C₂×C₃×D₃×C₇ #7003

order	252
isoclinism	isoclinic class of D₃
automorphism	Aut(G) = C₂³×C₃×D₃ (order 144) · Inn(G) = D₃ · Out(G) = C₂³×C₃
homology	Abelianization (H₁) = C₂²×C₃×C₇ · Schur multiplier (H₂) = C₂

Aliases

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

	subgroup	G/H
center, hypercenter	C₂×C₃×C₇	D₃
commutator, cosocle, hypocenter	C₃	C₂²×C₃×C₇
Frattini	C₁	G
Fitting, socle	C₂×C₃²×C₇	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₇

44 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 84

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(252, 42);