C₃×C₅×#1321 #14274355

order	1920
nilpotency class	4
isoclinism	isoclinic class of #353
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₅×C₈ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₃×C₅×#1321

Subgroups

	subgroup	G/H
center	C₃×C₄×C₅	#99
commutator	C₂×C₄	C₂×C₃×C₅×C₈
Frattini, cosocle	C₂×M₁₆	C₂²×C₃×C₅
socle	C₂×C₃×C₅	#323
prime radical	C₂×C₃×C₅×#41	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄×C₅ ◁ C₂×C₃×C₄×C₅ ◁ C₂×C₃×C₅×M₁₆ ◁ G

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

normal subgroups only

This group has 224 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)(9,10,13,18,14,17,28,34,15,19,29,36,12,24,26,35)(11,20,22,37,21,16,30,31,23,33,27,39,25,32,38,40);(9,11)(10,16)(12,25)(13,27)(14,21)(15,23)(17,33)(18,31)(19,32)(20,24)(22,29)(26,30)(28,38)(34,39)(35,37)(36,40)

Conjugacy classes & character table

390 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 53);