[GAP Forum] Small Groups which are not a direct product of a smaller group and a cyclic group

[Jacek M. Holeczek]

Hi,
I am looking for a bullet-proof method of identifying Small Groups that 
cannot be written as a direct product of a smaller group and a cyclic 
group of some size (I do not care about the size of this cyclic group).

Some people use the output of the "StructureDescription(G)" method 
(where "G:=SmallGroup(o,i);", of course) and, as soon as they see 
"C<size> x SomeThing", they assume that the group G is such a "genuine" 
direct product (denoted by the "x") of a "genuine" cyclic group 
"C<size>" and some smaller group described by "SomeThing".

However, the manual explicitly says that the output of this method 
should be used for "educational" purposes only as it really provides a 
partial description of the structure only, especially for orders higher 
than 100.
I think I have even read somewhere that the "C<size>" does not 
necessarily represent a "genuine" cyclic group and that the "x" does not 
necessarily represent a "genuine" direct product operation.

Thanks in advance,
Best regards,
Jacek.