GROUP OF ORDER 32 #31

GROUP #31

The MAGMA library number for G is 11

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

The center of G is abelian of type [ 4 ] .
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, 4, 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 [ 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, 2, 2, 3, 4 ] , by the ideal generated by the relations
z2 ,
zy ,
zx ,
zw + yw ,
x2 ,
xw + yv ,
zv + yv ,
xv ,
v2 .

The Hilbert series for the cohomology ring is
1 / t2 -2t+ 1.
Its denominator factors as ( t-1 )2 .

The Krull dimension of the cohomology ring is 2.
The cohomology ring is Cohen-Macaulay and the longest regular sequence consists of the generators
u , y2 + w .

Restrictions to Maximal Elementary Abelian Subgroups

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

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 ,
0 ,
0 ,
z2 ,
0 ,
z2y2 + y4
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 .

Subgroup E #2
Generated by [ 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 ,
z2y2 + y4
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 .

The nilradical of the cohomology of G is generated by
z , x , v .

It is nilpotent of degree 3.

Restrictions to Maximal Subgroups

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

The images of the generators of the cohomology of G restricted to H are
z ,
0 ,
zy ,
zy + y2 ,
x ,
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.2, G.4, G.3, G.5 ] .

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

of type AlmostExtraSpecial(16) .

The images of the generators of the cohomology of G restricted to H are
0 ,
x ,
z2 + zx ,
z2 + y2 + x2 ,
z3 + zy2 + zx2 ,
w
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.4, G.1 * G.2 * G.4 * G.5, G.3, G.5 ] .

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
z ,
z ,
zy ,
x ,
zx + yx + zw ,
xw + w2
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 ,
w
in the cohomology of G.

The kernel of the inflation to G of the cohomology of Q is generated by
y2 + w + v ,
yv .

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 .

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 .

CHECKS

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

THE COMPUTATION IS COMPLETE