C₅×#13489 #11936209

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13392
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#13489

Subgroups

	subgroup	G/H
center	C₄×C₅	#482
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂×C₅	#2073
prime radical	C₂×C₅×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#405 ◁ G

278 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 94 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)(8,20,24,14)(9,25,29,49)(10,17)(11,18)(12,19)(13,16)(15,27,41,36)(21,52)(22,48,35,38)(23,51,57,39)(26,59)(28,46,62,69)(30,44)(31,63,64,45)(32,65,66,42)(33,47)(34,43)(37,53)(40,60)(50,54)(55,67)(56,68)(58,61);(6,8)(7,14)(9,26)(10,22)(11,31)(12,35)(13,24)(15,40)(16,20)(17,38)(18,45)(19,48)(21,28)(23,33)(25,58)(27,60)(29,61)(30,57)(32,53)(34,64)(36,56)(37,42)(39,47)(41,68)(43,63)(44,51)(46,67)(49,59)(50,65)(52,69)(54,66)(55,62);(6,9)(7,15,16,41)(8,21)(10,27)(11,32)(12,36)(13,29)(14,37,20,50)(17,25,19,49)(18,46,43,69)(22,53)(23,56)(24,55)(26,31)(28,33)(30,62)(34,66)(35,54)(38,67,48,52)(39,59,51,58)(40,63,68,45)(42,44,65,47)(57,60)(61,64)

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, 6477);