C₅×#13669 #11936389

order	1280
nilpotency class	5
isoclinism	isoclinic class of #13669
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#13669

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#2184
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₅
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#131 ◁ C₅×#504 ◁ G

288 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 69
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 69 points:

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

Conjugacy classes & character table

185 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6657);