C₅×#69246 #14318157

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#69246

Splittings

Hall	C₃×C₅ ⋊ #2230
Sylow	C₃ ⋊ C₅×#2230

Subgroups

	subgroup	G/H
center	C₂×C₅	#4761
commutator, cosocle	C₂×C₃×C₈	C₂³×C₅
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₃×C₅×#360	C₂
socle	C₂×C₃×C₅	#476
prime radical	C₃×C₅×#135	C₂²
hypercenter	C₅×#360	D₃
hypocenter	C₃	C₅×#2230
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₈ ◁ C₅×#360

252 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 90 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)(6,7,8)(9,11)(10,16)(12,22)(13,23)(14,20)(15,21)(17,24)(18,19);(7,8)(9,10,15,12,14,18,13,17)(11,16,21,22,20,19,23,24);(10,12)(11,19,20,16)(13,15)(17,18)(21,22,23,24)

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, 43855);