C₃×C₅×#1709 #14274743

order	1920
nilpotency class	3
isoclinism	isoclinic class of #1705
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃×C₅ · Schur multiplier (H₂) = C₂×C₈

Aliases

direct product

C₃×C₅×#1709

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃×C₅	#127
commutator, Frattini, cosocle	C₄²	C₂³×C₃×C₅
prime radical	C₃×C₅×#493	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₃×C₄²×C₅ ◁ G

424 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 160 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) = 32
transitive degree	μ_t(G) = 960

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2,3)(4,5,6,7,8)(10,11)(12,13)(15,16)(17,18)(19,25)(20,28)(21,23)(22,26)(24,29)(27,31)(30,32);(9,10,13,15,14,16,12,11)(17,19,20,26,21,27,29,32)(18,22,24,25,23,30,28,31);(9,11,12,16,14,15,13,10)(18,23)(22,30)(24,28)(25,31)

Conjugacy classes & character table

570 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 441);