GROUP OF ORDER 8 # 5
GROUP OF ORDER 8 # 5
Quaternion(8)
The MAGMA library number for G is 4
GrpPC : G of order 8 = 2^3
PC-Relations:
G.1^2 = G.3,
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 a unique maximal elementary abelian subgroup which is central in
G and has order 2
The cohomology ring of G is a quotient of the polynomial ring in the
variables [
z,
y,
x
]
in degrees [ 1, 1, 4 ]
by the ideal of relations
[
z^2 + z*y + y^2,
y^3
]
The Hilbert series for the cohomology ring is (-t^2 - t - 1)/(t^3 - t^2 + t -
1)
Its numerator factors as [
]
Its numerator factors as
( t^2 + t + 1
)
Its denominator factors as
( t - 1
)
( t^2 + 1
)
The Krull dimension of cohomology ring is 1
The cohomology ring is Cohen-Macaulay and the longest regular sequence consists
of the generators [
x
]
Restriction to the Maximal Elementary Abelian Subgroup
Generated by [ G.3 ]
The cohomology ring of E is a Polynomial ring of rank 1 over GF(2)
Graded Reverse Lexicographical Order
Variables: z
with generators in
degrees [ 1 ]
and relations
[]
The generators of the cohomology of G restrict to
the elements
[
0,
0,
z^4
]
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
]
This ideal is also the nilradical of the cohomology of G.