C₅×#3500⋊C₃ #14511350

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃×C₅ · Schur multiplier (H₂) = C₂³

solvable

Aliases

direct product	C₅×#3500⋊C₃
unambiguous semidirect product	#3500⋊(C₃×C₅)
unambiguous semidirect product	(C₅×#3500)⋊C₃

Splittings

Fitting, Hall	C₅×#3500 ⋊ C₃
Sylow	#3500 ⋊ C₃×C₅

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂³×A₄
commutator, cosocle	C₂×Q₈	C₂³×C₃×C₅
Frattini	C₂²	C₂³×C₅×A₄
Fitting	C₅×#3500	C₃
hypercenter	C₅×#124	A₄
hypocenter	Q₈	C₂×C₃×C₅×D₄
LCS	G ▷ C₂×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×#124

730 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 730 conjugacy classes of subgroups — too many to draw. Only its 120 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) = 41
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 41 points:

(1,2,3,4,5)(10,12,13,21)(11,17,16,20)(14,38,15,36,26,37,24,39,25,22,27,23)(18,40,19,34,32,35,28,41,29,30,33,31);(6,7,8,9)(10,27,13,26)(11,32,16,33)(12,39,21,38)(14,25,24,15)(17,40,20,41)(18,19,28,29)(22,37,36,23)(30,31,34,35);(7,9)(10,11,13,16)(12,20,21,17)(14,18,24,28)(15,19,25,29)(22,34,36,30)(23,35,37,31)(26,32,27,33)(38,41,39,40)

Conjugacy classes & character table

260 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 237048);