D₃×#226 #64200

order	288
automorphism	Aut(G) = C₂²×D₃×S₄ (order 576) · Inn(G) = D₃×S₄ · Out(G) = C₂²
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

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

83 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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 848);