C₂×C₅×D₃×C₇ #88312

order	420
isoclinism	isoclinic class of D₃
automorphism	Aut(G) = C₂²×C₃×C₄×D₃ (order 288) · Inn(G) = D₃ · Out(G) = C₂²×C₃×C₄
homology	Abelianization (H₁) = C₂²×C₅×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×C₅×D₃×C₇
unambiguous semidirect product	C₃⋊(C₂²×C₅×C₇)
unambiguous semidirect product	(C₂×C₃)⋊(C₂×C₅×C₇)

	subgroup	G/H
center, hypercenter	C₂×C₅×C₇	D₃
commutator, cosocle, hypocenter	C₃	C₂²×C₅×C₇
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₅×C₇	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₂×C₅×C₇

40 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 210

A minimal faithful permutation representation — 2 generators on 17 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(420, 37);