C₂×D₅×A₄ #6837

order	240
isoclinism	isoclinic class of D₅×A₄
automorphism	Aut(G) = C₂×AGL(1,𝔽₅)×S₄ (order 960) · Inn(G) = D₅×A₄ · Out(G) = C₂³
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product	C₂×D₅×A₄
unambiguous semidirect product	C₅⋊(C₂²×A₄)
unambiguous semidirect product	(C₂×C₅)⋊(C₂×A₄)
unambiguous semidirect product	(C₂³×D₅)⋊C₃

	subgroup	G/H
center, hypercenter	C₂	D₅×A₄
commutator, cosocle, hypocenter	C₂²×C₅	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂³×C₅	C₂×C₃
LCS	G ▷ C₂²×C₅
UCS	C₁ ◁ C₂

78 conjugacy classes of subgroups.

minimal degree	μ(G) = 11
transitive degree	μ_t(G) = 30

A minimal faithful permutation representation — 2 generators on 11 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 198);