Group cohomology of symmetric group:S3

Classifying space and corresponding chain complex
The homology and cohomology groups are the same as the respective homology and cohomology groups of the configuration space of three unordered points in a countable-dimensional real projective space. For more on the topological perspective, see configuration space of unordered points of a countable-dimensional real vector space on the Topology Wiki.

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

$$\! H_p(S_3;\mathbb{Z}) = \left \lbrace \begin{array}{rl} \mathbb{Z}, & \qquad p = 0 \\ \mathbb{Z}/2\mathbb{Z}, & \qquad p \equiv 1 \pmod 4 \\ \mathbb{Z}/6\mathbb{Z}, & \qquad p \equiv 3 \pmod 4 \\ 0, & \qquad p \ne 0, p \ \operatorname{even} \\\end{array}\right.$$

Over an abelian group
The homology groups with coefficients in an abelian group are as follows:

$$H_p(S_3;M) = \left \lbrace \begin{array}{rl} M, & \qquad p = 0 \\ M/2M, & \qquad p \equiv 1 \pmod 4 \\ \operatorname{Ann}_M(2) & \qquad p \equiv 2 \pmod 4 \\ M/6M, & \qquad p \equiv 3 \pmod 4\\ \operatorname{Ann}_M(6), & \qquad p > 0, p \equiv 0 \pmod 4 \\\end{array}\right.$$

Here, $$\operatorname{Ann}_M(2)$$ denotes the 2-torsion subgroup of $$M$$ and $$\operatorname{Ann}_M(6)$$ denotes the 6-torsion subgroup of $$M$$.

Over the integers
The cohomology groups with coefficients in the ring of integers are as follows:

$$H^p(S_3;\mathbb{Z}) = \left \lbrace \begin{array}{rl} \mathbb{Z}, & \qquad p = 0 \\ \mathbb{Z}/2\mathbb{Z}, & \qquad p \equiv 2 \pmod 4 \\ \mathbb{Z}/6\mathbb{Z}, & \qquad p \ne 0, p \equiv 0 \pmod 4\\ 0, & \qquad p \ \operatorname{odd} \\\end{array}\right.$$

Over an abelian group
The cohomology groups with coefficients in an abelian group $$M$$ are as follows:

$$H^p(S_3;M) = \left \lbrace \begin{array}{rl} M, & \qquad p = 0 \\ \operatorname{Ann}_M(2), & \qquad p \equiv 1 \pmod 4 \\ M/2M & \qquad p \equiv 2 \pmod 4 \\ \operatorname{Ann}_M(6), & \qquad p \equiv 3 \pmod 4\\ M/6M, & \qquad p > 0, p \equiv 0 \pmod 4 \\\end{array}\right.$$

Here $$\operatorname{Ann}_M(2)$$ denotes the 2-torsion subgroup of $$M$$ and $$\operatorname{Ann}_M(6)$$ denotes the 6-torsion subgroup of $$M$$.