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D₄×D₁₅ #6818

order	240
automorphism	Aut(G) = C₂×D₃×D₄×AGL(1,𝔽₅) (order 1920) · Inn(G) = C₂²×D₁₅ · Out(G) = C₂²×C₄
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

D₄×D₁₅

Splittings

Fitting	C₃×C₅×D₄ ⋊ C₂
Hall	C₃×C₅ ⋊ C₂×D₄
Sylow	C₅ ⋊ D₃×D₄
Sylow	C₃ ⋊ D₄×D₅

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂²×D₁₅
commutator, socle, cosocle	C₂×C₃×C₅	C₂³
Fitting	C₃×C₅×D₄	C₂
hypercenter	D₄	D₁₅
hypocenter	C₃×C₅	C₂×D₄
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂ ◁ D₄

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:

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

Conjugacy classes & character table

45 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#91481, #91487, #91498, …
maximal normal subgroup of	#91481, #91487, #91498, …
maximal quotient of	#91767, C₂²⋊C₄×D₁₅, #91770, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 179);