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C₃×#704 #6755

order	240
isoclinism	isoclinic class of #700
automorphism	Aut(G) = C₂×D₄×AGL(1,𝔽₅) (order 320) · Inn(G) = C₂×AGL(1,𝔽₅) · Out(G) = C₂³
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₁

Aliases

Splittings

Hall	C₃×C₅ ⋊ M₁₆
Sylow	C₅ ⋊ C₃×M₁₆

Subgroups

	subgroup	G/H
center	C₂×C₃	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂×C₃×C₄
Frattini	C₂	C₂×C₃×AGL(1,𝔽₅)
Fitting, prime radical	C₂²×C₃×C₅	C₄
socle	C₂×C₃×C₅	C₂×C₄
cosocle	Dic₅	C₂²×C₃
hypercenter	C₂²×C₃	AGL(1,𝔽₅)
hypocenter	C₅	C₃×M₁₆
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃ ◁ C₂²×C₃

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 120

A minimal faithful permutation representation — 2 generators on 16 points:

(1,2,3,5,4)(7,10)(11,13)(14,15,16);(2,5,4,3)(6,11,12,7,9,13,8,10)

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#91174, #91175, C₃×#4005, …
maximal normal subgroup of	#91174, #91175, C₃×#4005, …
maximal quotient of	C₃×#4002, C₃×#4004, C₃×C₂²⋊#170, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 116);