C₃×C₅×#1722 #14274756

order	1920
nilpotency class	3
isoclinism	isoclinic class of #1714
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃×C₅ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₃×C₅×#1722

Subgroups

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

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

normal subgroups only

This group has 396 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)(10,13)(11,14)(15,16)(17,18,21,24)(19,25,26,32)(20,28,29,23)(22,30,31,27);(17,19,20,27,21,26,29,30)(18,22,23,25,24,31,28,32);(9,10,14,16,12,15,11,13)(19,26)(22,31)(25,32)(27,30)

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, 454);