C₅×#13499 #11936219

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13312
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#13499

Subgroups

	subgroup	G/H
center	C₄×C₅	#469
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)(6,7,10,15,30,16,46,43,12,18,32,45,11,17,31,44)(8,13,20,35,21,47,53,68,23,37,55,65,22,36,54,64)(9,24,27,40,33,58,62,39,26,57,61,41,28,42,29,14)(19,48,50,66,25,59,60,63,49,67,69,38,51,56,52,34);(6,8)(7,13)(9,25)(10,20)(11,22)(12,23)(14,38)(15,35)(16,47)(17,36)(18,37)(19,28)(21,30)(24,56)(26,51)(27,60)(29,50)(31,54)(32,55)(33,49)(34,40)(39,66)(41,63)(42,67)(43,68)(44,64)(45,65)(46,53)(48,58)(52,61)(57,59)(62,69);(6,9,11,33,12,26,30,28)(7,14,17,40,18,39,16,41)(8,19,22,25,23,49,21,51)(10,29,31,27,32,62,46,61)(13,34,36,66,37,63,47,38)(15,42,44,24,45,58,43,57)(20,52,54,50,55,60,53,69)(35,56,64,48,65,59,68,67)

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, 6487);