C₃×C₅×#1407 #14274441

order	1920
nilpotency class	5
isoclinism	isoclinic class of #1406
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄×C₅ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₃×C₅×#1407

Subgroups

	subgroup	G/H
center, socle	C₂×C₃×C₅	#353
commutator	C₄²	C₂×C₃×C₄×C₅
Frattini, cosocle	#124	C₂²×C₃×C₅
prime radical	C₂×C₃×C₅×D₄	D₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₅ ◁ C₂²×C₃×C₅ ◁ C₂×C₃×C₄×C₅ ◁ C₃×C₅×#124 ◁ G

196 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 56 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)(4,5,6,7,8)(9,10)(11,19,21,34)(12,23,24,39)(13,26,28,17)(14,18)(15,20,29,36)(16,25)(22,32,27,35)(30,37,31,38)(33,40);(9,11,14,22)(10,15,18,31)(12,20,24,38)(13,27,28,21)(16,32,33,19)(17,29,26,30)(23,35,39,34)(25,36,40,37)

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, 139);