C₅×#69252 #14316307

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

Aliases

direct product

C₅×#69252

Splittings

Fitting	C₃×C₅×#360 ⋊ C₂
Hall	C₃×C₅ ⋊ #2190
Sylow	C₃ ⋊ C₅×#2190

Subgroups

	subgroup	G/H
center	C₂×C₅	#4773
commutator, cosocle	C₂×C₃×C₈	C₂³×C₅
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₃×C₅×#360	C₂
socle	C₂×C₃×C₅	#447
hypercenter	C₅×#360	D₃
hypocenter	C₃	C₅×#2190
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₈ ◁ C₅×#360

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

normal subgroups only

This group has 452 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) = 24
transitive degree	μ_t(G) = 240

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

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

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, 42005);