C₅×C₃⋊(#127⋊C₃) #13086228

order	1440
isoclinism	isoclinic class of C₃⋊(#127⋊C₃)
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₅ · Schur multiplier (H₂) = C₄

Aliases

direct product	C₅×C₃⋊(#127⋊C₃)
unambiguous semidirect product	C₃⋊(C₅×#127⋊C₃)
unambiguous semidirect product	#874⋊(C₃×C₅)
unambiguous semidirect product	(C₅×#874)⋊C₃

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

	subgroup	G/H
center, hypercenter	C₅	C₃⋊(#127⋊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

118 conjugacy classes of subgroups; only the 22 normal subgroups are shown.

normal subgroups only

This group has 118 conjugacy classes of subgroups — too many to draw. Only its 22 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

A minimal faithful permutation representation — 2 generators on 20 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1440, 2545);