C₂²×C₃×D₃×C₇ #92772

order	504
isoclinism	isoclinic class of D₃
automorphism	Aut(G) = C₂²×C₃×D₃×S₄ (order 1728) · Inn(G) = D₃ · Out(G) = C₂²×C₃×S₄
homology	Abelianization (H₁) = C₂³×C₃×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product	C₂²×C₃×D₃×C₇
unambiguous semidirect product	C₃⋊(C₂³×C₃×C₇)
unambiguous semidirect product	(C₂×C₃)⋊(C₂²×C₃×C₇)

	subgroup	G/H
center, hypercenter	C₂²×C₃×C₇	D₃
commutator, cosocle, hypocenter	C₃	C₂³×C₃×C₇
Frattini	C₁	G
Fitting, socle	C₂²×C₃²×C₇	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₂²×C₃×C₇

138 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 168

A minimal faithful permutation representation — 3 generators on 17 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 198);