C₃×#857 #63749

order	288
isoclinism	isoclinic class of #857
automorphism	Aut(G) = C₂×#857⋊C₂ (order 384) · Inn(G) = #857 · Out(G) = C₂²
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂

Aliases

Fitting	C₃×C₄² ⋊ D₃
derived	C₄²⋊C₃ ⋊ C₂×C₃

	subgroup	G/H
center, hypercenter	C₃	#857
commutator, cosocle, hypocenter	C₄²⋊C₃	C₂×C₃
Frattini	C₂²	C₃×S₄
Fitting	C₃×C₄²	D₃
socle	C₂²×C₃	S₄
LCS	G ▷ C₄²⋊C₃
UCS	C₁ ◁ C₃

51 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 36

A minimal faithful permutation representation — 2 generators on 15 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 397);