GROUP OF ORDER 32 #47

GROUP #47

The MAGMA library number for G is 7

GrpPC : G of order 32 = 2^5 PC-Relations: G.1^2 = G.3, G.3^2 = G.5, G.2^G.1 = G.2 * G.4, G.3^G.2 = G.3 * G.5, G.4^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 8.
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, 3, 3, 4, 4 ] , by the ideal generated by the relations
z2 ,
zy ,
zx ,
zw ,
y2x + x2 ,
xw + yv ,
zv ,
zu ,
y2v + xv ,
xu + yt ,
zt ,
w3 + ywu + y2s + u2 ,
ywv + v2 ,
y2t + xt ,
vu + wt ,
w2v + yxs + vt + ut ,
ywt + vt ,
yut + t2 .

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 longest regular sequence consists of the generators s .
A homogeneous set of parameters is the set s , y2 , w of degrees [ 4, 2, 2 ] .

The hypercohomolgy spectral sequence has E2 term:

ROW (1 1) : [] [ z ]
ROW (0 1) : [] [ z ] [] [ yx ]
ROW (1 0) : [] [ z ]
ROW (0 0) : [1] [ z, y ] [ x ] [ u, v, yx ] [ t, yu ] [ yt ]

Restrictions to Maximal Elementary Abelian Subgroups

Subgroup E #1
Generated by [ G.4, G.2 * G.4 * G.5, 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 ,
z ,
0 ,
zx + x2 ,
0 ,
z2y + zy2 + z2x + x3 ,
0 ,
z2y2 + y4 + z2yx + zy2x + 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 ,
x ,
v ,
t .

Subgroup E #2
Generated by [ G.4, G.2 * G.3 * G.4 * G.5, 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 ,
z ,
z2 ,
zx + x2 ,
z2x + zx2 ,
z3 + z2y + zy2 + z2x + x3 ,
z4 + z3y + z2y2 + z3x + zx3 ,
z4 + 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 ,
yw + v ,
yu + t .

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

The Group H is Isomorphic to the Group of Order 16 Number11 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.3, $.3^2 = $.4, $.2^$.1 = $.2 * $.4 Generated by [ G.1, G.4 * G.5 ] .

The images of the generators of the cohomology of G restricted to H are
z ,
0 ,
zy ,
zy + y2 ,
x ,
y3 ,
yx ,
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 #2 Generated by [ G.3, G.2, G.5, 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.3 * G.4, G.3 * G.4 * G.5, G.2 * G.3 * G.5 ]

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

The images of the generators of the cohomology of G restricted to H are
0 ,
z + y ,
z2 + y2 + x2 ,
zy ,
z2y + zy2 + z2x + zx2 ,
z3 + z2y + y3 + z2x + yx2 + zw + yw ,
z4 + z2y2 + zy3 + y4 + z3x + z2yx + zy2x + y2x2 + z2w + y2w + x2w ,
z4 + z3y + zy3 + y4 + z3x + zy2x + y2x2 + 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 #3 Generated by [ G.1 * G.2 * G.4, G.3, G.5, G.4 * G.5 ] .

The Group H is Isomorphic to the Group of Order 16 Number11 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.3, $.3^2 = $.4, $.2^$.1 = $.2 * $.4 Generated by [ G.1 * G.2 * G.4, G.4 * G.5 ] .

The images of the generators of the cohomology of G restricted to H are
z ,
z ,
zy ,
zy + y2 ,
x ,
y3 + x ,
yx ,
w
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.5 .

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

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

The kernel of the inflation to G of the cohomology of Q is generated by
x + v ,
zw + yw .

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 1.



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


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

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

The annihilator of the depth-essential cohomology has dimension 1 .

The depth-essential cohomology is a free module over the polynomial subring Q of the cohomology ring of G generated by s
The depth-essential cohomology is generated as a module over Q by the elements
[ z ]

Action of Automorphisms

The groups of outer automorphisms of G has order 8, and is generated by 3 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 ,
yw + u ,
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 ,
y ,
y2 + x ,
w ,
yw + v ,
y3 + u ,
y4 + y2x + yu + t ,
y4 + y2w + 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

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

CHECKS

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

THE COMPUTATION IS COMPLETE