C₅×#71378 #14291802

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#71378

Splittings

Hall	C₃×C₅ ⋊ #1676
Sylow	C₃ ⋊ C₅×#1676

Subgroups

	subgroup	G/H
center	C₂²×C₅	#938
commutator, cosocle	C₃×C₄²	C₂³×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#500	C₂
socle, prime radical	C₂²×C₃×C₅	#121
hypercenter	C₅×#500	D₃
hypocenter	C₃	C₅×#1676
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#500

288 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 1920

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

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

Conjugacy classes & character table

255 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 17500);