C₃×#983 #64253

order	288
automorphism	Aut(G) = C₂³×S₄ (order 192) · Inn(G) = C₂×S₄ · Out(G) = C₂²
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

C₂×C₃×Q₈ ⋊ D₃

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

85 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 2 generators on 19 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 901);