GROUP OF ORDER 8 # 4
GROUP OF ORDER 8 # 4
Dihedral(8)
The MAGMA library number for G is 3
GrpPC : G of order 8 = 2^3
PC-Relations:
G.2^2 = G.3,
G.2^G.1 = G.2 * G.3
The center of G is abelian of type [ 2 ]
The orders of the terms of the lower central series are [ 8, 2, 1 ]
The orders of the terms of the upper central series are [ 1, 2, 8 ]
The order of the Frattini subgroup is 2
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 [ 4, 4 ]
The orders of the centralizers of the maximal elementary abelian subgroups are
[ 4, 4 ]
The orders of the normalizers of the maximal elementary abelian subgroups are [
8, 8 ]
The cohomology ring of G is a quotient of the polynomial ring in the
variables [
z,
y,
x
]
in degrees [ 1, 1, 2 ]
by the ideal of relations
[
z*y + y^2
]
The Hilbert series for the cohomology ring is 1/(t^2 - 2*t + 1)
Its numerator factors as []
Its denominator factors as
( t - 1
)^2
The Krull dimension of cohomology ring is 2
The cohomology ring is Cohen-Macaulay and the longest regular sequence consists
of the generators [
z^2,
x
]
Restrictions to Maximal Elementary Abelian Subgroups
Subgroup E # 1
Generated by [ G.3, G.1 * G.2 * G.3 ]
The cohomology ring of E is a Polynomial ring of rank 2 over GF(2)
Graded Reverse Lexicographical Order
Variables: z, y
with generators in
degrees [ 1, 1 ]
and relations
[]
The generators of the cohomology of G restrict to
the elements
[
z,
z,
z*y + y^2
]
in the cohomology of E
The kernel of the restriction of the cohomology of G to the
cohomology of E is generated by
[
z + y
]
Subgroup E # 2
Generated by [ G.1 * G.3, G.3 ]
The cohomology ring of E is a Polynomial ring of rank 2 over GF(2)
Graded Reverse Lexicographical Order
Variables: z, y
with generators in
degrees [ 1, 1 ]
and relations
[]
The generators of the cohomology of G restrict to
the elements
[
z,
0,
z*y + y^2
]
in the cohomology of E
The kernel of the restriction of the cohomology of G to the
cohomology of E is generated by
[
y
]
The nilradical of the cohomology of G is zero