A Hajós group is a group for which all factorizations of the form (say) 
have 
or 
periodic, where the period is a divisor of 
. Hajós groups arose after solution of Minkowski's conjecture
on about tiling space with nonoverlapping cuboids. The classification of Hajós
finite Abelian groups was achieved by Sands in the 1980s.
For example, 
(mod 12), so while the first factor is acyclic, the second factor has period three.
Since this turns out to be the case for all tilings of 
, it is a Hajós group. The smallest case where
this is false is 
,
followed by 
.
The cyclic group of order 
is a Hajós group if 
is of the form 
,
, 
, 
, 
, or 
, where 
, 
,
, and 
are distinct primes and 
and 
arbitrary integers. Non-Hajós groups therefore have orders 72, 108, 120, 144,
168, 180, 200, 216, ... (OEIS A102562).
