C₃×C₅×#1411 #14274445

order	1920
nilpotency class	5
isoclinism	isoclinic class of #1408
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄×C₅ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₃×C₅×#1411

Subgroups

	subgroup	G/H
center, socle	C₂×C₃×C₅	#353
commutator	C₄²	C₂×C₃×C₄×C₅
Frattini, cosocle	C₄⋊Q₈	C₂²×C₃×C₅
prime radical	C₂²×C₃×C₅	#99
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₅ ◁ C₂²×C₃×C₅ ◁ C₂×C₃×C₄×C₅ ◁ C₃×C₅×C₄⋊Q₈ ◁ G

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

normal subgroups only

This group has 188 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)(9,10,12,17,14,19,26,36)(11,20,21,34,23,31,38,39)(13,18,28,35,29,16,25,33)(15,24,30,27,32,37,40,22);(9,11,14,23)(10,15,19,32)(12,24,26,37)(13,27,29,22)(16,31,18,20)(17,34,36,39)(21,25,38,28)(30,35,40,33)

Conjugacy classes & character table

255 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 143);