C₅×#13503 #11936223

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

Aliases

direct product

C₅×#13503

Subgroups

	subgroup	G/H
center	C₄×C₅	#482
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂×C₅	#2077
prime radical	C₂×C₅×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#131 ◁ G

272 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 98 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)(6,7,13,35,32,16,11,17,12,19,37,45,10,18,34,47)(8,20,36,66,22,14,23,50,25,44,54,39,24,40,56,69)(9,26,33,43,28,62,29,42,30,59,63,15,31,46,64,48)(21,51,55,68,57,58,53,38,27,60,65,49,52,67,61,41);(6,8)(7,14)(9,27)(10,24)(11,23)(12,25)(13,36)(15,41)(16,44)(17,39)(18,20)(19,40)(21,30)(22,32)(26,58)(28,52)(29,61)(31,57)(33,65)(34,56)(35,50)(37,54)(38,43)(42,49)(45,69)(46,51)(47,66)(48,68)(53,64)(55,63)(59,67)(60,62);(6,9,10,28,12,30,32,31)(7,15,16,42,19,43,18,48)(8,21,22,52,25,27,24,57)(11,33,13,29,34,63,37,64)(14,38,20,49,40,41,44,68)(17,46,45,59,47,62,35,26)(23,55,54,61,56,65,36,53)(39,60,50,67,66,51,69,58)

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, 6491);