C₅×C₃⋊(#126⋊C₃) #13086229

order	1440
isoclinism	isoclinic class of C₃⋊(#126⋊C₃)
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₅×C₃⋊(#126⋊C₃)
unambiguous semidirect product	C₃⋊(C₅×#126⋊C₃)
unambiguous semidirect product	#876⋊(C₃×C₅)
unambiguous semidirect product	(C₅×#876)⋊C₃

Fitting	C₃×C₄²×C₅ ⋊ C₂×C₃
derived	C₃×C₄² ⋊ C₂×C₃×C₅

	subgroup	G/H
center, hypercenter	C₅	C₃⋊(#126⋊C₃)
commutator, cosocle, hypocenter	C₃×C₄²	C₂×C₃×C₅
Frattini	C₂²	C₅×D₃×A₄
Fitting	C₃×C₄²×C₅	C₂×C₃
socle	C₂²×C₃×C₅	C₂×A₄
LCS	G ▷ C₃×C₄²
UCS	C₁ ◁ C₅

86 conjugacy classes of subgroups.

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 240

A minimal faithful permutation representation — 2 generators on 24 points:

(1,2,3,4,5)(7,8)(10,21,23,14,15,19)(11,17,24,18,16,22)(12,13,20);(6,7,8)(9,22,20,15)(10,19,14,18)(11,17,23,16)(12,24,13,21)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1440, 2546);