# Second cohomology group for trivial group action of D8 on V4

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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 problemArithmetic 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 for the trivial group action congruence classes of central extensions with the specified subgroup and quotient group .

This descends to a correspondence:

Orbits for the group action of on 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 | 2 | 2 | 4 | ||

? | direct product of SmallGroup(16,3) and Z2 | 22 | No | No | 2 | 2 | 3 | |||

? | direct product of SmallGroup(16,4) and Z2 | 23 | No | No | 2 | 2 | 3 | |||

? | SmallGroup(32,2) | 2 | No | ? | 2 | 2 | 2 | |||

? | SmallGroup(32,9) | 9 | No | ? | 3 | 2 | 2 | |||

? | SmallGroup(32,10) | 10 | No | ? | 3 | 2 | 2 | |||

? | semidirect product of Z8 and Z4 of semidihedral type | 13 | No | ? | 3 | 2 | 2 | |||

? | semidirect product of Z8 and Z4 of dihedral type | 14 | No | ? | 3 | 2 | 2 | |||

? | direct product of D16 and Z2 | 39 | No | No | 3 | 2 | 3 | |||

? | direct product of SD16 and Z2 | 40 | No | No | 3 | 2 | 3 | |||

? | direct product of Q16 and Z2 | 41 | No | No | 3 | 2 | 3 | |||

Total (11 rows) | 64 | -- | -- | -- | -- | -- | -- | -- | -- |