C₂×C₃×D₃×D₅ #67221

order	360
isoclinism	isoclinic class of D₃×D₅
automorphism	Aut(G) = C₂³×D₃×AGL(1,𝔽₅) (order 960) · Inn(G) = D₃×D₅ · Out(G) = C₂⁴
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product	C₂×C₃×D₃×D₅
unambiguous semidirect product	(C₅×D₃)⋊(C₂²×C₃)
unambiguous semidirect product	(C₃×D₅)⋊(C₂²×C₃)
unambiguous semidirect product	D₁₅⋊(C₂²×C₃)

	subgroup	G/H
center, hypercenter	C₂×C₃	D₃×D₅
commutator, cosocle, hypocenter	C₃×C₅	C₂³×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₃²×C₅	C₂²
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₃

138 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 60

A minimal faithful permutation representation — 3 generators on 13 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 151);