GROUP OF ORDER 16 #13

GROUP #13

Semidihedral(16)

The MAGMA library number for G is 8

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

The center of G is abelian of type [ 2 ] .
The orders of the terms of the lower central series are [ 16, 4, 2, 1 ] .
The orders of the terms of the upper central series are [ 1, 2, 4, 16 ] .
The order of the Frattini subgroup is 4.
The exponent of G is 8.
G has one conjugate class of maximal elementary abelian subgroups. Any element of the class has order 4. Its centralizer has order 4 and its normalizer has order 8.

The cohomology ring of G is the quotient of a polynomial ring in the variables
[ z, y, x, w ] in degrees [ 1, 1, 3, 4 ] , by the ideal generated by the relations
z2 + zy ,
zy2 ,
zx ,
zyw + y2w + x2 .

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

The Krull dimension of the cohomology ring is 2.
The longest regular sequence consists of the generators w .
A homogeneous set of parameters is the set w , y2 of degrees [ 4, 2 ] .

The hypercohomolgy spectral sequence has E2 term:

ROW (1) : [] [ z ] [ zy ]
ROW (0) : [1] [ y, z ] [ zy ] [ x ] [ yx ]

Restriction to the Maximal Elementary Abelian Subgroup

Generated by [ G.4, G.2 ] 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 ,
z2y + zy2 ,
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 .

This ideal is also the nilradical of the cohomology of G.

It is nilpotent of degree 3.

Restrictions to Maximal Subgroups

Maximal Subgroup H #1

The group H is abelian of type [ 8 ]

The cohomology ring of H is a the quotient of a polynomial ring in the variables
[ z, y ] , in degrees [ 1, 2 ] , by the ideal of relations
z2 .

The images of the generators of the cohomology of G restricted to H are
z ,
0 ,
zy ,
y2
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

The Group H is Isomorphic to the Group of Order 8 Number4 GrpPC of order 8 = 2^3 PC-Relations: $.2^$.1 = $.2 * $.3 Generated by [ G.2 * G.3 * G.4, G.2 ]

of type Dihedral(8) .

The images of the generators of the cohomology of G restricted to H are
0 ,
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 .

Maximal Subgroup H #3

The Group H is Isomorphic to the Group of Order 8 Number5 GrpPC of order 8 = 2^3 PC-Relations: $.1^2 = $.3, $.2^2 = $.3, $.2^$.1 = $.2 * $.3 Generated by [ G.3, G.1 * G.2 * G.3 ]

of type Quaternion(8) .

The images of the generators of the cohomology of G restricted to H are
y ,
y ,
zy2 ,
x
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 8 Number4 GrpPC of order 8 = 2^3 PC-Relations: $.2^$.1 = $.2 * $.3

of type Dihedral(8) .

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

The images of the generators of the cohomology of Q inflated to G are
z ,
z + y ,
z2
in the cohomology of G.

The kernel of the inflation to G of the cohomology of Q is generated by
z2 + x ,
zx .

Action of Automorphisms

The groups of outer automorphisms of G has order 2, and is generated by 1 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 .

CHECKS

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

THE COMPUTATION IS COMPLETE