Second cohomology group for trivial group action of D8 on V4

From Groupprops
Jump to: navigation, search
This article gives information about the second cohomology group for trivial group action (i.e., the second cohomology group with trivial action) of the group dihedral group:D8 on Klein four-group. The elements of this classify the group extensions with Klein four-group in the center and dihedral group:D8 the corresponding quotient group. Specifically, these are precisely the central extensions with the given base group and acting group.
The value of this cohomology group is elementary abelian group:E64.
Get more specific information about dihedral group:D8 |Get more specific information about Klein four-group|View other constructions whose value is elementary abelian group:E64

Description of the group

This article describes the second cohomology group for trivial group action of dihedral group:D8 on the Klein four-group. The value of the cohomology group is elementary abelian group:E64, i.e., it has order 64.

Elements

FACTS TO CHECK AGAINST (second cohomology group for trivial group action):
Background reading on relationship with extension groups: Group extension problem
Arithmetic functions of extension group:
order (thus all extension groups have the same order): order of extension group is product of order of normal subgroup and quotient group
nilpotency class: nilpotency class of extension group is between nilpotency class of quotient group and one more for central extension
derived length: derived length of extension group is bounded by sum of derived length of normal subgroup and quotient group
minimum size of generating set: minimum size of generating set of extension group is bounded by sum of minimum size of generating set of normal subgroup and quotient group|minimum size of generating set of quotient group is at most minimum size of generating set of group
WHAT'S THE TABLE BELOW?: Recall that there is a correspondence:
Elements of the group H^2(G;A) for the trivial group action \leftrightarrow congruence classes of central extensions with the specified subgroup A and quotient group G.
This descends to a correspondence:
Orbits for the group action of \operatorname{Aut}(G) \times \operatorname{Aut}(A) on H^2(G;A) \leftrightarrow pseudo-congruence classes of central extensions.
The table below breaks down the second cohomology group as a union of these orbits, with (as a general rule) each row describing one orbit, i.e., one "cohomology class type", aka one "pseudo-congruence class" of central extensions. The number of rows is the number of pseudo-congruence classes of central extensions.


Cohomology class type Number of cohomology classes Corresponding group extension Second part of GAP ID (order is 32) Stem extension? Base characteristic in whole group? Hall-Senior family (equivalence class up to being isoclinic) Nilpotency class of whole group (at least 2, at most 3) Derived length of whole group (always exactly 2) Minimum size of generating set of whole group (at least 2, at most 4) Subgroup information on base in whole group
trivial 1 direct product of D8 and V4 46 No No \Gamma_2 2 2 4
 ? direct product of SmallGroup(16,3) and Z2 22 No No \Gamma_2 2 2 3
 ? direct product of SmallGroup(16,4) and Z2 23 No No \Gamma_2 2 2 3
 ? SmallGroup(32,2) 2 No  ? \Gamma_2 2 2 2
 ? SmallGroup(32,9) 9 No  ? \Gamma_3 3 2 2
 ? SmallGroup(32,10) 10 No  ? \Gamma_3 3 2 2
 ? semidirect product of Z8 and Z4 of semidihedral type 13 No  ? \Gamma_3 3 2 2
 ? semidirect product of Z8 and Z4 of dihedral type 14 No  ? \Gamma_3 3 2 2
 ? direct product of D16 and Z2 39 No No \Gamma_3 3 2 3
 ? direct product of SD16 and Z2 40 No No \Gamma_3 3 2 3
 ? direct product of Q16 and Z2 41 No No \Gamma_3 3 2 3
Total (11 rows) 64 -- -- -- -- \Gamma_2, \Gamma_3 -- -- -- --