C₅×#13500 #11936220

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13392
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#13500

Subgroups

	subgroup	G/H
center	C₄×C₅	#482
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂×C₅	#2076
prime radical	C₂×C₅×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#131 ◁ G

276 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 98 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)(6,7,11,16,31,45,62,58,12,17,34,48,10,15,30,44)(8,18,21,50,20,39,55,46,23,38,32,66,22,13,35,37)(9,24,27,43,29,64,65,42,28,59,63,41,26,47,33,14)(19,51,53,36,52,57,67,68,25,60,61,40,54,69,56,49);(6,8)(7,13)(9,25)(10,22)(11,32)(12,23)(14,40)(15,38)(16,46)(17,39)(18,45)(19,28)(20,31)(21,34)(24,57)(26,52)(27,53)(29,54)(30,55)(33,56)(35,62)(36,42)(37,48)(41,68)(43,49)(44,50)(47,51)(58,66)(59,69)(60,64)(61,63)(65,67);(6,9,12,28)(7,14,17,42)(8,19)(10,29,31,26)(11,33,34,65)(13,36)(15,43,45,41)(16,47,48,64)(18,49)(20,54)(21,56)(22,52)(23,25)(24,58,59,44)(27,62,63,30)(32,67)(35,53)(37,51)(38,68)(39,40)(46,60)(50,69)(55,61)(57,66)

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6488);