C₅×#3487⋊C₃ #14511344

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

solvable

Aliases

direct product	C₅×#3487⋊C₃
unambiguous semidirect product	#3487⋊(C₃×C₅)
unambiguous semidirect product	(C₅×#3487)⋊C₃

Splittings

Fitting, Hall	C₅×#3487 ⋊ C₃
Sylow	#3487 ⋊ 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×#3487	C₃
hypercenter	C₅×#123	A₄
hypocenter	Q₈	C₂×C₃×C₅×D₄
LCS	G ▷ C₂×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×#123

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

normal subgroups only

This group has 732 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)(11,17)(13,14,23)(16,27)(18,31,32,30,19,33)(20,21,34)(24,26,25)(28,39,40,38,29,41)(35,37,36);(7,9)(10,34)(11,40)(12,25)(13,37)(14,20)(15,36)(16,33)(17,41)(18,39)(19,28)(21,24)(22,23)(26,35)(27,32)(29,30)(31,38);(6,7)(8,9)(10,11,15,17)(12,16,22,27)(13,18,24,30)(14,19,26,31)(20,28,35,38)(21,29,37,39)(23,32,25,33)(34,40,36,41)

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, 237042);