C₅×#2736⋊C₃ #14511012

order	1920
automorphism	?
homology	Abelianization (H₁) = C₃×C₅ · Schur multiplier (H₂) = C₂×C₄²

Aliases

direct product	C₅×#2736⋊C₃
unambiguous semidirect product	#2736⋊(C₃×C₅)
unambiguous semidirect product	(C₅×#2736)⋊C₃

Splittings

Fitting, Hall	C₅×#2736 ⋊ C₃
derived, Sylow	#2736 ⋊ C₃×C₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₅	#5464
commutator, cosocle, hypocenter	#2736	C₃×C₅
Frattini	C₂³	C₅×C₂²⋊A₄
Fitting	C₅×#2736	C₃
socle	C₂³×C₅	C₂²⋊A₄
LCS	G ▷ #2736
UCS	C₁ ◁ C₂×C₅

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

normal subgroups only

This group has 270 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.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 29 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236710);