C₅×#67547 #14279536

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#67547

Splittings

Hall	C₃×C₅ ⋊ #1414
Sylow	C₃ ⋊ C₅×#1414

Subgroups

	subgroup	G/H
center	C₂×C₅	#4473
commutator	C₃×C₄²	C₂×C₄×C₅
Frattini	#124	C₂×C₅×D₃
Fitting	C₃×C₅×#355	C₂
socle	C₂×C₃×C₅	#354
prime radical	C₂×C₃×C₅×D₄	D₄
cosocle	C₃×#124	C₂²×C₅
hypercenter	C₅×#355	D₃
hypocenter	C₃	C₅×#1414
LCS	G ▷ C₃×C₄² ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#124 ◁ C₅×#355

168 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 46 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 — 2 generators on 40 points:

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

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 5234);