C₃×C₅×#1419 #14274453

order	1920
nilpotency class	5
isoclinism	isoclinic class of #1418
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₃×C₅×#1419

Subgroups

	subgroup	G/H
center, socle	C₂×C₃×C₅	#357
commutator	C₁₆	C₂×C₃×C₄×C₅
Frattini, cosocle	C₂×C₁₆	C₂²×C₃×C₅
prime radical	C₂×C₃×C₅×D₁₆	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₅ ◁ C₂×C₃×C₄×C₅ ◁ C₂×C₃×C₅×C₈ ◁ C₂×C₃×C₅×C₁₆ ◁ G

156 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 72 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)(9,10,12,18,15,20,14,31)(11,21,24,38,27,16,26,32)(13,28,23,35,30,36,29,19)(17,34,33,22,37,40,39,25);(10,16)(11,22)(12,14)(13,29)(17,35)(18,38)(19,37)(20,21)(23,30)(24,34)(25,27)(26,40)(28,33)(31,32)(36,39)

Conjugacy classes & character table

390 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 151);