C₅×#13666 #11936386

order	1280
nilpotency class	5
isoclinism	isoclinic class of #13661
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#13666

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₅
prime radical	C₂×C₅×#508	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂²×C₅×C₈ ◁ C₂²×C₅×C₁₆ ◁ G

288 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 73
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 3 generators on 73 points:

(1,2,3,4,5)(6,7)(8,9)(10,11,16,22)(12,23,28,54)(13,18,31,21)(14,32,35,66)(15,36,38,44)(17,39,42,25)(19,45,48,34)(20,49,51,30)(24,53,27,60)(26,58,59,72)(29,61,64,65)(33,50,68,73)(37,46,56,71)(40,57,52,55)(41,62,69,67)(43,70,47,63);(10,12)(11,17)(13,24)(14,33)(15,37)(16,28)(18,40)(19,46)(20,50)(21,52)(22,42)(23,32)(25,34)(26,30)(27,31)(29,62)(35,68)(36,41)(38,56)(39,45)(43,58)(44,69)(47,72)(48,71)(49,59)(51,73)(53,61)(54,66)(55,63)(57,70)(60,65)(64,67);(7,9)(10,13,16,31)(11,18,22,21)(12,24,28,27)(14,29,35,64)(15,30,38,49)(17,40,42,52)(19,43,48,47)(20,44,51,36)(23,53,54,60)(25,55,39,57)(26,56,59,37)(32,61,66,65)(33,62,68,67)(34,63,45,70)(41,50,69,73)(46,58,71,72)

Conjugacy classes & character table

245 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6654);