Determine order of $\langle(12345),(2354)\rangle$.
Consider the integers modulo $5$. Let $\alpha$ be the operation $\alpha\colon x\mapsto x+1$. Let $\beta$ be the operation $\beta\colon x\mapsto 2x$.
Then $\alpha,\beta$ generate all affine automorphisms of $\mathbb{Z}/5\mathbb{Z}$, which has size $20$: $$x\mapsto ax+b,$$ with $a\in \{1,2,3,4\}$ and $b\in \{0,1,2,3,4,5\}$.
As pointed out by @JeanMarie, we have shifted the indices by $-1$ here, so $\alpha$ is the permutation $(01234)$ and $\beta$ is the permutation $(1243)$.