C₃×C₅×#1342 #14274376

order	1920
nilpotency class	4
isoclinism	isoclinic class of #1339
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₃×C₅×#1342

Subgroups

	subgroup	G/H
center, socle	C₂×C₃×C₅	#327
commutator	C₂×C₈	C₂×C₃×C₄×C₅
Frattini, cosocle	C₂×M₁₆	C₂²×C₃×C₅
prime radical	C₂³×C₃×C₅	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₅ ◁ C₂×C₃×C₄×C₅ ◁ C₂×C₃×C₅×M₁₆ ◁ G

200 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 72 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)(9,10,13,18,15,19,14,30)(11,20,22,34,24,37,23,40)(12,25,27,17,29,35,28,36)(16,26,31,39,33,38,32,21);(9,11,15,24)(10,16,19,33)(12,26,29,38)(13,23,14,22)(17,34,36,40)(18,31,30,32)(20,35,37,25)(21,28,39,27)

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, 74);