Facto-RLE
Task
Given a non-empty string containing non-numeric printable ASCII characters, compute its facto-run-length encoded version.
Definition
Let \$S\$ be a non-empty string of length \$l\in\mathbb{N}^+\$ containing non-numeric printable ASCII characters. For each positive factor \$n|l\$ one can represent \$S\$ as a \$n\times(l/n)\$ matrix of characters. Let \$M^n\$ denote these matrices.
For a given \$n\$ and \$j\in\{1,\dots,l/n\}\$, let \$C^n_j\$ be the string representing the \$j\$-th column of \$M^n\$. Let \$E^n_j := \text{RLE}(C^n_j)\$ denote the array of this string's run-length encoded version.
Let \$R^n := \text{CNCT}(\text{ZIP}(E^n_1,\dots,E^n_{l/n}))\$ denote the string representation of all \$E^n_j\$ zipped together.
The facto-run-length encoding of \$S\$ is defined as the string array \$\text{FRLE}(S):=[R^n:n|l]\$.
For a family of \$k\in\mathbb{N}^+\$ arrays \$A^i\$ with respective lengths \$l^i, i\in\{1,\dots,k\}\$ and elements \$A^i_j,j\in\{1,\dots,l^i\}\$, let \$Z_j:=[A^i_j:i\in\{1,\dots,k\}\land j\leq l_i]\$ denote the array of elements with index \$j\$, if present. Furthermore, define \$\text{ZIP}(A^1,\dots,A^k) := Z_1\Vert\dots\Vert Z_{\max\{l_i\}}\$ as the concatenated array of all \$Z_j\$.
For an array of strings \$A\$ of length \$j\$, define \$\text{CNCT(A)}:=A_1\Vert \dots\Vert A_j\$ as the concatenation of all strings of \$A\$.
Example
Let \$S:=\text{"Hello world!"}\$, therefore \$l=12\$ with positive factors \$\{1,2,3,4,6,12\}\$.
$$
M^2=\begin{pmatrix}
\text{H}&\text{e}&\text{l}&\text{l}&\text{o}&\text{ }\\
\text{w}&\text{o}&\text{r}&\text{l}&\text{d}&\text{!}
\end{pmatrix},
\\C^2_1=\text{"Hw"},
C^2_2=\text{"eo"},
C^2_3=\text{"lr"},
\\C^2_4=\text{"ll"},
C^2_5=\text{"od"},
C^2_6=\text{" !"},
\\E^2_1=[\text{"H"},\text{"w"}],
E^2_2=[\text{"e"},\text{"o"}],
E^2_3=[\text{"l"},\text{"r"}],
\\E^2_4=[\text{"l2"}],
E^2_5=[\text{"o"},\text{"d"}],
E^2_6=[\text{" "},\text{"!"}],
\\R^2=\text{"Hell2o word!"}
$$
$$
M^4=\begin{pmatrix}
\text{H}&\text{e}&\text{l}\\
\text{l}&\text{o}&\text{ }\\
\text{w}&\text{o}&\text{r}\\
\text{l}&\text{d}&\text{!}\\
\end{pmatrix},
\\C^4_1=\text{"Hlwl"},
C^4_2=\text{"eood"},
C^4_3=\text{"l r!"},
\\E^4_1=[\text{"H"}, \text{"l"}, \text{"w"}, \text{"l"}],
E^4_2=[\text{"e"}, \text{"o2"}, \text{"d"}],
E^4_1=[\text{"l"}, \text{" "}, \text{"r"}, \text{"!"}],\\
R^4=\text{STR}([\text{"H"},\text{"e"},\text{"l"},\text{"l"},\text{"o2"},\text{" "},\text{"w"},\text{"d"},\text{"r"},\text{"l"},\text{"!"}])=\text{"Hello2 wdrl!"}
$$
$$
M^{12}=\begin{pmatrix}
\text{H}\\
\text{e}\\
\text{l}\\
\text{l}\\
\text{o}\\
\text{ }\\
\text{w}\\
\text{o}\\
\text{r}\\
\text{l}\\
\text{d}\\
\text{!}\\
\end{pmatrix},
\\C^{12}_1=\text{"Hello world!"},
\\E^{12}_1=[\text{"H"},\text{"e"},\text{"l2"},\text{"o"},\text{" "},\text{"w"},\text{"o"},\text{"r"},\text{"l"},\text{"d"},\text{"!"}],
\\R^{12}=\text{"Hel2o world!"}
$$
Therefore the following follows.
$$
\text{FRLE}(\text{"Hello world!"}) = [\dots, \text{"Hell2o word!"}, \dots, \text{"Hello2 wdrl!"}, \dots, \text{"Hel2o world!"}]
$$
