C₅×#70987 #14316910

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

Aliases

direct product

C₅×#70987

Splittings

Fitting	C₃×C₅×#454 ⋊ C₂
Hall	C₃×C₅ ⋊ #2202
Sylow	C₃ ⋊ C₅×#2202

Subgroups

	subgroup	G/H
center	C₂×C₅	#5032
commutator, cosocle	C₂×C₃×Q₈	C₂³×C₅
Frattini	C₂×Q₈	C₂²×C₅×D₃
Fitting	C₃×C₅×#454	C₂
socle	C₂×C₃×C₅	B(4,𝔽₂)
hypercenter	C₅×#454	D₃
hypocenter	C₃	C₅×#2202
LCS	G ▷ C₂×C₃×Q₈ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₅ ◁ C₂×C₅×Q₈ ◁ C₅×#454

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

normal subgroups only

This group has 496 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)(7,12)(9,18)(11,20)(13,21)(14,15)(17,19)(22,23,24);(7,11)(8,16)(9,18)(12,20)(13,15)(14,21)(23,24);(6,7)(8,11)(9,12)(10,15)(13,16)(14,18)(17,20)(19,21)

Conjugacy classes & character table

165 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 42608);