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C₄×D₃×D₅ #6774

order	240
isoclinism	isoclinic class of D₃×D₅
automorphism	Aut(G) = C₂³×D₃×AGL(1,𝔽₅) (order 960) · Inn(G) = D₃×D₅ · Out(G) = C₂⁴
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

C₄×D₃×D₅

Splittings

Fitting	C₃×C₄×C₅ ⋊ C₂²
derived, Hall	C₃×C₅ ⋊ C₂²×C₄
Sylow	C₅ ⋊ C₂×C₄×D₃
Sylow	C₃ ⋊ C₂×C₄×D₅

Subgroups

	subgroup	G/H
center, hypercenter	C₄	D₃×D₅
commutator, hypocenter	C₃×C₅	C₂²×C₄
Frattini	C₂	C₂×D₃×D₅
Fitting	C₃×C₄×C₅	C₂²
socle, cosocle	C₂×C₃×C₅	C₂³
prime radical	C₂×D₃×D₅	C₂
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₄

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

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×C₈×D₅, D₅×ΓL(1,𝔽₂₅), D₃×#676, …
maximal normal subgroup of	D₃×C₈×D₅, D₅×ΓL(1,𝔽₂₅), D₃×#676, …
maximal quotient of	D₃×C₈×D₅, D₅×ΓL(1,𝔽₂₅), D₃×#676, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 135);