C₅×#13447 #11936167

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13312
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#13447

Subgroups

	subgroup	G/H
center	C₄×C₅	#469
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂×C₅	#2054
prime radical	C₅×#3414	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#405 ◁ G

386 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 106 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)(9,27)(10,34)(11,37)(12,17)(13,39)(14,20)(15,23)(16,30)(18,36)(19,35)(22,41)(24,51)(25,43)(26,50)(28,52)(29,60)(31,49)(32,63)(33,57)(38,45)(40,42)(44,56)(46,58)(47,61)(48,62)(53,65)(54,68)(55,67)(59,69)(64,66);(6,8)(7,15)(9,28)(10,24)(11,38)(12,23)(13,40)(14,26)(16,46)(17,42)(18,50)(19,51)(20,45)(21,34)(22,31)(25,35)(27,41)(29,54)(30,55)(32,53)(33,59)(36,56)(37,43)(39,44)(47,68)(48,69)(49,65)(52,60)(57,67)(58,61)(62,64)(63,66);(6,9,10,29,35,61,11,30,14,33,36,62,13,32,12,31)(7,16,17,47,39,60,18,27,20,49,37,63,19,48,34,57)(8,22,23,53,40,64,56,59,26,55,38,58,25,54,24,28)(15,41,21,52,51,68,43,46,45,67,50,69,44,66,42,65)

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6435);