C₅×#13668 #11936388

order	1280
nilpotency class	5
isoclinism	isoclinic class of #13667
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#13668

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#2184
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₅
prime radical	C₅×#3414	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂²×C₅×C₈ ◁ C₂×C₅×M₃₂ ◁ G

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

normal subgroups only

This group has 284 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)(6,7,12,17)(8,18,20,47)(9,23,25,53)(10,29,32,63)(11,33,31,16)(13,35,36,19)(14,39,41,27)(15,43,45,26)(21,51,50,68)(22,52,34,48)(24,44,56,69)(28,61,59,55)(30,40,64,67)(37,62,65,58)(38,49,46,57)(42,60,54,66);(6,8)(7,13)(9,24)(10,30)(11,34)(12,20)(14,40)(15,44)(16,46)(17,36)(18,23)(19,27)(21,26)(22,31)(25,56)(28,62)(29,37)(32,64)(33,38)(35,39)(41,67)(42,68)(43,50)(45,69)(47,53)(48,61)(49,66)(51,54)(52,55)(57,60)(58,59)(63,65);(9,25)(14,41)(18,47)(19,35)(23,53)(24,56)(27,39)(28,59)(40,67)(42,54)(48,52)(49,57)(51,68)(55,61)(58,62)(60,66)

Conjugacy classes & character table

155 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6656);