# Group cohomology of groups of order 16

From Groupprops

This article gives specific information, namely, group cohomology, about a family of groups, namely: groups of order 16.

View group cohomology of group families | View group cohomology of groups of a particular order |View other specific information about groups of order 16

View these in a broader context: group cohomology of groups of prime-fourth order | group cohomology of groups of order 2^n

## Homology groups for trivial group action

FACTS TO CHECK AGAINST(homology group for trivial group action):

First homology group: first homology group for trivial group action equals tensor product with abelianization

Second homology group: formula for second homology group for trivial group action in terms of Schur multiplier and abelianization|Hopf's formula for Schur multiplier

General: universal coefficients theorem for group homology|homology group for trivial group action commutes with direct product in second coordinate|Kunneth formula for group homology

### Over the integers

Group | GAP ID second part | Hall-Senior number | Nilpotency class | (= abelianization) | (= Schur multiplier) | |||
---|---|---|---|---|---|---|---|---|

cyclic group:Z16 | 1 | 5 | 1 | 0 | 0 | |||

direct product of Z4 and Z4 | 2 | 3 | 1 | |||||

SmallGroup(16,3) | 3 | 9 | 2 | |||||

nontrivial semidirect product of Z4 and Z4 | 4 | 10 | 2 | |||||

direct product of Z8 and Z2 | 5 | 4 | 1 | |||||

M16 | 6 | 11 | 2 | 0 | ||||

dihedral group:D16 | 7 | 12 | 3 | |||||

semidihedral group:SD16 | 8 | 13 | 3 | 0 | ||||

generalized quaternion group:Q16 | 9 | 14 | 3 | 0 | 0 | |||

direct product of Z4 and V4 | 10 | 2 | 1 | |||||

direct product of D8 and Z2 | 11 | 6 | 2 | |||||

direct product of Q8 and Z2 | 12 | 7 | 2 | |||||

central product of D8 and Z4 | 13 | 8 | 2 | |||||

elementary abelian group:E16 | 14 | 1 | 1 |

## Second cohomology groups and extensions

### Schur multiplier and Schur covering groups

FACTS TO CHECK AGAINSTfor Schur multiplier of group of prime power orderABELIAN CASE: Schur multiplier of finite abelian group is its exterior squareUPPER BOUNDS: upper bound on size of Schur multiplier for group of prime power order based on prime-base logarithm of order |upper bound on size of Schur multiplier for group of prime power order based on prime-base logarithm of order and exponent of centerLOWER BOUNDS: lower bound on size of Schur multiplier for group of prime power order based on minimum size of generating set