C₃×C₅×#1663 #14274697

order	1920
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×C₅×#1663

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₃×C₅	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₃×C₅
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₃×C₄²×C₅ ◁ G

344 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 120 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) = 1920

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

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

Conjugacy classes & character table

300 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 395);