMA library number for G is 33

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

The center of G is abelian of type [ 2, 2 ] .
The orders of the terms of the lower central series are [ 32, 4, 1 ] .
The orders of the terms of the upper central series are [ 1, 4, 32 ] .
The order of the Frattini subgroup is 4.
The exponent of G is 4.
G has a unique maximal elementary abelian subgroup which is normal and has order 8. Its centralizer 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, 3, 3, 3, 4, 4 ] , by the ideal generated by the relations
z2 + zx + yx ,
zy + y2 + zx ,
y2x + zx2 ,
zx2 ,
zw + xv + yu ,
yw + zu ,
zv + yu ,
yv + xv + zu + yu ,
y2u + zxu + yxu ,
y2t + x2s + w2 + u2 ,
zxt + zxs + yxs + x2s + w2 + wv + vu + u2 ,
yxt + zxs + wv ,
x2t + zxs + yxs + w2 + v2 ,
y2s + zxs + x2s + w2 + v2 + vu + u2 .

The Hilbert series for the cohomology ring is
t4 -t3 -1 / t7 -3t6+ 5t5 -7t4+ 7t3 -5t2+ 3t -1.
Its numerator factors as ( t4-t3-1 ) .
Its denominator factors as ( t-1 )3 ( t2+1 )2 .

The Krull dimension of the cohomology ring is 3.
The longest regular sequence consists of the generators t , s .
A homogeneous set of parameters is the set t , s , x2 of degrees [ 4, 4, 2 ] .

The hypercohomolgy spectral sequence has E2 term:

ROW (1) : [] [ z ] [ zx, y2, yx ] [ yx2 ] [ yu, zu, xv ] [ yxu, zxu ]
ROW (0) : [1] [ z, y, x ] [ zx, y2, yx ] [ u, v, w ] [ zu, xw, yu, xu, xv ] [ yxu ] [ wu ] [ xwu ]

Restriction to the Maximal Elementary Abelian Subgroup

Generated by [ G.5, G.4, G.3 * 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 ,
z ,
z2y + zy2 ,
0 ,
z2y + zy2 + z2x + zx2 ,
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 ,
v .

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

The Group H is Isomorphic to the Group of Order 16 Number10 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.3, $.2^2 = $.3, $.2^$.1 = $.2 * $.4 Generated by [ G.1 * G.3 * G.4, G.2 ] .

The images of the generators of the cohomology of G restricted to H are
y ,
z ,
y ,
zw ,
yx + zw ,
zx + yw ,
x2 + w2 ,
w2
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.1, G.5, G.4, G.2 ] .

The Group H is Isomorphic to the Group of Order 16 Number10 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.3, $.2^2 = $.3, $.2^$.1 = $.2 * $.4 Generated by [ G.1 * G.2 * G.4 * G.5, G.1 ] .

The images of the generators of the cohomology of G restricted to H are
z + y ,
z ,
0 ,
yx + zw ,
zx + yw ,
zx + yx + zw ,
x2 ,
z2x + x2 + 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 #3 Generated by [ G.5, G.4, G.3, G.2 ] .

The Group H is Isomorphic to the Group of Order 16 Number9 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.3 * $.4, $.2^2 = $.3, $.2^$.1 = $.2 * $.4 Generated by [ G.2, G.2 * G.3 * G.5 ] .

The images of the generators of the cohomology of G restricted to H are
0 ,
z + y ,
z ,
zx + zw + yw + yv ,
zw + yw + zv + yv ,
zy2 + yw ,
y2v + v2 ,
zy3 + y2v + w2 + 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 #4 Generated by [ G.1, G.5, G.4, G.3 ] .

The Group H is Isomorphic to the Group of Order 16 Number9 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.3 * $.4, $.2^2 = $.3, $.2^$.1 = $.2 * $.4 Generated by [ G.1, G.1 * G.3 * G.4 ] .

The images of the generators of the cohomology of G restricted to H are
z + y ,
0 ,
z ,
zy2 + yw ,
zw + yw + zv + yv ,
zy2 + zx + yw + zv ,
zy3 + w2 ,
y2v + 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 #5 Generated by [ G.1, G.5, G.4, G.2 * G.3 * G.5 ] .

The Group H is Isomorphic to the Group of Order 16 Number10 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.3, $.2^2 = $.3, $.2^$.1 = $.2 * $.4 Generated by [ G.2 * G.3 * G.5, G.1 * G.2 * G.3 * G.5 ] .

The images of the generators of the cohomology of G restricted to H are
z ,
z + y ,
z + y ,
zx + yx + yw ,
yx + zw ,
zw ,
y2w + w2 ,
z2x + y2w + x2
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 #6 Generated by [ G.5, G.4, G.3, G.1 * G.2 ] .

The Group H is Isomorphic to the Group of Order 16 Number9 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.3 * $.4, $.2^2 = $.3, $.2^$.1 = $.2 * $.4 Generated by [ G.1 * G.2 * G.3 * G.4 * G.5, G.1 * G.2 ] .

The images of the generators of the cohomology of G restricted to H are
z + y ,
z + y ,
z ,
zy2 + zw + zv + yv ,
zw + yw ,
zx + zw + yw + yv ,
zy3 + w2 ,
zy3 + y2v + w2 + v2
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.5, G.4, G.2 * G.3 * G.5, G.1 * G.2 ] .

The group H is abelian of type [ 4, 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
y ,
z + y ,
z ,
zx + yx + zw + yw ,
yx + zw ,
yw ,
x2 ,
zyx + zyw + w2
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
yx2 ,
zxu ,
yxu .

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 t , s .
The essential cohomology is generated as a module over Q by the elements
[] [] [ yx2 ] [] [ zxu, yxu ]

Inflations from Maximal Quotient Groups

Maximal Quotient Group Q #1

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

The Group Q is Isomorphic to the Group of Order 16 Number8 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.3^2 = $.4, $.3^$.2 = $.3 * $.4

of type AlmostExtraSpecial(16) .

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

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

The kernel of the inflation to G of the cohomology of Q is generated by
zy + y2 + zx + yx + x2 ,
zx2 + x3 .

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 16 Number8 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.3^2 = $.4, $.3^$.2 = $.3 * $.4

of type AlmostExtraSpecial(16) .

The generators of G have images [ pcy.2, pcy.1 * pcy.2, 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 ,
y + x ,
z ,
t
in the cohomology of G.

The kernel of the inflation to G of the cohomology of Q is generated by
zy + zx + yx ,
yx2 + x3 .

Maximal Quotient Group Q #3

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

The Group Q is Isomorphic to the Group of Order 16 Number8 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.3^2 = $.4, $.3^$.2 = $.3 * $.4

of type AlmostExtraSpecial(16) .

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

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

The kernel of the inflation to G of the cohomology of Q is generated by
zy + zx + yx ,
yx2 + x3 .

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 is only one conjugacy class of subgroups which are centralizers of elementary abelian subgroups of rank 3. It is represented by the subgroup generated by
[ G.3, G.3 * G.4 * G.5, G.3 * G.5 ] .


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

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 t , s .
The depth-essential cohomology is generated as a module over Q by the elements
[ z, y ] [ zx, y2, yx ] [ yx2, v ] [ zu, yu, xv ] [ zxu, yxu ]

Action of Automorphisms

The groups of outer automorphisms of G has order 24, and is generated by 4 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.4
    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 ,
z2x + w ,
v ,
z2x + 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.4 * G.5
    G.2 * G.4 * 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 ,
z2x + 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.4
    G.2 * G.4
    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 ,
v ,
z2x + u ,
t ,
s .

Automorphism #4

The order of the class of the automorphism in the outer automorphism group is 6. The images of the generators of G are

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

The images of the generators of the cohomology of G under the map induced by the automorphism are
z + y ,
z ,
z + x ,
u ,
v ,
z2x + w + v + u ,
zw + t + s ,
zw + xv + t .

CHECKS

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

THE COMPUTATION IS COMPLETE