C₅×C₃⋊#857 #13086212

order	1440
isoclinism	isoclinic class of C₃⋊#857
automorphism	?
homology	Abelianization (H₁) = C₂×C₅ · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product	C₅×C₃⋊#857
unambiguous semidirect product	C₃⋊(C₅×#857)

Fitting	C₃×C₄²×C₅ ⋊ D₃
derived	C₃×C₄²⋊C₃ ⋊ C₂×C₅

	subgroup	G/H
center, hypercenter	C₅	C₃⋊#857
commutator, cosocle, hypocenter	C₃×C₄²⋊C₃	C₂×C₅
Frattini	C₂²	C₅×C₃⋊S₄
Fitting	C₃×C₄²×C₅	D₃
socle	C₂²×C₃×C₅	S₄
LCS	G ▷ C₃×C₄²⋊C₃
UCS	C₁ ◁ C₅

120 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 180

A minimal faithful permutation representation — 3 generators on 20 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1440, 2529);