# Group cohomology of semidihedral groups

## Contents

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:

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