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C₃×#688 #6685

order	240
isoclinism	isoclinic class of #686
automorphism	Aut(G) = C₂²×D₄×AGL(1,𝔽₅) (order 640) · Inn(G) = C₂²⋊D₅ · Out(G) = C₂²×C₄
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₁

Aliases

Splittings

Fitting	C₃×C₅×Q₈ ⋊ C₂
Hall	C₃×C₅ ⋊ SD₁₆
Sylow	C₅ ⋊ C₃×SD₁₆

Subgroups

	subgroup	G/H
center	C₂×C₃	C₂²⋊D₅
commutator, cosocle	C₄×C₅	C₂²×C₃
Frattini	C₄	C₂×C₃×D₅
Fitting	C₃×C₅×Q₈	C₂
socle	C₂×C₃×C₅	D₄
prime radical	C₃×D₂₀	C₂
hypercenter	C₃×Q₈	D₅
hypocenter	C₅	C₃×SD₁₆
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄ ◁ C₃×Q₈

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:

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

Conjugacy classes & character table

51 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#688, #91504, #91505, …
maximal normal subgroup of	D₃×#688, #91504, #91505, …
maximal quotient of	C₃×#3940, C₃×#3941, C₃×#3967, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 46);