C₅×#13491 #11936211

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13490
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#13491

Subgroups

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

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

normal subgroups only

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

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6479);