C₅×#70988 #14316968

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

Aliases

direct product

C₅×#70988

Splittings

Fitting	C₃×C₅×#454 ⋊ C₂
Hall	C₃×C₅ ⋊ #2204
Sylow	C₃ ⋊ C₅×#2204

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×#2204
LCS	G ▷ C₂×C₃×Q₈ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₅ ◁ C₂×C₅×Q₈ ◁ C₅×#454

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

normal subgroups only

This group has 472 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)(6,8)(9,16)(10,11)(12,13)(14,18)(17,19)(22,23,24);(7,12)(8,10)(9,17)(13,15)(14,20)(18,21)(23,24);(6,7)(8,14)(9,12)(10,18)(11,15)(13,17)(16,21)(19,20)

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, 42666);