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D₃×C₂²⋊D₅ #6789

order	240
automorphism	Aut(G) = C₂³×D₃×AGL(1,𝔽₅) (order 960) · Inn(G) = C₂×D₃×D₅ · Out(G) = C₂×C₄
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

D₃×C₂²⋊D₅

Splittings

Fitting	C₂²×C₃×C₅ ⋊ C₂²
Hall	C₃×C₅ ⋊ C₂×D₄
Sylow	C₅ ⋊ D₃×D₄
Sylow	C₃ ⋊ C₂×C₂²⋊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:

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

Conjugacy classes & character table

39 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#709, #92018, D₃×D₄×D₅, …
maximal normal subgroup of	D₃×#709, #92018, D₃×D₄×D₅, …
maximal quotient of	#91329, #91351, #91352, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 150);