C₅×C₃⋊D₉ #63198

order	270
isoclinism	isoclinic class of C₃⋊D₉
automorphism	?
homology	Abelianization (H₁) = C₂×C₅ · Schur multiplier (H₂) = C₃

Aliases

direct product	C₅×C₃⋊D₉
unambiguous semidirect product	C₃⋊(C₅×D₉)
unambiguous semidirect product	C₉⋊(C₅×D₃)

Fitting, Hall	C₃×C₅×C₉ ⋊ C₂
derived, Sylow	C₃×C₉ ⋊ C₂×C₅

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

40 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 135

A minimal faithful permutation representation — 3 generators on 17 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(270, 16);