Classification of groups of order a product of three distinct primes

This page deals with the classification of groups of order $n=pqr$ a product of three distinct primes $p$, $q$, $r$. Say $p<q<r$.

Classification

(Some, but not all possible cases, are listed here, for now. If you can add more, please do)

Note that some of these cases may overlap for certain $n$. They give the same results, perhaps with different expressions for the groups, but they are the same up to isomorphism.

Case 1: one of the primes is even

Further information: Classification of groups of order two times a product of two distinct odd primes

Say instead now that $n=2pq$, $p<q$.

In all cases, we have the cyclic group ${\mathbb{Z}}_{2pq}$, the dihedral group $D_{2pq}$, and the direct products ${\mathbb{Z}}_q\times D_{2p}$ and ${\mathbb{Z}}_p\times D_{2q}$.

Furthermore, if $q\equiv 1modp$, then there are two semidirect products, $({\mathbb{Z}}_q⋊{\mathbb{Z}}_p)\times {\mathbb{Z}}_2$ and ${\mathbb{Z}}_q⋊{\mathbb{Z}}_{2p}$.

Thus, there are 4 groups in total if $q\equiv ̸1modp$, otherwise there are 6.

Case 2, q does not divide r-1

Case 2.1: p divides neither q-1 nor r-1, q does not divide r-1

If $p\nmid q-1$, $p\nmid r-1$, then the only group of this order is ${\mathbb{Z}}_{pqr}$, the cyclic group.

Case 2.2: p divides exactly one of q-1, r-1, q does not divide r-1

If $p\nmid q-1$, $p\mid r-1$ or if $p\mid q-1$, $p\nmid r-1$, then there are $2$ groups of this order, ${\mathbb{Z}}_{pqr}$, the cyclic group, and a semidirect product ${\mathbb{Z}}_p⋊({\mathbb{Z}}_q\times {\mathbb{Z}}_r)$.

Case 2.3: p divides both of q-1, r-1, q does not divide r-1

If $p\mid q-1$, $p\mid r-1$, then there are $p+2$ groups of this order, ${\mathbb{Z}}_{pqr}$, the cyclic group, and $p+1$ non-isomorphic semidirect products of the form ${\mathbb{Z}}_p⋊({\mathbb{Z}}_q\times {\mathbb{Z}}_r)$.

Other cases

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See also

• Classification of groups of order a product of two distinct primes
• Classification of groups of order a product of four distinct primes
• Hölder's formula for the number of groups of squarefree order up to isomorphism

References

Classification of some groups of order pqr, Adam Burley