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.
