C₅×#71170 #14288599

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#71170

Splittings

Hall	C₃×C₅ ⋊ #1615
Sylow	C₃ ⋊ C₅×#1615

Subgroups

	subgroup	G/H
center	C₂²×C₅	#937
commutator, cosocle	C₂²×C₃×C₄	C₂³×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#481	C₂
socle	C₂²×C₃×C₅	#120
prime radical	C₃×C₅×#121	C₂²
hypercenter	C₅×#481	D₃
hypocenter	C₃	C₅×#1615
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#481

416 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 40
transitive degree	μ_t(G) = 480

A minimal faithful permutation representation — 3 generators on 40 points:

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

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 14297);