Group cohomology of semidihedral groups

This article gives specific information, namely, group cohomology, about a family of groups, namely: semidihedral group.
View group cohomology of group families | View other specific information about semidihedral group

We describe here the homology and cohomology groups of the semidihedral group of order $2^{n}$, which is obtained as the external semidirect product of a cyclic group of order $2^{n-1}$ and cyclic group:Z2 where the non-identity element acts via multiplication by $2^{n-2}-1$.

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 ring of integers are given as follows:

$H_{q}(SD_{2^{n}};\mathbb {Z} )=\left\lbrace {\begin{array}{rl}\mathbb {Z} ,&q=0\\(\mathbb {Z} /2\mathbb {Z} )^{(q+7)/4},&q\equiv 1{\pmod {4}}\\(\mathbb {Z} /2\mathbb {Z} )^{(q-2)/4},&q\equiv 2{\pmod {4}}\\\mathbb {Z} /2^{n-1}\mathbb {Z} \oplus (\mathbb {Z} /2\mathbb {Z} )^{(q+1)/4},&q\equiv 3{\pmod {4}}\\(\mathbb {Z} /2\mathbb {Z} )^{q/4},&q\equiv 0{\pmod {4}},q>0\\\end{array}}\right.$