GROUP OF ORDER 32 #46

GROUP #46

The MAGMA library number for G is 6

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

The center of G is abelian of type [ 2 ] .
The orders of the terms of the lower central series are [ 32, 4, 2, 1 ] .
The orders of the terms of the upper central series are [ 1, 2, 8, 32 ] .
The order of the Frattini subgroup is 8.
The exponent of G is 4.
G has 2 conjugacy classes of maximal elementary abelian p-subgroups. The orders of the maximal elementary abelian subgroups are [ 8, 8 ] .
The orders of the centralizers of the maximal elementary abelian subgroups are [ 8, 8 ] .
The orders of the normalizers of the maximal elementary abelian subgroups are [ 32, 32 ] .

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, 2, 2, 2, 3, 3, 4 ] , by the ideal generated by the relations
z2 ,
zy ,
y3 + yx + zv ,
zw + zv ,
yw + zv ,
y2v + w2 + xv ,
zu ,
yu ,
zt ,
y2t + xu + vu + xt + wt ,
wu + vu + wt ,
y2x2 + w2v + v3 + yvt + y2s + u2 + t2 ,
xw2 + w2v + u2 ,
xwv + wv2 + u2 + ut .

The Hilbert series for the cohomology ring is
-1 / t4 -2t3+ 2t -1.
Its denominator factors as ( t-1 )3 ( t+1 ) .

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

The hypercohomolgy spectral sequence has E2 term:

ROW (1) : [] [] [] [ zv ]
ROW (0) : [1] [ z, y ] [ w, y2 ] [ u, t ] [ yt ] [ wt ]

Restrictions to Maximal Elementary Abelian Subgroups

Subgroup E #1
Generated by [ G.4 * G.5, G.2 * G.3, 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 ,
z ,
z2 ,
0 ,
zx + x2 ,
0 ,
z3 + z2y + zy2 + z2x + x3 ,
z2y2 + y4 + z3x + z2yx + zy2x + z2x2 + zyx2 + y2x2
in the cohomology of E.

The kernel of the restriction to E (Minimal Prime) of the cohomology of G is generated by
z ,
y2 + x ,
w ,
u .

Subgroup E #2
Generated by [ G.4 * G.5, G.2 * G.4, 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 ,
z2 ,
zx ,
x2 ,
z2x + zx2 ,
z2x + x3 ,
z2y2 + y4 + z3x + z2yx + zy2x + z2x2 + zyx2 + y2x2
in the cohomology of E.

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

The nilradical of the cohomology of G is generated by
z

It is nilpotent of degree 2.

Restrictions to Maximal Subgroups

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

The Group H is Isomorphic to the Group of Order 16 Number6 GrpPC of order 16 = 2^4 PC-Relations: $.3^2 = $.4, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.4 Generated by [ G.2 * G.4, G.3, G.2 * G.4 * G.5 ]

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

The images of the generators of the cohomology of G restricted to H are
0 ,
x ,
z2 + y2 ,
z2 + zy + yx + x2 ,
z2 + yx + x2 ,
z2y + zy2 + z2x + zx2 ,
zy2 + z2x + yx2 + xw ,
z3y + zy3 + zyw + w2
in the cohomology of H.

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

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

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.4 * G.5 ] .

The images of the generators of the cohomology of G restricted to H are
z + y ,
0 ,
v ,
x ,
y2 ,
zx + zw + yw + zv ,
zy2 + zw + yw + zv ,
zy3 + zyx + xw + w2 + xv + wv
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.2, G.4, G.1 * G.3, G.5 ] .

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.1 * G.3 ] .

The images of the generators of the cohomology of G restricted to H are
z + y ,
z + y ,
y2 + v ,
x + v ,
v ,
zx + zw + yw + zv ,
zx ,
zy3 + y2v + xw + w2 + xv + wv
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 zero.

Inflations from Maximal Quotient Groups

Maximal Quotient Group Q #1

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

The Group Q 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 .

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

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

The kernel of the inflation to G of the cohomology of Q is generated by
y2 + x ,
yx + zw + yw + zv ,
zw2 + yw2 + zwv + ywv .

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 are 2 conjugacy classes of subgroups which are centralizers of elementary abelian subgroups of rank 3. They are represented by the subgroups generated by


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

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

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 x , s .
The depth-essential cohomology is generated as a module over Q by the elements
[ z ] [] [ zv ]

Action of Automorphisms

The groups of outer automorphisms of G has order 4, and is generated by 2 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

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

The images of the generators of the cohomology of G under the map induced by the automorphism are
z ,
z + y ,
x ,
y2 + x + w ,
y2 + x + v ,
u ,
u + t ,
s .

CHECKS

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

THE COMPUTATION IS COMPLETE