C₂×D₃×SL(2,𝔽₃) #64274

order	288
isoclinism	isoclinic class of C₃⋊(#41⋊C₃)
automorphism	Aut(G) = D₃×D₄×S₄ (order 1152) · Inn(G) = D₃×A₄ · Out(G) = C₂×D₄
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×D₃×SL(2,𝔽₃)
unambiguous semidirect product	C₃⋊(C₂²×SL(2,𝔽₃))
unambiguous semidirect product	(C₂×C₃)⋊(C₂×SL(2,𝔽₃))
unambiguous semidirect product	(C₂×D₃×Q₈)⋊C₃

Fitting	C₂×C₃×Q₈ ⋊ C₂×C₃
derived	C₃×Q₈ ⋊ C₂²×C₃

	subgroup	G/H
center, hypercenter	C₂²	D₃×A₄
commutator, cosocle, hypocenter	C₃×Q₈	C₂²×C₃
Frattini	C₂	C₂×D₃×A₄
Fitting	C₂×C₃×Q₈	C₂×C₃
socle	C₂²×C₃	C₂×A₄
LCS	G ▷ C₃×Q₈
UCS	C₁ ◁ C₂²

129 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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 922);