A group is G perfect if G=G', that is if its commutator subgroup is the whole
group. This is the same as to say that G has no non-trivial cyclic
(equivalently abelian, equivalently soluble) quotients.
All non-abelian simple and quasisimple groups are perfect.
A perfect group may have cyclic composition factors,
for example SL2(𝔽5)=C2.A5 is perfect.
Groups of order 1
Groups of order 60
|A5||Alternating group on 5 letters; = SL2(𝔽4) = L2(5) = L2(4) = icosahedron/dodecahedron rotations; 1st non-abelian simple||5||3+||A5||60,5|
Groups of order 120
Groups of order 168
Groups of order 336
Groups of order 360