C₅×#71517 #14316376

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

Aliases

direct product

C₅×#71517

Splittings

Fitting	C₃×C₅×#509 ⋊ C₂
Hall	C₃×C₅ ⋊ #2192
Sylow	C₃ ⋊ C₅×#2192

Subgroups

	subgroup	G/H
center	C₂×C₅	#5156
commutator, cosocle	C₂×C₃×C₈	C₂³×C₅
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₃×C₅×#509	C₂
socle	C₂×C₃×C₅	#447
hypercenter	C₅×#509	D₃
hypocenter	C₃	C₅×#2192
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₈ ◁ C₅×#509

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

normal subgroups only

This group has 440 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)(9,11)(10,16)(12,21)(13,23)(14,24)(15,17)(18,34)(19,35)(20,30)(22,27)(25,32)(26,28)(29,38)(31,39)(33,36)(37,40);(6,7,8)(9,11)(10,15)(12,26)(13,22)(14,24)(16,17)(18,31)(19,33)(20,25)(21,28)(23,27)(29,37)(30,32)(34,39)(35,36)(38,40);(9,10)(11,20)(12,25)(13,30)(14,19)(15,21)(16,22)(17,28)(18,27)(23,35)(24,34)(26,31)(29,39)(32,37)(33,38)(36,40)

Conjugacy classes & character table

195 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 42074);