C₅×#13693 #11936413

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

Aliases

direct product

C₅×#13693

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#2206
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₅
prime radical	C₅×#508	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₄×C₅×C₈ ◁ C₄×C₅×C₁₆ ◁ G

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

normal subgroups only

This group has 196 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)(6,7,14,21)(8,19,25,40)(9,26,29,58)(10,20,13,17)(11,31,35,63)(12,15,24,41)(16,42,44,28)(18,46,49,33)(22,52,53,69)(23,45,55,48)(27,59,37,57)(30,39,34,62)(32,50,64,67)(36,47,56,68)(38,65,66,61)(43,60,51,54);(6,8,13,24,14,25,10,12)(7,15,20,40,21,41,17,19)(9,22,37,56,29,53,27,36)(11,23,34,46,35,55,30,33)(16,38,51,67,44,66,43,50)(18,39,48,63,49,62,45,31)(26,47,59,69,58,68,57,52)(28,32,54,65,42,64,60,61);(6,9)(7,16)(8,22)(10,27)(11,32)(12,36)(13,37)(14,29)(15,38)(17,43)(18,47)(19,50)(20,51)(21,44)(23,54)(24,56)(25,53)(26,31)(28,33)(30,61)(34,65)(35,64)(39,59)(40,67)(41,66)(42,46)(45,52)(48,69)(49,68)(55,60)(57,62)(58,63)

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, 6681);