C₂×C₃×#41⋊C₃ #64335

order	288
isoclinism	isoclinic class of SL(2,𝔽₃)
automorphism	Aut(G) = D₃×D₄×S₄ (order 1152) · Inn(G) = A₄ · Out(G) = C₂×D₃×D₄
homology	Abelianization (H₁) = C₂²×C₃² · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product	C₂×C₃×#41⋊C₃
unambiguous semidirect product	(C₂×#41)⋊C₃²
unambiguous semidirect product	(C₂×C₃×#41)⋊C₃

Fitting	C₂×C₃×#41 ⋊ C₃
Sylow	C₂×#41 ⋊ C₃²

	subgroup	G/H
center, hypercenter	C₂×C₃×C₄	A₄
commutator, cosocle, hypocenter	Q₈	C₂²×C₃²
Frattini	C₂	C₂²×C₃×A₄
Fitting	C₂×C₃×#41	C₃
socle	C₂²×C₃	C₂×A₄
LCS	G ▷ Q₈
UCS	C₁ ◁ C₂×C₃×C₄

124 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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 983);