C₃×C₅×#1358 #14274392

order	1920
nilpotency class	4
isoclinism	isoclinic class of #1357
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₃×C₅×#1358

Subgroups

	subgroup	G/H
center, socle	C₂×C₃×C₅	#331
commutator	C₄²	C₂×C₃×C₄×C₅
Frattini, cosocle	C₄×C₈	C₂²×C₃×C₅
prime radical	C₂²×C₃×C₅	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₅ ◁ C₂×C₃×C₄×C₅ ◁ C₃×C₄×C₅×C₈ ◁ G

184 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 80 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)(9,10,12,17,13,16,25,32,14,19,26,35,28,18,36,34)(11,20,21,37,22,31,38,40,23,30,39,33,27,15,24,29);(9,11,14,23)(10,15,19,31)(12,24,26,38)(13,27,28,22)(16,30,18,20)(17,33,35,37)(21,25,39,36)(29,32,40,34)

Conjugacy classes & character table

345 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 90);