Group cohomology of dihedral group:D16

This article gives specific information, namely, group cohomology, about a particular group, namely: dihedral group:D16.
View group cohomology of particular groups | View other specific information about dihedral group:D16

This article describes the homology and cohomology group of dihedral group:D16, the dihedral group of order 16 and degree 8.

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

The homology groups with coefficients in the integers are given as follows:

$H_{q}(D_{16};\mathbb {Z} )=\left\lbrace {\begin{array}{rl}\mathbb {Z} ,&q=0\\(\mathbb {Z} /2\mathbb {Z} )^{(q+3)/2},&q\equiv 1{\pmod {4}}\\(\mathbb {Z} /2\mathbb {Z} )^{(q+1)/2}\oplus \mathbb {Z} /8\mathbb {Z} ,&q\equiv 3{\pmod {4}}\\(\mathbb {Z} /2\mathbb {Z} )^{q/2},&q{\mbox{ even }},q>0\\\end{array}}\right.$

The first few homology groups are as follows:

$q$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$
$H_{q}$	$\mathbb {Z}$	$\mathbb {Z} /2\mathbb {Z} \oplus \mathbb {Z} /2\mathbb {Z}$	$\mathbb {Z} /2\mathbb {Z}$	$\mathbb {Z} /2\mathbb {Z} \oplus \mathbb {Z} /2\mathbb {Z} \oplus \mathbb {Z} /8\mathbb {Z}$	$\mathbb {Z} /2\mathbb {Z} \oplus \mathbb {Z} /2\mathbb {Z}$	$(\mathbb {Z} /2\mathbb {Z} )^{4}$	$(\mathbb {Z} /2\mathbb {Z} )^{3}$	$(\mathbb {Z} /2\mathbb {Z} )^{4}\oplus \mathbb {Z} /8\mathbb {Z}$	$(\mathbb {Z} /2\mathbb {Z} )^{4}$