C₅×#13700 #11936420

order	1280
nilpotency class	5
isoclinism	isoclinic class of #13700
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#13700

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#2215
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₅
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#131 ◁ C₅×#504 ◁ G

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

normal subgroups only

This group has 188 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)(7,14)(8,21)(9,15)(10,12)(11,25)(16,42)(17,36)(18,48)(19,34)(20,37)(22,52)(23,54)(24,29)(26,40)(27,39)(28,41)(30,31)(32,43)(33,35)(38,50)(44,67)(45,60)(46,59)(47,62)(49,61)(53,64)(55,63)(57,66)(58,65)(68,69);(6,7,15,35,11,18,26,34,10,17,39,61,30,47,28,37,13,20,41,62,31,49,27,36,12,19,40,48,25,33,9,14)(8,16,38,58,23,44,55,60,29,46,64,69,56,68,53,59,24,45,63,67,54,65,50,42,21,43,52,66,51,57,22,32);(6,8)(7,16)(9,22)(10,29)(11,23)(12,21)(13,24)(14,32)(15,38)(17,46)(18,44)(19,43)(20,45)(25,51)(26,55)(27,50)(28,53)(30,56)(31,54)(33,57)(34,60)(35,58)(36,42)(37,59)(39,64)(40,52)(41,63)(47,68)(48,66)(49,65)(61,69)(62,67)

Conjugacy classes & character table

230 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6688);