I am looking for a reference for a complete list of 3-cocycle representatives for $H^3(D_{2n},\mathbb{C}^\times)$, where $$ D_{2n}=\langle a, b\mid a^2=b^2=(ab)^n=e\rangle $$ is the dihedral group of order $2n$.

Here's what I believe so far, but I do not know much group cohomology.

First, I believe there are isomorphisms $$ H^3(D_{2n},\mathbb{C}^\times)\cong H^4(D_{2n},\mathbb{Z})\cong H_3(D_{2n},\mathbb{Z}). $$ I calculated these groups in GAP for $2\leq n\leq 10$, and I got the following formula: $$ H_3(D_{2n},\mathbb{Z}) \cong \begin{cases} \mathbb{Z}/2 \oplus \mathbb{Z}/2 \oplus \mathbb{Z}/n & \text{if $n$ is even}\\ \mathbb{Z}/2 \oplus \mathbb{Z}/n & \text{if $n$ is odd.} \end{cases} $$ Are these formulas correct? Is there a reference for this, or does it follow by some spectral sequence calculation? Why is there an extra factor of $\mathbb{Z}/2$ when $n$ is even?

Second, I believe I have a formula for 3-cocycles representing the $\mathbb{Z}/n$ factor, which is the factor I most care about. To give this formula, I use the following notation. First, represent the elements of $D_{2n}$ by the alternating words in $a,b$ with length at most $2n-1$ which always start with $a$. For such a word $w$, let $|w|$ be its length.

If $\zeta_n=\exp(2\pi i / n)$, then for $k=0,\dots, n-1$, I believe the maps $$ \lambda_k(x,y,z) = \begin{cases} (\zeta_n^k)^{(-1)^{|x|} \lfloor (|x|+1)/2\rfloor } & \text{if $|y|$ is even and $|y|+|z|\geq 2n$}\\ (\zeta_n^k)^{(-1)^{|x|+1} \lfloor (|x|+1)/2\rfloor } & \text{if $|y|$ is odd and $|z|>|y|$}\\ 1 & \text{else} \end{cases} $$ give a complete list of normalized 3-cocycle representatives for the $\mathbb{Z}/n$ factor of $H^3(D_{2n},\mathbb{C}^\times)$. Certainly the $\lambda_k$ act like $\mathbb{Z}/n$ under multiplication, but is there an easy way to see they are not cohomologous? Is there an easier way to express these formulas, perhaps by choosing a different presentation of $D_{2n}$?

Finally, the formulas above yield the trivial 3-cocycle when restricted to the two copies of $\mathbb{Z}/2$ generated by $a$ and $b$. To see this, just note $\lambda_k$ is normalized, $\lambda_k(a,a,a) = 1$, and if $w=(ab)^{n-1}a$, which is equal to $b$, then $\lambda_k(w,w,w) = 1$. (In both cases, $|y|$ is odd and $|y|=|z|$.) I believe any 3-cocycle representatives for the other cohomology classes should not restrict trivially to at least one of these copies of $\mathbb{Z}/2$.

Partial answers would be greatly appreciated as well!