C₅×#71894 #14308996

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#71894

Splittings

Hall	C₃×C₅ ⋊ #2030
Sylow	C₃ ⋊ C₅×#2030

Subgroups

	subgroup	G/H
center	C₂²×C₅	#953
commutator, cosocle	C₂⁴×C₃	C₂³×C₅
Frattini	C₂⁴	C₂²×C₅×D₃
Fitting	C₃×C₅×#543	C₂
socle	C₂²×C₃×C₅	#120
prime radical	C₂⁴×C₃×C₅	C₂³
hypercenter	C₅×#543	D₃
hypocenter	C₃	C₅×#2030
LCS	G ▷ C₂⁴×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂⁴×C₅ ◁ C₅×#543

580 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 142 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 — 3 generators on 32 points:

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

Conjugacy classes & character table

240 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 34694);