D₃²×D₅ #67207

order	360
isoclinism	isoclinic class of D₃²×D₅
automorphism	Aut(G) = AGL(1,𝔽₅)×GO⁺(4,𝔽₂) (order 1440) · Inn(G) = G · Out(G) = C₂²
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

direct product	D₃²×D₅
unambiguous semidirect product	(C₃×D₃×D₅)⋊C₂
unambiguous semidirect product	(D₅×C₃⋊D₃)⋊C₂
unambiguous semidirect product	(D₃×D₁₅)⋊C₂

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃²×C₅	C₂³
LCS	G ▷ C₃²×C₅
UCS	C₁

138 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 3 generators on 11 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 137);