C₅×#72113 #14316343

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

Aliases

direct product

C₅×#72113

Splittings

Hall	C₃×C₅ ⋊ #2191
Sylow	C₃ ⋊ C₅×#2191

Subgroups

	subgroup	G/H
center	C₂×C₅	#5217
commutator, cosocle	C₂×C₃×C₈	C₂³×C₅
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting, prime radical	C₃×C₅×#575	C₂
socle	C₂×C₃×C₅	#447
hypercenter	C₅×#575	D₃
hypocenter	C₃	C₅×#2191
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×M₁₆ ◁ C₅×#575

432 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 106 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)(6,7,8)(12,25)(16,35)(19,37)(20,30)(23,39)(24,40)(27,33)(34,38);(9,11)(10,15)(12,24)(13,29)(14,21)(16,34)(17,36)(18,31)(19,23)(20,27)(22,26)(25,40)(28,32)(30,33)(35,38)(37,39);(7,8)(9,10,14,18)(11,19,21,37)(12,23,25,39)(13,28,26,17)(15,30,31,20)(16,33,35,27)(22,38,29,34)(24,36,40,32)

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, 42041);