C₃×C₅×D₉ #63190

order	270
isoclinism	isoclinic class of D₉
automorphism	Aut(G) = C₂×C₄×AGL(1,ℤ/9) (order 432) · Inn(G) = D₉ · Out(G) = C₂×C₃×C₄
homology	Abelianization (H₁) = C₂×C₃×C₅ · Schur multiplier (H₂) = C₁

Aliases

C₃×C₅×D₉

	subgroup	G/H
center, hypercenter	C₃×C₅	D₉
commutator, cosocle, hypocenter	C₉	C₂×C₃×C₅
Frattini	C₃	C₃×C₅×D₃
Fitting	C₃×C₅×C₉	C₂
socle	C₃²×C₅	D₃
LCS	G ▷ C₉
UCS	C₁ ◁ C₃×C₅

28 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 90

A minimal faithful permutation representation — 2 generators on 17 points:

(1,2,3)(4,5,6,7,8)(10,12)(11,15)(13,16)(14,17);(9,14,12,11,16,13,15,10,17)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(270, 8);