Groups of order 12

Statistics at a glance
The number 12 has prime factorization $$12 = 2^2 \cdot 3$$.

2-Sylow subgroups
Here is the occurrence summary:

Note that the number of 2-Sylow subgroups is either 1 or 3. The former happens if and only if we have a normal Sylow subgroup for the prime 2. The latter happens if and only if we have a self-normalizing Sylow subgroup for the prime 2.

The only non-abelian (and also the only non-nilpotent) example where there is a normal Sylow subgroup is the case of alternating group:A4.

GAP implementation
gap> SmallGroupsInformation(12);

There are 5 groups of order 12. 1 is of type 6.2. 2 is of type c12. 3 is of type A4. 4 is of type D12. 5 is of type 2^2x3.

The groups whose order factorises in at most 3 primes have been classified by O. Hoelder. This classification is used in the SmallGroups library.

This size belongs to layer 1 of the SmallGroups library. IdSmallGroup is available for this size.