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D₅×#67 #6788

order	240
automorphism	Aut(G) = C₂³×D₃×AGL(1,𝔽₅) (order 960) · Inn(G) = C₂×D₃×D₅ · Out(G) = C₂³
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

Splittings

Fitting	C₂²×C₃×C₅ ⋊ C₂²
Hall	C₃×C₅ ⋊ C₂×D₄
Sylow	C₅ ⋊ C₂×#67
Sylow	C₃ ⋊ D₄×D₅

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₃×D₅
commutator, socle, cosocle	C₂×C₃×C₅	C₂³
Fitting	C₂²×C₃×C₅	C₂²
hypercenter	C₂²	D₃×D₅
hypocenter	C₃×C₅	C₂×D₄
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂ ◁ C₂²

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 60

A minimal faithful permutation representation — 3 generators on 12 points:

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

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#67, #67⋊AGL(1,𝔽₅), D₅×#235, …
maximal normal subgroup of	AGL(1,𝔽₅)×#67, #67⋊AGL(1,𝔽₅), D₅×#235, …
maximal quotient of	#91326, #91327, #91348, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 149);