C₅×#69267 #14318556

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

Aliases

direct product

C₅×#69267

Splittings

Fitting	C₃×C₅×#362 ⋊ C₂
Hall	C₃×C₅ ⋊ #2239
Sylow	C₃ ⋊ C₅×#2239

Subgroups

	subgroup	G/H
center	C₂×C₅	#4777
commutator, cosocle	C₂×C₃×C₈	C₂³×C₅
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₃×C₅×#362	C₂
socle	C₂×C₃×C₅	#481
prime radical	C₂²×C₅×D₃	SD₁₆
hypercenter	C₅×#362	D₃
hypocenter	C₃	C₅×#2239
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₈ ◁ C₅×#362

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

normal subgroups only

This group has 248 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) = 40
transitive degree	μ_t(G) = 480

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,3,4,5)(6,7,8)(9,11,13,29,14,20,26,22)(10,15,17,36,18,31,28,32)(12,25,27,30,24,39,33,21)(16,35,23,37,34,38,40,19);(7,8)(12,24)(16,34)(19,37)(21,30)(23,40)(25,39)(27,33)(35,38);(9,10,14,18)(11,19,20,37)(12,23,24,40)(13,28,26,17)(15,30,31,21)(16,33,34,27)(22,35,29,38)(25,32,39,36)

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, 44254);