C₅×D₃×D₇ #88300

order	420
isoclinism	isoclinic class of D₃×D₇
automorphism	Aut(G) = C₄×D₃×AGL(1,𝔽₇) (order 1008) · Inn(G) = D₃×D₇ · Out(G) = C₃×C₄
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₅×D₃×D₇
unambiguous semidirect product	(D₃×C₇)⋊(C₂×C₅)
unambiguous semidirect product	(C₃×D₇)⋊(C₂×C₅)
unambiguous semidirect product	D₂₁⋊(C₂×C₅)

	subgroup	G/H
center, hypercenter	C₅	D₃×D₇
commutator, cosocle, hypocenter	C₃×C₇	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₇	C₂²
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₅

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(420, 25);