C₅×#71002 #14316730

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

Aliases

direct product

C₅×#71002

Splittings

Fitting	C₃×C₅×#458 ⋊ C₂
Hall	C₃×C₅ ⋊ #2198
Sylow	C₃ ⋊ C₅×#2198

Subgroups

	subgroup	G/H
center	C₂×C₅	#5032
commutator, cosocle	C₂×C₃×D₄	C₂³×C₅
Frattini	C₂×D₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#458	C₂
socle	C₂×C₃×C₅	B(4,𝔽₂)
prime radical	C₅×#5241	C₂
hypercenter	C₅×#458	D₃
hypocenter	C₃	C₅×#2198
LCS	G ▷ C₂×C₃×D₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₅ ◁ C₂×C₅×D₄ ◁ C₅×#458

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

normal subgroups only

This group has 492 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)(6,9,11,8)(7,13,15,16)(10,17,19,18)(12,14,20,21)(22,23,24);(6,8,11,9)(7,12,15,20)(10,17,19,18)(13,14,16,21)(23,24);(6,7)(8,16)(9,13)(10,14)(11,15)(12,17)(18,20)(19,21)

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, 42428);