C₃×C₅×#1353 #14274387

order	1920
nilpotency class	4
isoclinism	isoclinic class of #1343
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₃×C₅×#1353

Subgroups

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

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

normal subgroups only

This group has 232 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) = 32
transitive degree	μ_t(G) = 480

A minimal faithful permutation representation — 2 generators on 32 points:

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

Conjugacy classes & character table

345 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 85);