Non-soluble groups

See soluble groups.

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
S₅	Symmetric group on 5 letters; = PGL₂(𝔽₅) = Aut(A₅) = 5-cell symmetries; almost simple	5	4+	S5	120,34
SL₂(𝔽₅)	Special linear group on 𝔽₅²; = C₂ .A₅ = 2I = <2,3,5>	24	2-	SL(2,5)	120,5
C₂×A₅	Direct product of C₂ and A₅; = icosahedron/dodecahedron symmetries	10	3+	C2xA5	120,35

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 180

		d	ρ	Label	ID
C₃×A₅	Direct product of C₃ and A₅; = GL₂(𝔽₄)	15	3	C3xA5	180,19

Groups of order 240

		d	ρ	Label	ID
CSU₂(𝔽₅)	Conformal special unitary group on 𝔽₅²; = C₂ .₂ S₅	48	4-	CSU(2,5)	240,89
A₅⋊C₄	The semidirect product of A₅ and C₄ acting via C₄/C₂=C₂	12	4	A5:C4	240,91
C₄.A₅	The central extension by C₄ of A₅	24	2	C4.A5	240,93
C₂.S₅	2^nd central stem extension by C₂ of S₅	40	4-	C2.S5	240,90
C₂×S₅	Direct product of C₂ and S₅; = O₃(𝔽₅)	10	4+	C2xS5	240,189
C₄×A₅	Direct product of C₄ and A₅	20	3	C4xA5	240,92
C₂²×A₅	Direct product of C₂² and A₅	20		C2^2xA5	240,190
C₂×SL₂(𝔽₅)	Direct product of C₂ and SL₂(𝔽₅)	48		C2xSL(2,5)	240,94

Groups of order 300

		d	ρ	Label	ID
C₅×A₅	Direct product of C₅ and A₅; = U₂(𝔽₄)	25	3	C5xA5	300,22

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
PGL₂(𝔽₇)	Projective linear group on 𝔽₇²; = GL₃(𝔽₂)⋊C₂ = Aut(GL₃(𝔽₂)); almost simple	8	6+	PGL(2,7)	336,208
C₂×GL₃(𝔽₂)	Direct product of C₂ and GL₃(𝔽₂)	14	3	C2xGL(3,2)	336,209

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
ΓL₂(𝔽₄)	Semilinear group on 𝔽₄²; = C₃ ⋊S₅	15	6	GammaL(2,4)	360,120
C₃×S₅	Direct product of C₃ and S₅	15	4	C3xS5	360,119
S₃×A₅	Direct product of S₃ and A₅	15	6+	S3xA5	360,121
C₆×A₅	Direct product of C₆ and A₅	30	3	C6xA5	360,122
C₃×SL₂(𝔽₅)	Direct product of C₃ and SL₂(𝔽₅)	72	2	C3xSL(2,5)	360,51

Groups of order 420

		d	ρ	Label	ID
C₇×A₅	Direct product of C₇ and A₅	35	3	C7xA5	420,13

Groups of order 480

		d	ρ	Label	ID
GL₂(𝔽₅)	General linear group on 𝔽₅²; = SL₂(𝔽₅)⋊₁ C₄ = Aut(C₅²)	24	4	GL(2,5)	480,218
C₄⋊S₅	The semidirect product of C₄ and S₅ acting via S₅/A₅=C₂	20	6	C4:S5	480,944
C₂²⋊S₅	The semidirect product of C₂² and S₅ acting via S₅/A₅=C₂	20	6+	C2^2:S5	480,951
A₅⋊Q₈	The semidirect product of A₅ and Q₈ acting via Q₈/C₄=C₂	24	6	A5:Q8	480,945
A₅⋊C₈	The semidirect product of A₅ and C₈ acting via C₈/C₄=C₂	40	4	A5:C8	480,217
C₄.₃S₅	3^rd non-split extension by C₄ of S₅ acting via S₅/A₅=C₂	40	4	C4.3S5	480,948
C₈.A₅	The central extension by C₈ of A₅	48	2	C8.A5	480,221
D₄.A₅	The non-split extension by D₄ of A₅ acting through Inn(D₄)	48	4-	D4.A5	480,957
Q₈.A₅	The non-split extension by Q₈ of A₅ acting through Inn(Q₈)	48	4+	Q8.A5	480,959
C₄.₆S₅	3^rd central extension by C₄ of S₅	48	4	C4.6S5	480,946
C₄.S₅	2^nd non-split extension by C₄ of S₅ acting via S₅/A₅=C₂	48	4	C4.S5	480,947
C₂².S₅	The non-split extension by C₂² of S₅ acting via S₅/A₅=C₂	48	4-	C2^2.S5	480,953
C₂².₂S₅	1^st central extension by C₂² of S₅	96		C2^2.2S5	480,219
C₄×S₅	Direct product of C₄ and S₅; = CO₃(𝔽₅)	20	4	C4xS5	480,943
D₄×A₅	Direct product of D₄ and A₅	20	6+	D4xA5	480,956
C₂²×S₅	Direct product of C₂² and S₅	20		C2^2xS5	480,1186
C₈×A₅	Direct product of C₈ and A₅	40	3	C8xA5	480,220
Q₈×A₅	Direct product of Q₈ and A₅	40	6-	Q8xA5	480,958
C₂³×A₅	Direct product of C₂³ and A₅	40		C2^3xA5	480,1187
C₄×SL₂(𝔽₅)	Direct product of C₄ and SL₂(𝔽₅)	96		C4xSL(2,5)	480,222
C₂×CSU₂(𝔽₅)	Direct product of C₂ and CSU₂(𝔽₅)	96		C2xCSU(2,5)	480,949
C₂²×SL₂(𝔽₅)	Direct product of C₂² and SL₂(𝔽₅)	96		C2^2xSL(2,5)	480,960
C₂×A₅⋊C₄	Direct product of C₂ and A₅⋊C₄	24		C2xA5:C4	480,952
C₂×C₄×A₅	Direct product of C₂×C₄ and A₅	40		C2xC4xA5	480,954
C₂×C₄.A₅	Direct product of C₂ and C₄.A₅	48		C2xC4.A5	480,955
C₂×C₂.S₅	Direct product of C₂ and C₂.S₅	80		C2xC2.S5	480,950

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