= Groups of order seventy
{title2=$C_{70},\ D_{70},\ D_{14}\times C_5,\ D_{10}\times C_7$}
These are all the <groups> of <group order> $70$ up to <group isomorphism>, with $D_n$ denoting a <dihedral group> of order $n$. The <Sylow theorems> make the <Sylow subgroups> of orders five and seven unique: the possible counts divide fourteen and ten, respectively, and only one has the required residue. These <normal subgroups of coprime order commute>, giving a <normal subgroup> $N\cong C_{35}$. <Cauchy's theorem for finite groups> provides an element of order two outside $N$, so $G=N\rtimes C_2$.
The <automorphism of a finite cyclic group> defining this <semidirect product> is a <unit> $a$ modulo $35$ with $a^2\equiv1$. The <Chinese remainder theorem> makes its restrictions modulo five and seven independently $+1$ or $-1$. The resulting exponents are $1,34,6,29$, giving the four displayed <groups> in order. Their <center of a group> orders are $70,1,5,7$, so no two are isomorphic. Every case exists as the specified <semidirect product>.
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