Group cohomology of dicyclic groups

We consider here the dicyclic group $$\operatorname{Dic}_{4n}$$ of degree $$n$$ and order $$4n$$.

Over the integers for odd degree
The homology groups with coefficients in the ring of integers are as follows when the degree $$n$$ is odd.

$$\! H_q(\operatorname{Dic}_{4n};\mathbb{Z}) = \left\lbrace \begin{array}{rl} \mathbb{Z}, & \qquad q = 0 \\ \mathbb{Z}/4\mathbb{Z}, & \qquad q \equiv 1 \pmod 4 \\ \mathbb{Z}/4n\mathbb{Z}, & \qquad q \equiv 3 \pmod 4 \\ 0, & \qquad q \ne 0, q \ \operatorname{even}\\\end{array}\right.$$

Over the integers for even degree
The homology groups with coefficients in the ring of integers are as follows when the degree $$n$$ is even.

$$\! H_q(\operatorname{Dic}_{4n};\mathbb{Z}) = \left\lbrace \begin{array}{rl} \mathbb{Z}, & \qquad q = 0 \\ \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}, & \qquad q \equiv 1 \pmod 4 \\ \mathbb{Z}/4n\mathbb{Z}, & \qquad q \equiv 3 \pmod 4 \\ 0, & \qquad q \ne 0, q \ \operatorname{even}\\\end{array}\right.$$