C₅×#68049 #14280664

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₅ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₅×#68049

Splittings

Fitting	C₃×C₅×C₈² ⋊ C₂
Hall	C₃×C₅ ⋊ #1454
Sylow	C₃ ⋊ C₅×#1454

Subgroups

	subgroup	G/H
center	C₄²×C₅	C₂²×D₃
commutator	C₂²×C₃	C₂×C₄²×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×C₈²	C₂
socle	C₂²×C₃×C₅	C₂×C₄²
prime radical	C₂²×C₅×D₃	C₄²
cosocle	C₃×C₄²	C₂³×C₅
hypercenter	C₅×C₈²	D₃
hypocenter	C₃	C₅×#1454
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₄²×C₅ ◁ C₅×C₈²

356 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 178 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) = 960

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

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

Conjugacy classes & character table

600 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 6362);