D₃×C₇×D₅ #88302

order	420
isoclinism	isoclinic class of D₃×D₅
automorphism	Aut(G) = C₂×C₃×D₃×AGL(1,𝔽₅) (order 720) · Inn(G) = D₃×D₅ · Out(G) = C₂²×C₃
homology	Abelianization (H₁) = C₂²×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product	D₃×C₇×D₅
unambiguous semidirect product	(C₅×D₃)⋊(C₂×C₇)
unambiguous semidirect product	(C₃×D₅)⋊(C₂×C₇)
unambiguous semidirect product	D₁₅⋊(C₂×C₇)

	subgroup	G/H
center, hypercenter	C₇	D₃×D₅
commutator, cosocle, hypocenter	C₃×C₅	C₂²×C₇
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₇	C₂²
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₇

40 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 105

A minimal faithful permutation representation — 2 generators on 15 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(420, 27);