C₃×C₅×#1328 #14274362

order	1920
nilpotency class	4
isoclinism	isoclinic class of AΣL(1,𝔽₁₆)
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₅×C₈ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₃×C₅×#1328

Subgroups

	subgroup	G/H
center	C₃×C₄×C₅	#99
commutator	C₂³	C₂×C₃×C₅×C₈
Frattini, cosocle	C₂×M₁₆	C₂²×C₃×C₅
socle	C₂×C₃×C₅	#324
prime radical	C₂³×C₃×C₅	C₂×C₈
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄×C₅ ◁ C₂×C₃×C₄×C₅ ◁ C₂×C₃×C₅×M₁₆ ◁ G

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

normal subgroups only

This group has 180 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)(9,10,12,17,13,18,23,31,14,16,22,28,24,29,35,38)(11,15,20,25,21,27,34,36,19,26,32,30,33,37,40,39);(9,11)(10,15,16,26)(12,20)(13,21)(14,19)(17,30,28,25)(18,27,29,37)(22,32)(23,34)(24,33)(31,39,38,36)(35,40)

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, 60);