C₅×#13702 #11936422

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

Aliases

direct product

C₅×#13702

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#2215
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₅
prime radical	C₅×#2221	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#131 ◁ C₅×#504 ◁ G

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

normal subgroups only

This group has 184 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)(9,25)(10,12)(11,26)(15,29)(16,43)(17,37)(18,49)(19,35)(20,38)(22,53)(23,56)(24,30)(27,41)(28,42)(31,32)(33,44)(34,36)(39,48)(40,57)(45,64)(46,62)(47,61)(50,63)(51,54)(55,65)(59,67)(60,66)(68,69);(6,7,15,39,11,18,28,37,10,17,41,49,31,48,9,14,13,20,25,36,32,50,27,35,12,19,42,63,26,34,29,38)(8,16,40,64,23,45,51,43,30,47,53,67,58,68,22,33,24,46,54,60,56,66,57,62,21,44,65,69,52,59,55,61);(6,8)(7,16)(9,22)(10,30)(11,23)(12,21)(13,24)(14,33)(15,40)(17,47)(18,45)(19,44)(20,46)(25,54)(26,52)(27,57)(28,51)(29,55)(31,58)(32,56)(34,59)(35,62)(36,60)(37,43)(38,61)(39,64)(41,53)(42,65)(48,68)(49,67)(50,66)(63,69)

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, 6690);