C₅×#13691 #11936411

order	1280
nilpotency class	5
isoclinism	isoclinic class of #13691
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#13691

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#2206
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₅
prime radical	C₅×#2263	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₄×C₅×C₈ ◁ C₄×C₅×C₁₆ ◁ G

200 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 70 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)(8,18)(9,24)(10,29)(11,32)(12,36)(13,19)(14,21)(15,26)(16,31)(17,34)(20,49)(22,53)(23,48)(25,58)(27,57)(28,59)(30,39)(33,47)(35,65)(37,45)(38,62)(40,60)(41,66)(42,54)(43,61)(44,55)(46,52)(50,56)(51,69)(63,64)(67,68);(6,8,10,21,13,23,31,12)(7,14,16,18,19,36,29,48)(9,20,25,50,28,51,27,37)(11,22,30,52,35,55,63,34)(15,38,40,47,43,41,42,68)(17,39,44,32,46,64,53,65)(24,56,57,49,59,45,58,69)(26,33,54,62,61,67,60,66);(6,9)(7,15)(8,20)(10,25)(11,33)(12,37)(13,28)(14,38)(16,40)(17,45)(18,47)(19,43)(21,50)(22,54)(23,51)(24,32)(26,34)(27,31)(29,42)(30,62)(35,67)(36,41)(39,58)(44,69)(46,56)(48,68)(49,53)(52,61)(55,60)(57,64)(59,65)(63,66)

Conjugacy classes & character table

185 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6679);