GROUP OF ORDER 32 #45

GROUP #45

The MAGMA library number for G is 44

GrpPC : G of order 32 = 2^5 PC-Relations: G.3^2 = G.5, G.4^2 = G.5, G.2^G.1 = G.2 * G.5, G.3^G.1 = G.3 * G.4, G.4^G.1 = G.4 * G.5, G.4^G.3 = 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 4.
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 [ 4, 4 ] .
The orders of the centralizers of the maximal elementary abelian subgroups are [ 16, 8 ] .
The orders of the normalizers of the maximal elementary abelian subgroups are [ 32, 16 ] .

The cohomology ring of G is the quotient of a polynomial ring in the variables
[ z, y, x, w, v, u ] in degrees [ 1, 1, 1, 5, 5, 8 ] , by the ideal generated by the relations
zx ,
zy2 + x3 ,
y2x3 ,
zw + zv ,
xv ,
x2w + y2v ,
z9y + z2u + v2 ,
y8x2 + y4xw + y5v + x2u + w2 + v2 ,
wv + v2 .

The Hilbert series for the cohomology ring is
t6+ t5+ t2+ t+ 1 / t8 -2t7+ 2t6 -2t5+ 2t4 -2t3+ 2t2 -2t+ 1.
Its numerator factors as ( t6+t5+t2+t+1 ) .
Its denominator factors as ( t-1 )2 ( t2+1 ) ( t4+1 ) .

The Krull dimension of the cohomology ring is 2.
The longest regular sequence consists of the generators u .
A homogeneous set of parameters is the set u , z2 + y2 of degrees [ 8, 2 ] .

The hypercohomolgy spectral sequence has E2 term:

ROW (1) : [] [] [] [ x3 ] [ yx3 ]
ROW (0) : [1] [ x, z, y ] [ yx, zy, y2, x2 ] [ y3, yx2, x3 ] [ yx3 ] [ w, v ] [ xw, zv, yw, yv ] [ zyv, y2v, yxw ] [ y3v ]

Restrictions to Maximal Elementary Abelian Subgroups

Subgroup E #1
Generated by [ G.5, G.2 * G.5 ]

The cohomology ring of E is a polynomial ring in the variables z , y .

The images of the generators of the cohomology of G restricted to E are
0 ,
z ,
0 ,
0 ,
0 ,
z4y4 + y8
in the cohomology of E.

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

Subgroup E #2
Generated by [ G.5, G.1 ]

The cohomology ring of E is a polynomial ring in the variables z , y .

The images of the generators of the cohomology of G restricted to E are
z ,
0 ,
0 ,
z3y2 + zy4 ,
z3y2 + zy4 ,
z4y4 + y8
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 + v .

The nilradical of the cohomology of G is generated by
x , zy , w + v , yv .

It is nilpotent of degree 4.

Restrictions to Maximal Subgroups

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

The images of the generators of the cohomology of G restricted to H are
z ,
y ,
z ,
y2x ,
zw ,
y5x + 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.2, G.4, G.5, G.1 ] .

The Group H 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 Generated by [ G.2, G.2 * G.4, G.1 * G.2 * G.4 ]

of type AlmostExtraSpecial(16) .

The images of the generators of the cohomology of G restricted to H are
x ,
z + y + x ,
0 ,
zy4 + z4x + z3x2 + xw ,
z4x + z3x2 + zyx3 + xw ,
zy7 + z6x2 + z5x3 + 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.2, G.3, 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.3, G.4 ]

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

The images of the generators of the cohomology of G restricted to H are
0 ,
x ,
z ,
z2yx2 + zyx3 + y2x3 + yx4 + zw ,
z2yx2 ,
z2x6 + y2x6 + yx7 + z2x2w + zx3w + x4w + 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 #4 Generated by [ G.3, G.4, G.5, G.1 ] .

The Group H is Isomorphic to the Group of Order 16 Number13 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.3 * $.4, $.3^2 = $.4, $.2^$.1 = $.2 * $.3, $.3^$.2 = $.3 * $.4 Generated by [ G.1 * G.3 * G.4, G.1 ]

of type Semidihedral(16) .

The images of the generators of the cohomology of G restricted to H are
z + y ,
0 ,
z ,
yw ,
zw + yw ,
w2
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.3, G.4, G.5, G.1 ] .

The Group H is Isomorphic to the Group of Order 16 Number13 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.3 * $.4, $.3^2 = $.4, $.2^$.1 = $.2 * $.3, $.3^$.2 = $.3 * $.4 Generated by [ G.1 * G.2 * G.3 * G.4 * G.5, G.1 ]

of type Semidihedral(16) .

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

The Group H is Isomorphic to the Group of Order 16 Number14 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.2^2 = $.3 * $.4, $.3^2 = $.4, $.2^$.1 = $.2 * $.3, $.3^$.1 = $.3 * $.4 Generated by [ G.1 * G.2 * G.3, G.3 ]

of type Quaternion(16) .

The images of the generators of the cohomology of G restricted to H are
y ,
y ,
z + y ,
zx ,
yx ,
x2
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.1 * G.2, G.2 * G.3, G.4, G.5 ] .

The Group H is Isomorphic to the Group of Order 16 Number14 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.2^2 = $.3 * $.4, $.3^2 = $.4, $.2^$.1 = $.2 * $.3, $.3^$.1 = $.3 * $.4 Generated by [ G.2 * G.3, G.1 * G.3 ]

of type Quaternion(16) .

The images of the generators of the cohomology of G restricted to H are
y ,
z ,
z + y ,
zx ,
yx ,
x2
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
yx3 .

It is nilpotent of degree 2.
The annihilator of the Essential Cohomology has dimension 1 .
The essential cohomology is a free module over the polynomial subring Q of the cohomology ring of G generated by u
The essential cohomology is generated as a module over Q by the elements
[] [] [] [ yx3 ]

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 Number6 GrpPC of order 16 = 2^4 PC-Relations: $.3^2 = $.4, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.4

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

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

The images of the generators of the cohomology of Q inflated to G are
z + y ,
y + x ,
x ,
zy + x2
in the cohomology of G.

The kernel of the inflation to G of the cohomology of Q is generated by
zy + y2 + yx + x2 + w ,
yx2 + yw + xw ,
y2xw .

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

The depth-essential cohomology of G is generated as an ideal by
x3 .

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 u
The depth-essential cohomology is generated as a module over Q by the elements
[] [] [ x3 ] [ yx3 ]

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

    G.1
    G.2 * G.5
    G.3 * G.5
    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 ,
y4x + w ,
v ,
y7x + y6x2 + u .

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.2 * G.3 * G.5
    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 ,
y + x ,
x ,
w ,
v ,
y2xw + y3v + u .

CHECKS

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

THE COMPUTATION IS COMPLETE