C₅×#71325 #14317308

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

Aliases

direct product

C₅×#71325

Splittings

Fitting	C₃×C₅×#495 ⋊ C₂
Hall	C₃×C₅ ⋊ #2212
Sylow	C₃ ⋊ C₅×#2212

Subgroups

	subgroup	G/H
center	C₂×C₅	#5072
commutator, cosocle	C₂×C₃×C₈	C₂³×C₅
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₃×C₅×#495	C₂
socle	C₂×C₃×C₅	#459
prime radical	C₅×#5147	C₂
hypercenter	C₅×#495	D₃
hypocenter	C₃	C₅×#2212
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₄×C₅×C₈ ◁ C₅×#495

308 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 94 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)(10,17)(11,21)(13,25)(14,31)(15,27)(16,32)(18,33)(19,36)(20,35)(22,38)(23,39)(24,30)(26,29)(28,34)(37,40);(6,7,8)(9,11,13,22,15,23,29,14)(10,16,18,19,20,30,28,37)(12,21,25,38,27,39,26,31)(17,32,33,36,35,24,34,40);(9,10)(11,19)(12,24)(13,28)(14,30)(15,20)(16,22)(17,21)(18,29)(23,37)(25,36)(26,40)(27,32)(31,33)(34,38)(35,39)

Conjugacy classes & character table

225 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 43006);