C₅×#2760⋊C₃ #14510924

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₅ · Schur multiplier (H₂) = C₂

solvable

Aliases

direct product	C₅×#2760⋊C₃
unambiguous semidirect product	#2760⋊(C₃×C₅)
unambiguous semidirect product	(C₅×#2760)⋊C₃

Splittings

Fitting, Hall	C₅×#2760 ⋊ C₃
Sylow	#2760 ⋊ C₃×C₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₅	#560⋊C₃
commutator, cosocle, hypocenter	#95	C₂²×C₃×C₅
Frattini	C₂³	C₂²×C₅×A₄
Fitting	C₅×#2760	C₃
socle	C₂³×C₅	C₂²×A₄
prime radical	C₅×#392⋊C₃	C₂
LCS	G ▷ #95
UCS	C₁ ◁ C₂×C₅

230 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degree	μ(G) = 29
transitive degree	μ_t(G) = 120

A minimal faithful permutation representation — 2 generators on 29 points:

(1,2,3,4,5)(6,16,18,10,9,25,11,17,26,8,19,24)(7,20,22,15,13,28,14,21,29,12,23,27);(6,15,11,12)(7,8,14,10)(9,23)(13,19)(16,21)(17,20)(18,28,26,27)(22,25,29,24)

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236622);