C₅×#8514 #11931234

order	1280
nilpotency class	3
isoclinism	isoclinic class of #8510
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#8514

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#379
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₅×#3568	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

648 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 138 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)(6,7,10,17,13,8,19,23)(9,25,28,61,30,51,47,60)(11,33,31,43,36,59,48,54)(12,18,32,15,37,24,49,21)(14,38,40,65,20,50,39,27)(16,44,46,56,22,55,58,35)(26,62,29,45,64,69,66,57)(34,41,67,42,68,52,63,53);(6,8)(7,13)(9,20)(10,15)(11,33)(12,24)(14,30)(16,35)(17,32)(18,37)(19,21)(22,56)(23,49)(25,34)(26,39)(27,45)(28,42)(29,52)(31,58)(36,59)(38,60)(40,64)(41,66)(43,55)(44,54)(46,48)(47,53)(50,61)(51,68)(57,65)(62,67)(63,69);(6,9)(7,14)(8,20)(10,28)(11,34)(12,29)(13,30)(15,42)(16,45)(17,40)(18,41)(19,47)(21,53)(22,57)(23,39)(24,52)(25,33)(26,49)(27,35)(31,67)(32,64)(36,68)(37,66)(38,44)(43,61)(46,69)(48,63)(50,55)(51,59)(54,60)(56,65)(58,62)

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, 1502);