C₃×C₅×D₅ #3877

order	150
isoclinism	isoclinic class of D₅
automorphism	Aut(G) = C₂×C₄×AGL(1,𝔽₅) (order 160) · Inn(G) = D₅ · Out(G) = C₂²×C₄
homology	Abelianization (H₁) = C₂×C₃×C₅ · Schur multiplier (H₂) = C₁

Aliases

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

	subgroup	G/H
center, hypercenter	C₃×C₅	D₅
commutator, cosocle, hypocenter	C₅	C₂×C₃×C₅
Frattini	C₁	G
Fitting, socle	C₃×C₅²	C₂
LCS	G ▷ C₅
UCS	C₁ ◁ C₃×C₅

20 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 30

A minimal faithful permutation representation — 2 generators on 13 points:

(1,2,3)(4,5,6,7,8)(10,11)(12,13);(9,10,12,13,11)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(150, 8);