By cyclic number, we mean positive integers $N$ such that $kN$ is a permutation (more precisely, a cycling) of the digits of $N$ for each $1\leqslant k<N$. We allow (somewhat necessary) leading zeros, without which the only such number would be $N=7$. For the moment I am thinking of base-$10$ cyclic numbers.
These numbers are formed from the block of digits which repeats (sometimes called the repetend) in the decimal expansion of unit fractions $1/q$ precisely when this repetend is of length $q-1$. This can be made more tangible by the fact that this occurs exactly when $q$ is a prime such that $10^m \not\equiv 1\pmod{q}$ for all $0<m<q-1$.
(for instance the repetend of $1/7$ is of length $6$ and $1/7=0.\overline{142857}$ so $N=142857$ is a cyclic number)
I am trying to figure out how this cycling occurs when we successively multiply $N$ by $k=2,3,4\dots $
There are various ways to talk about this, all a bit awkward it seems, but I will define the function $c_j(n)$ which takes the $j^{\text{th}}$ digit of $n$ from the left and cycles all digits so that this digit is now at the front (i.e. in first position).
(for instance $c_3(1992)=9219$ or $c_2(2019)=0192$)
Then, for any cyclic $N$, there is a bijection $i$ taking each $k\in\{1,\dots ,N-1\}$ to the value $i(k)\in \{1,\dots ,N-1\}$ such that $kN=c_{i(k)}(N)$. In other words, a function which tells us, for each $k$, at which point in the digits of $N$ the cycle $kN$ will begin.
Example, for the cyclic number $N=142857$, with the values for $i$ on the right:
$$\begin{aligned} 1\cdot \color{red}{1}42857 &=\color{blue}{1}42857\qquad & i(1)=1 \\ 2\cdot 14\color{red}{2}857 &=\color{blue}{2}85714 & i(2)=3 \\ 3\cdot 1\color{red}{4}2857 & =\color{blue}{4}28571 & i(3)=2 \\ 4\cdot 1428\color{red}{5}7&=\color{blue}{5}71248 & i(4)=5 \\ 5\cdot 14285\color{red}{7} &= \color{blue}714285 & i(5)=6 \\ 6\cdot 142\color{red}{8}57 &=\color{blue}{8}57142 & i(6)=4\end{aligned}$$
Can much be said about these bijections or any similar device informing us as to how these sequence of cycles operate?
I am having a hard time seeing whether my question is 'artificial', in the sense that progressively multiplying the original number by $1,2, 3,\dots$ may not 'naturally' relate to any sort of meaningful pattern in the resulting numbers, but I thought I would ask anyway to see if anyone knows about this/has comments.