Editing List of small groups (section)

==Classifying groups of small order==

Small groups of [[prime power]] order ''p''<sup>''n''</sup> are given as follows:
*Order ''p'': The only group is cyclic.
*Order ''p''<sup>2</sup>: There are just two groups, both abelian.
*Order ''p''<sup>3</sup>: There are three abelian groups, and two non-abelian groups. One of the non-abelian groups is the semidirect product of a normal cyclic subgroup of order ''p''<sup>2</sup> by a cyclic group of order ''p''. The other is the quaternion group for {{nowrap|1=''p'' = 2}} and a group of exponent ''p'' for {{nowrap|''p'' > 2}}. 
*Order ''p''<sup>4</sup>: The classification is complicated, and gets much harder as the exponent of ''p'' increases.

Most groups of small order have a Sylow ''p'' subgroup ''P'' with a [[normal p-complement|normal ''p''-complement]] ''N'' for some prime ''p'' dividing the order, so can be classified in terms of the possible primes ''p'', ''p''-groups ''P'', groups ''N'', and actions of ''P'' on ''N''. In some sense this reduces the classification of these groups to the classification of ''p''-groups. Some of the small groups that do not have a normal ''p''-complement include:
*Order 24: The symmetric group S<sub>4</sub>
*Order 48: The binary octahedral group and the product {{nowrap|S<sub>4</sub> × Z<sub>2</sub>}}
*Order 60: The alternating group A<sub>5</sub>.

The smallest order for which it is ''not'' known how many nonisomorphic groups there are is 2048 = 2<sup>11</sup>.<ref>{{cite book|url=https://www.quendi.de/data/papers/EHH2018-small-groups.pdf|first1= Bettina|last1=Eick|first2=Max|last2=Horn|first3=Alexander|last3=Hulpke|title=Constructing groups of Small Order: Recent results and open problems|year=2018|pages=199–211|doi=10.1007/978-3-319-70566-8_8|publisher=Springer|isbn=978-3-319-70566-8}}</ref>