C₅×#69249 #14315833

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#69249

Splittings

Fitting	C₃×C₅×#360 ⋊ C₂
Hall	C₃×C₅ ⋊ AGL(1,ℤ/16)
Sylow	C₃ ⋊ C₅×AGL(1,ℤ/16)

Subgroups

	subgroup	G/H
center	C₂×C₅	#4785
commutator	C₃×C₈	C₂²×C₄×C₅
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₃×C₅×#360	C₂
socle	C₂×C₃×C₅	#442
prime radical	C₅×#5193	C₂
cosocle	C₂×C₃×C₈	C₂³×C₅
hypercenter	C₅×#360	D₃
hypocenter	C₃	C₅×AGL(1,ℤ/16)
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₈ ◁ C₅×#360

308 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 24
transitive degree	μ_t(G) = 240

A minimal faithful permutation representation — 3 generators on 24 points:

(1,2,3,4,5)(7,8)(9,11,19,22,13,20,12,18)(10,15,17,14,16,24,21,23);(6,7,8)(9,10)(11,15)(12,21)(13,16)(14,22)(17,19)(18,23)(20,24);(10,14,16,23)(11,18)(12,19)(15,21,24,17)(20,22)

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, 41531);