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

		d	ρ	Label	ID
C₁	Trivial group	1	1+	C1	1,1

Groups of order 60

		d	ρ	Label	ID
A₅	Alternating group on 5 letters; = SL₂(𝔽₄) = L₂(5) = L₂(4) = icosahedron/dodecahedron rotations; 1^st non-abelian simple	5	3+	A5	60,5

Groups of order 120

		d	ρ	Label	ID
SL₂(𝔽₅)	Special linear group on 𝔽₅²; = C₂ .A₅ = 2I = <2,3,5>	24	2-	SL(2,5)	120,5

Groups of order 168

		d	ρ	Label	ID
GL₃(𝔽₂)	General linear group on 𝔽₂³; = Aut(C₂³) = L₃(2) = L₂(7); 2^nd non-abelian simple	7	3	GL(3,2)	168,42

Groups of order 336

		d	ρ	Label	ID
SL₂(𝔽₇)	Special linear group on 𝔽₇²; = C₂ .GL₃(𝔽₂)	16	4	SL(2,7)	336,114

Groups of order 360

		d	ρ	Label	ID
A₆	Alternating group on 6 letters; = PSL₂(𝔽₉) = L₂(9); 3^rd non-abelian simple	6	5+	A6	360,118

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