C₃×C₇×SL(2,𝔽₃) #92700

order	504
isoclinism	isoclinic class of SL(2,𝔽₃)
automorphism	Aut(G) = C₂×C₃×D₃×S₄ (order 864) · Inn(G) = A₄ · Out(G) = C₂²×C₃×D₃
homology	Abelianization (H₁) = C₃²×C₇ · Schur multiplier (H₂) = C₃

Aliases

direct product	C₃×C₇×SL(2,𝔽₃)
unambiguous semidirect product	Q₈⋊(C₃²×C₇)
unambiguous semidirect product	(C₃×Q₈)⋊(C₃×C₇)

	subgroup	G/H
center, socle, hypercenter	C₂×C₃×C₇	A₄
commutator, cosocle, hypocenter	Q₈	C₃²×C₇
Frattini	C₂	C₃×C₇×A₄
Fitting	C₃×C₇×Q₈	C₃
LCS	G ▷ Q₈
UCS	C₁ ◁ C₂×C₃×C₇

40 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 168

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 126);