C₅×#69313 #14318526

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

Aliases

direct product

C₅×#69313

Splittings

Fitting	C₃×C₅×#365 ⋊ C₂
Hall	C₃×C₅ ⋊ #2238
Sylow	C₃ ⋊ C₅×#2238

Subgroups

	subgroup	G/H
center	C₂×C₅	#4863
commutator, cosocle	C₂×C₃×C₈	C₂³×C₅
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₃×C₅×#365	C₂
socle	C₂×C₃×C₅	#481
prime radical	C₅×#4865	C₂
hypercenter	C₅×#365	D₃
hypocenter	C₃	C₅×#2238
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₈ ◁ C₅×#365

268 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 82 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)(9,10,12,21,13,15,20,16)(11,19,17,22,18,23,24,14);(7,8)(11,18)(14,22)(17,24)(19,23);(7,8)(10,14)(11,17)(12,20)(15,22)(16,19)(18,24)(21,23)

Conjugacy classes & character table

180 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 44224);