C₅×#69381 #14319278

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#69381

Splittings

Fitting	C₃×C₅×SD₆₄ ⋊ C₂
Hall	C₃×C₅ ⋊ #2264
Sylow	C₃ ⋊ C₅×#2264

Subgroups

	subgroup	G/H
center	C₂×C₅	D₃×D₁₆
commutator, cosocle	C₃×C₁₆	C₂³×C₅
Frattini	C₁₆	C₂²×C₅×D₃
Fitting	C₃×C₅×SD₆₄	C₂
socle	C₂×C₃×C₅	C₂×D₁₆
prime radical	C₅×#4912	C₂
hypercenter	C₅×SD₆₄	D₃
hypocenter	C₃	C₅×#2264
LCS	G ▷ C₃×C₁₆ ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₄×C₅ ◁ C₅×C₈ ◁ C₅×C₁₆ ◁ C₅×SD₆₄

208 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 70 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 — 3 generators on 72 points:

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

Conjugacy classes & character table

225 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 44976);