C₂×C₃×#226 #64251

order	288
isoclinism	isoclinic class of #226
automorphism	Aut(G) = C₂×D₄×S₄ (order 384) · Inn(G) = S₄ · Out(G) = C₂×D₄
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

	subgroup	G/H
center, socle, hypercenter	C₂²×C₃	S₄
commutator, cosocle, hypocenter	SL(2,𝔽₃)	C₂²×C₃
Frattini	C₂	C₂×C₃×S₄
Fitting	C₂×C₃×Q₈	D₃
prime radical	C₂×C₃×SL(2,𝔽₃)	C₂
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂²×C₃

85 conjugacy classes of subgroups.

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 2 generators on 21 points:

(3,4,5)(6,11,15,18,10,20,17,12)(7,16,19,9,13,8,21,14);(1,2)(8,14,17)(9,15,16)(11,18,21)(12,19,20)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 899);