C₃×C₅×#1658 #14274692

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×#1658

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃×C₅	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₃×C₅
prime radical	C₃×C₅×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₃×C₄²×C₅ ◁ G

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

normal subgroups only

This group has 364 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) = 960

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

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

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, 390);