imitating-kayla.php

<?php

$permutations = [
    ['a', 'c', 'f'],
    ['b', 'd'],
    ['a', 'b', 'd'],
    ['e', 'f'],
];

/*
$permutations = [
    ['a', 'c', 'f'],
    ['b', 'd'],
];
*/

$chars = array_unique(call_user_func_array('array_merge', $permutations));
$chars = array_combine($chars, $chars);

$out = [];
$i = 0;
while (count($chars) > 0) {
    $cycle = [];
    $first = $current = current($chars);

    do {
        unset($chars[$current]);
        foreach ($permutations as $permutation) {
            $current = apply_permutation($permutation, $current);
        }
        $cycle[] = $current;
    } while ($current !== $first);

    $out[] = $cycle;
}

var_dump($out);

function apply_permutation($permutation, $current) {
   foreach ($permutation as $i => $char) {
       if ($char === $current) {
           return $permutation[($i + 1) % count($permutation)];
       }
   }
   return $current;
}