C₅×#67584 #14279664

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#67584

Splittings

Hall	C₃×C₅ ⋊ #1420
Sylow	C₃ ⋊ C₅×#1420

Subgroups

	subgroup	G/H
center	C₂×C₅	#4491
commutator	C₃×C₁₆	C₂×C₄×C₅
Frattini	C₂×C₁₆	C₂×C₅×D₃
Fitting	C₃×C₅×#368	C₂
socle	C₂×C₃×C₅	#357
prime radical	C₂²×C₃×C₅	SD₃₂
cosocle	C₂×C₃×C₁₆	C₂²×C₅
hypercenter	C₅×#368	D₃
hypocenter	C₃	C₅×#1420
LCS	G ▷ C₃×C₁₆ ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₈ ◁ C₂×C₅×C₁₆ ◁ C₅×#368

140 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 72
transitive degree	μ_t(G) = 960

A minimal faithful permutation representation — 2 generators on 72 points:

(1,2,3,4,5)(7,8)(9,11,16,27)(10,17,21,43)(12,29,34,58)(13,26,15,24)(14,36,38,56)(18,45,48,54)(19,49,39,41)(20,50,51,52)(22,28,55,65)(23,32,59,69)(25,30,61,37)(31,67,33,66)(35,42,46,71)(40,44,57,68)(47,60,64,72)(53,70,62,63);(6,7,8)(9,10,13,19,16,21,15,39)(11,22,24,53,27,55,26,62)(12,28,31,63,34,65,33,70)(14,35,30,51,38,46,37,20)(17,40,41,23,43,57,49,59)(18,44,47,69,48,68,64,32)(25,60,56,45,61,72,36,54)(29,50,66,71,58,52,67,42)

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, 5362);