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C₃×#713 #6801

order	240
automorphism	Aut(G) = C₂²×AGL(1,𝔽₅)×S₄ (order 1920) · Inn(G) = C₂²×D₅ · Out(G) = C₂³×D₃
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

Splittings

Fitting	C₃×C₅×Q₈ ⋊ C₂
Hall	C₃×C₅ ⋊ #41
Sylow	C₅ ⋊ C₃×#41

Subgroups

	subgroup	G/H
center	C₂×C₃	C₂²×D₅
commutator, cosocle	C₂×C₅	C₂³×C₃
Frattini	C₂	C₂²×C₃×D₅
Fitting	C₃×C₅×Q₈	C₂
socle	C₂×C₃×C₅	C₂³
hypercenter	C₃×Q₈	D₅
hypocenter	C₅	C₃×#41
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃ ◁ C₃×Q₈

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 16 points:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#4010, #91239, #91502, …
maximal normal subgroup of	C₃×#4010, #91239, #91502, …
maximal quotient of	C₃×#4036, C₃×#4038, C₃×#4039, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 162);