GROUP OF ORDER 32 #38

GROUP #38

The MAGMA library number for G is 30

GrpPC : G of order 32 = 2^5 PC-Relations: G.1^2 = G.4 * G.5, G.2^2 = 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 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 [ 16, 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 ] in degrees [ 1, 1, 1, 2, 3, 3, 4 ] , by the ideal generated by the relations
z2 + zy + zx ,
y2 + zx + yx ,
zyx + zx2 ,
zx3 + yv + xu ,
zv + yu + xu ,
zu + yu + xu ,
yx3w + zxt + yxt + x2t + v2 + vu + u2 ,
yx2u + x3u + ywv + xwu + vu + u2 ,
zyt + yxt + x2t + v2 + u2 .

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

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

The hypercohomolgy spectral sequence has E2 term:

ROW (1) : [] [] [ zy+ zx ]
ROW (0) : [1] [ x, z, y ] [ zx, yx, zy ] [ u, v ] [ yu, xu, xv ] [ yxu ]

Restrictions to Maximal Elementary Abelian Subgroups

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

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 ,
z ,
y2 ,
z2y ,
z2y ,
z4 + 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 + x ,
v + u .

Subgroup E #2
Generated by [ G.5, G.4 * G.5, G.3 ]

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 ,
zy + y2 ,
z2y + zy2 + z2x + zx2 ,
0 ,
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 ,
u .

The nilradical of the cohomology of G is generated by
z

It is nilpotent of degree 3.

Restrictions to Maximal Subgroups

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

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

The images of the generators of the cohomology of G restricted to H are
z + y ,
0 ,
z ,
y2 + x + w + v ,
zx + zv ,
zv + yv ,
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 #2 Generated by [ G.5, G.2 * G.3 * G.5, G.4, G.1 ] .

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 ] .

The images of the generators of the cohomology of G restricted to H are
z + y ,
z ,
z ,
v ,
zx ,
zx + zw + yw + zv + yv ,
y2v + w2 + v2
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 #3 Generated by [ G.2, G.5, G.4, G.1 ] .

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.1 * G.2 ] .

The images of the generators of the cohomology of G restricted to H are
z ,
z + y ,
0 ,
z2 + x ,
zx + yx + zw + yw ,
zx + zw ,
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 #4 Generated by [ G.2, G.5, G.4, G.3 ] .

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

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

The images of the generators of the cohomology of G restricted to H are
0 ,
z + y + x ,
z + y ,
z2 + yx + x2 ,
z3 + zy2 + zx2 + xw ,
z3 + zy2 + zx2 ,
z4 + y4 + z3x + z2yx + y2x2 + 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 #5 Generated by [ G.2, G.1 * G.3 * G.4, G.5, G.4 ] .

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.1 * G.3 * G.4 ] .

The images of the generators of the cohomology of G restricted to H are
y ,
z ,
y ,
z2 + x ,
zw + yw ,
yw ,
z2x + y2w + 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 #6 Generated by [ G.5, G.4, G.3, G.1 * G.2 * 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.2 * G.3, G.1 * G.2 * G.4 * G.5 ] .

The images of the generators of the cohomology of G restricted to H are
z + y ,
z + y ,
z ,
y2 + w ,
zy2 + zx + yw + yv ,
zw + yw + zv + yv ,
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.1 * G.3 * G.4, G.5, G.4, G.1 * G.2 * G.4 * 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
z ,
y ,
z + y ,
zx + x2 ,
zy2 + zyx + y2x + zw ,
y2x + zw ,
y4 + y2w + 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 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 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 ,
zy2x + yx3 + zyw + zxw + t
in the cohomology of G.

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

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 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.4, 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 ,
z ,
z2 + w
in the cohomology of G.

The kernel of the inflation to G of the cohomology of Q is generated by
z2 + zy + yx ,
yx2 .

Maximal Quotient Group Q #3

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.1 * pcy.2, pcy.2 * pcy.4 ] in the quotient group.

The images of the generators of the cohomology of Q inflated to G are
y + x ,
z + y ,
x ,
zy2x + yx3 + zxw + yxw + w2 + zv + t
in the cohomology of G.

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

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

The depth-essential cohomology of G is generated as an ideal by
zy + zx .

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 w , t .
The depth-essential cohomology is generated as a module over Q by the elements
[] [ zy+ zx ]

Action of Automorphisms

The groups of outer automorphisms of G has order 16, 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

The images of the generators of the cohomology of G under the map induced by the automorphism are
z ,
y ,
x ,
z2 + zy + y2 + w ,
y3 + zyx + v ,
y3 + u ,
t .

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 ,
x ,
w ,
v ,
u ,
t .

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 ,
y ,
x ,
w ,
zyx + v ,
zyx + u ,
t .

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

The images of the generators of the cohomology of G under the map induced by the automorphism are
z ,
z + y ,
z + x ,
w ,
yw + xw + v ,
zw + u ,
zyw + yxw + w2 + zv + t .

CHECKS

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

THE COMPUTATION IS COMPLETE