C₂×C₃×GL(2,𝔽₃) #64252

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

Fitting	C₂×C₃×Q₈ ⋊ D₃
derived	SL(2,𝔽₃) ⋊ C₂²×C₃

	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
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂²×C₃

109 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 2 generators on 13 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 900);