C₅×#67484 #14278048

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

Aliases

direct product

C₅×#67484

Splittings

Hall	C₃×C₅ ⋊ #1342
Sylow	C₃ ⋊ C₅×#1342

Subgroups

	subgroup	G/H
center	C₂×C₅	#4451
commutator	C₂×C₃×C₈	C₂×C₄×C₅
Frattini	C₂×M₁₆	C₂×C₅×D₃
Fitting	C₃×C₅×#338	C₂
socle	C₂×C₃×C₅	#327
prime radical	C₂³×C₃×C₅	SD₁₆
cosocle	C₂×C₃×M₁₆	C₂²×C₅
hypercenter	C₅×#338	D₃
hypocenter	C₃	C₅×#1342
LCS	G ▷ C₂×C₃×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×M₁₆ ◁ C₅×#338

200 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 62 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)(7,8)(9,11,15,24)(10,16,19,33)(12,26,29,38)(13,23,14,22)(17,34,36,40)(18,31,30,32)(20,35,37,25)(21,28,39,27);(6,7,8)(9,10,13,18,15,19,14,30)(11,20,22,34,24,37,23,40)(12,25,27,17,29,35,28,36)(16,26,31,39,33,38,32,21)

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 3746);