C₅×#68515 #14293368

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

Aliases

direct product

C₅×#68515

Splittings

Hall	C₃×C₅ ⋊ #1704
Sylow	C₃ ⋊ C₅×#1704

Subgroups

	subgroup	G/H
center	C₂²×C₅	#884
commutator, cosocle	C₃×C₄²	C₂³×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#333	C₂
socle, prime radical	C₂²×C₃×C₅	#121
hypercenter	C₅×#333	D₃
hypocenter	C₃	C₅×#1704
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#333

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

normal subgroups only

This group has 280 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) = 1920

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

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

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, 19066);