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The fact that every such group has the structure of the form that I've explained in my answer, is due to Zassenhaus (hence the name Zassenhaus metacyclic groups or $Z$-groups for short). The statement that for every divisor of $|G|$ there is a corresponding subgroup, follows easily from this description (write this divisor as $m'n'$ with $m' \mid m$ and $n' \mid n$, and take $r' = r^{n / n'}$). Alternatively, in your specific case you can simply invoke Hall's theorem saying that a solvable group $G$ has a Hall $\pi$-subgroup for every set $\pi$ of primes.
The relations are: $\gcd(m,n) = 1$, $\gcd(m, r-1) = 1$ and $r^n \equiv 1 \pmod{m}$. A possible reference is Huppert's "Endliche Gruppen I", Chapter IV, Satz 2.11 (p. 420).
The local complementation w.r.t. a given vertex v is the operation that changes the subgraph induced on the neighbors of v (not including v) into its complement. In particular, it is an involution, so the "local complementation group" is a transitive group generated by involutions.
By the way, the "$b$" in my previous comment is really the "$b$" from the comment below my question, so $q=a/b$ with $a,b$ coprime. So only looking at $n=bc$ with $b,c$ coprime really is a restriction.
@Francois: What I'm doing at the moment is a combination of recursively writing $n = bc$ with $b,c$ coprime (maybe the coprimeness is too restrictive, but it makes the computation easier), together with a fast lookup-table containing the values for $\delta(n)$ for $n$ going from $1$ to $10^7$. But I find this somehow rather down-to-the-earth, and I was wondering whether there exist more advanced methods instead.
Thanks a lot! Are there more recent developments since Pomerance's paper (which is from 1977)? Do you know of any kind of efficient algorithms to try to compute, for example, the smallest element in $\delta^{-1}(q)$ for given $q$ (up to some given upper bound)?
Yes: if $A$ is a $\pi$-group (where $\pi$ is a set of primes), then the fact that $A \cap F(G) > 1$ gives $A \cap O_\pi(G) > 1$. Now proceed as in the previous problem with $F(G)$ replaced by $O_\pi(G)$.