Integral cohomology of the dihedral group of order ten
= Integral cohomology of the dihedral group of order ten
For the dihedral group $D_{10}=C_5\rtimes C_2$,
$$
H^n(D_{10},\mathbb Z)\cong
\begin{cases}
\mathbb Z,&n=0,\\
0,&n\text{ odd},\\
\mathbb Z/2,&n\equiv2\pmod4,\\
\mathbb Z/10,&n>0\text{ and }n\equiv0\pmod4.
\end{cases}
$$
The Lyndon–Hochschild–Serre spectral sequence proves this from the inversion action of $C_2$ on $H^2(C_5,\mathbb Z)$.