C₅×#71003 #14316871

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

Aliases

direct product

C₅×#71003

Splittings

Hall	C₃×C₅ ⋊ #2201
Sylow	C₃ ⋊ C₅×#2201

Subgroups

	subgroup	G/H
center	C₂×C₅	#5032
commutator, cosocle	C₂×C₃×D₄	C₂³×C₅
Frattini	C₂×D₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#458	C₂
socle	C₂×C₃×C₅	B(4,𝔽₂)
prime radical	C₃×C₅×H(5,𝔽₂)	C₂²
hypercenter	C₅×#458	D₃
hypocenter	C₃	C₅×#2201
LCS	G ▷ C₂×C₃×D₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₅ ◁ C₂×C₅×D₄ ◁ C₅×#458

452 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 82 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)(6,9,11,22)(7,13,15,29)(8,17,19,20)(10,23,24,21)(12,25,26,27)(14,28,30,37)(16,31,32,36)(18,34,35,33)(38,39,40);(6,8,11,19)(7,12,15,26)(9,20,22,17)(10,18,24,35)(13,27,29,25)(14,16,30,32)(21,34,23,33)(28,36,37,31)(39,40);(6,7)(8,16)(9,13)(10,14)(11,15)(12,18)(17,31)(19,32)(20,36)(21,28)(22,29)(23,37)(24,30)(25,33)(26,35)(27,34)

Conjugacy classes & character table

165 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 42569);