View source for Group cohomology of semidihedral groups

{{group family-specific information|
information type = group cohomology|
group family = semidihedral group|
connective = of}}

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

==Homology groups for trivial group action==

{{homology groups for trivial group action facts to check against}}

===Over the integers===

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

<math>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.</math>