C₃×C₅×#1710 #14274744

order	1920
nilpotency class	3
isoclinism	isoclinic class of #1705
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃×C₅ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₃×C₅×#1710

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₃×C₅	#127
commutator, Frattini, cosocle	C₄²	C₂³×C₃×C₅
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₃×C₄²×C₅ ◁ G

336 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 160 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)(9,10,13,15)(11,17,19,24)(12,20,21,16)(14,22,23,18)(25,26)(27,33)(28,30)(29,37)(31,35)(32,36)(34,39)(38,40);(9,11,12,18,13,19,21,22)(10,14,16,17,15,23,20,24)(26,30)(31,38)(32,37)(33,39);(10,15)(14,23)(16,20)(17,24)(25,27,29,35,28,34,36,40)(26,31,32,33,30,38,37,39)

Conjugacy classes & character table

570 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 442);