Groups of order 69

This article gives information about, and links to more details on, groups of order 69
See pages on algebraic structures of order 69 | See pages on groups of a particular order

Up to isomorphism, there is a unique group of order 69, namely cyclic group:Z69, which is also the external direct product of cyclic group:Z3 and cyclic group:Z23.

The fact of uniqueness follows from the classification of groups of order a product of two distinct primes. Since $69=3\cdot 23$ and $3$ does not divide $(23-1)$, the number $69$ falls in the one isomorphism class case.

Another way of viewing this is that $69$ is a cyclicity-forcing number, i.e., any group of order 69 is cyclic. See the classification of cyclicity-forcing numbers to see the necessary and sufficient condition for a natural number to be cyclicity-forcing.