Given distinct primes $p,q,r$, how many groups of order $n=pqr$ do we have given that:
I) $q, r = 1\pmod p$ and $r = 1 \pmod q$
II) $q,r=1\pmod p$ but $r \neq 1 \pmod q$
III) $q = 1\pmod p$ and $r = 1\pmod q$ but $r\neq 1\pmod p$
In any given case, there is always the trivial cyclic group $\Bbb Z_{n}$. Since $n$ is squarefree, this is the only abelian group of order $n$. So the interest here is the possible non-abelian groups of order $n$.
In all three cases, since $q=1\pmod p$, we have the non-abelian group given by
$\Bbb Z_{r} \times (\Bbb Z_{q} \rtimes \Bbb Z_{p})$
In case I) and III) there is the group:
$\Bbb Z_{p} \times (\Bbb Z_{r} \rtimes \Bbb Z_{q})$
In cases I) and II), since $qr=1\pmod p$, we have the groups
$(\Bbb Z_{q} \times \Bbb Z_{r}) \rtimes \Bbb Z_{p}$
$\Bbb Z_{q} \times (\Bbb Z_{r} \rtimes \Bbb Z_{p})$
There should be more groups, at least in the first case though, so which ones, if any am I missing?
For example, when $n=903$, there should be $7$ non-isomorphic groups of order $n$, according to this paper.
Trivially, we have $\Bbb Z_{903}$, but there are also the groups
$\Bbb Z_{43} \times (\Bbb Z_{7} \rtimes \Bbb Z_{3})$
$\Bbb Z_{3} \times (\Bbb Z_{43} \rtimes \Bbb Z_{7})$
$(\Bbb Z_{7} \times \Bbb Z_{43}) \rtimes \Bbb Z_{3}$
$\Bbb Z_{7} \times (\Bbb Z_{43} \rtimes \Bbb Z_{3})$
This only accounts for $5$ of the $7$ different non-isomorphic groups. $n$ also follows I) one where $p=3$, $q=7$, and $r=43$. Thanks for any help or hints.
Best Answer
First, there is a group of the form $C_{43}\rtimes C_{21}$, which is the unique index $2$ subgroup of $\mathrm{AGL}(1,43)$. Next, there are actually two non-isomorphic groups of the form $(C_7\times C_{43})\rtimes C_3$. (Roughly speaking, there are two "different" ways an element of order $3$ can act on each of the parts. By choosing the generator, you can choose whichever way on one part, but then on the other part it's determined.)
More generally, in your case I, there are $p+4$ isomorphism classes. See p.6-7 of https://www.math.auckland.ac.nz/~obrien/research/gnu.pdf for a brief outline.