Group cohomology of dihedral group:D8

Over the integers
The homology groups over the integers are given as follows:

$$H_q(D_8;\mathbb{Z}) = \left \lbrace \begin{array}{rl} \mathbb{Z}, & q = 0 \\(\mathbb{Z}/2\mathbb{Z})^{(q + 3)/2}, & q \equiv 1 \pmod 4\\ (\mathbb{Z}/2\mathbb{Z})^{(q + 1)/2} \oplus \mathbb{Z}/4\mathbb{Z}, & q \equiv 3 \pmod 4 \\(\mathbb{Z}/2\mathbb{Z})^{q/2}, & q \mbox{ even }, q > 0 \\ \end{array}\right.$$

The first few homology groups are given below:

Over an abelian group
The homology groups over an abelian group $$M$$ are given as follows:

$$H_q(D_8;M) = \left \lbrace \begin{array}{rl} M, & q = 0 \\(M/2M)^{(q + 3)/2} \oplus (\operatorname{Ann}_M(2))^{(q - 1)/2}, & q \equiv 1 \pmod 4\\ (M/2M)^{q/2} \oplus (\operatorname{Ann}_M(2))^{(q + 2)/2}, & q \equiv 2 \pmod 4 \\(M/2M)^{(q + 1)/2} \oplus M/4M \oplus (\operatorname{Ann}_M(2))^{(q - 1)/2}, & q \equiv 3 \pmod 4 \\(M/2M)^{q/2} \oplus (\operatorname{Ann}_M(2))^{q/2} \oplus \operatorname{Ann}_M(4), & q \equiv 0 \pmod 4, q > 0 \\ \end{array}\right.$$ The first few homology groups with coefficients in an abelian group $$M$$ are given below:

Over the integers
The cohomology groups over the integers are given as follows:

$$H^q(D_8;\mathbb{Z}) = \left \lbrace \begin{array}{rl} \mathbb{Z}, & q =0 \\(\mathbb{Z}/2\mathbb{Z})^{(q - 1)/2}, & q \equiv 1 \pmod 4\\ (\mathbb{Z}/2\mathbb{Z})^{(q/2} \oplus \mathbb{Z}/4\mathbb{Z}, & q \equiv 0 \pmod 4, q \ne 0\\(\mathbb{Z}/2\mathbb{Z})^{(q+2)/2}, & q \equiv 2 \pmod 4 \\ \end{array}\right.$$

The first few cohomology groups are given below:

Over an abelian group
The cohomology groups over an abelian group $$M$$ are given as follows:

$$H^q(D_8;M) = \left \lbrace \begin{array}{rl} M, & q = 0 \\(\operatorname{Ann}_M(2))^{(q + 3)/2} \oplus (M/2M)^{(q - 1)/2}, & q \equiv 1 \pmod 4\\ (\operatorname{Ann}_M(2))^{q/2} \oplus (M/2M)^{(q + 2)/2}, & q \equiv 2 \pmod 4 \\(\operatorname{Ann}_M(2))^{(q + 1)/2} \oplus \operatorname{Ann}_M(4) \oplus (M/2M)^{(q - 1)/2}, & q \equiv 3 \pmod 4 \\(\operatorname{Ann}_M(2))^{q/2} \oplus (M/2M)^{q/2} \oplus M/4M, & q \equiv 0 \pmod 4, q > 0 \\ \end{array}\right.$$

The first few cohomology groups with coefficients in an abelian group $$M$$ are:

Schur multiplier
The Schur multiplier, defined as second cohomology group for trivial group action $$H^2(G;\mathbb{C}^\ast)$$, and also as the second homology group $$H_2(G;\mathbb{Z})$$, is cyclic group:Z2.

This has implications for projective representation theory of dihedral group:D8.

Schur covering groups
The three possible Schur covering groups for dihedral group:D8 are: dihedral group:D16, semidihedral group:SD16, and generalized quaternion group:Q16. For more, see second cohomology group for trivial group action of D8 on Z2, where these correspond precisely to the stem extensions.

Computation of integral homology
The homology groups for trivial group action with coefficients in $$\mathbb{Z}$$ can be computed in GAP using the GroupHomology function in the HAP package, which can be loaded by the command LoadPackage("hap"); if it is installed but not loaded. The function outputs the orders of cyclic groups for which the homology or cohomology group is the direct product of these (more technically, it outputs the elementary divisors for the homology or cohomology group that we are trying to compute).

Here are computations of the first few homology groups:

Computation of first homology group
gap> GroupHomology(DihedralGroup(8),1); [ 2, 2 ]

The way this is to be interpreted is that the first homology group (the abelianization) is the direct sum of cyclic groups of the orders listed, so in this case we get that $$H_1(D_8;\mathbb{Z})$$ is $$\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$$, which is the Klein four-group.

Computation of second homology group
gap> GroupHomology(DihedralGroup(8),2); [ 2 ]

This says that the second homology group (the Schur multiplier) is cyclic group:Z2.

Computation of first few homology groups
To compute the first eight homology groups, do:

gap> List([1,2,3,4,5,6,7,8],i->[i,GroupHomology(DihedralGroup(8),i)]); [ [ 1, [ 2, 2 ] ], [ 2, [ 2 ] ], [ 3, [ 2, 2, 4 ] ], [ 4, [ 2, 2 ] ], [ 5, [ 2, 2, 2, 2 ] ], [ 6, [ 2, 2, 2 ] ], [ 7, [ 2, 2, 2, 2, 4 ] ], [ 8, [ 2, 2, 2, 2 ] ] ]