C₅×#13312 #11936032

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13312
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#13312

Subgroups

	subgroup	G/H
center	C₄×C₅	#469
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂×C₅	#1999
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#131 ◁ G

520 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 114 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)(8,20)(9,26)(10,31)(11,18)(12,17)(13,19)(14,22)(15,29)(16,34)(21,50)(23,56)(24,44)(25,49)(27,59)(28,61)(30,42)(32,47)(33,46)(35,45)(36,43)(37,66)(38,54)(39,55)(40,51)(41,64)(48,52)(53,69)(57,60)(58,63)(62,67)(65,68);(6,8)(7,14)(9,27)(10,32)(11,23)(12,24)(13,25)(15,40)(16,44)(17,38)(18,47)(19,39)(20,31)(21,29)(22,34)(26,57)(28,62)(30,52)(33,55)(35,54)(36,53)(37,42)(41,48)(43,67)(45,49)(46,56)(50,58)(51,64)(59,61)(60,63)(65,66)(68,69);(6,9)(7,15)(8,21)(10,28)(11,36)(12,29)(13,30)(14,37)(16,41)(17,42)(18,26)(19,43)(20,48)(22,51)(23,27)(24,52)(25,53)(31,58)(32,60)(33,65)(34,63)(35,64)(38,67)(39,57)(40,47)(44,69)(45,68)(46,61)(49,59)(50,56)(54,66)(55,62)

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, 6300);