C₃×D₃×D₅ #4381

order	180
isoclinism	isoclinic class of D₃×D₅
automorphism	Aut(G) = C₂×D₃×AGL(1,𝔽₅) (order 240) · Inn(G) = D₃×D₅ · Out(G) = C₂²
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₃×D₃×D₅
unambiguous semidirect product	(C₅×D₃)⋊(C₂×C₃)
unambiguous semidirect product	(C₃×D₅)⋊(C₂×C₃)
unambiguous semidirect product	D₁₅⋊(C₂×C₃)

	subgroup	G/H
center, hypercenter	C₃	D₃×D₅
commutator, cosocle, hypocenter	C₃×C₅	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₃²×C₅	C₂²
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₃

44 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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(180, 26);