C₅×(C₃⋊D₃)⋊D₇ #11929557

order	1260
isoclinism	isoclinic class of (C₃⋊D₃)⋊D₇
automorphism	Aut(G) = C₄×AGL(1,𝔽₇)×GO⁺(4,𝔽₂) (order 12096) · Inn(G) = (C₃⋊D₃)⋊D₇ · Out(G) = C₂²×C₃×C₄
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂

Aliases

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

	subgroup	G/H
center, hypercenter	C₅	(C₃⋊D₃)⋊D₇
commutator, cosocle, hypocenter	C₃²×C₇	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₃²×C₅×C₇	C₂²
LCS	G ▷ C₃²×C₇
UCS	C₁ ◁ C₅

88 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 210

A minimal faithful permutation representation — 2 generators on 18 points:

(1,2,3,4,5)(6,7)(10,11)(12,13,15,17,18,16,14);(7,8)(9,10,11)(13,14)(15,16)(17,18)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1260, 109);