# Perfect groups

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

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C1Trivial group11+C11,1

### Groups of order 60

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A5Alternating group on 5 letters; = SL2(𝔽4) = L2(5) = L2(4) = icosahedron/dodecahedron rotations; 1st non-abelian simple53+A560,5

### Groups of order 120

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SL2(𝔽5)Special linear group on 𝔽52; = C2.A5 = 2I = <2,3,5>242-SL(2,5)120,5

### Groups of order 168

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GL3(𝔽2)General linear group on 𝔽23; = Aut(C23) = L3(2) = L2(7); 2nd non-abelian simple73GL(3,2)168,42

### Groups of order 336

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SL2(𝔽7)Special linear group on 𝔽72; = C2.GL3(𝔽2)164SL(2,7)336,114

### Groups of order 360

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A6Alternating group on 6 letters; = PSL2(𝔽9) = L2(9); 3rd non-abelian simple65+A6360,118
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