C₅×#13075 #11935795

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13074
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#13075

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1893
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂×C₅×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#131 ◁ G

458 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 146 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)(8,18)(9,24,26,14)(10,28)(11,33)(12,16)(13,21)(15,31)(17,30)(19,41)(20,42,49,62)(22,45)(23,55)(25,58,29,60)(27,39,59,57)(32,34)(35,48,56,51)(36,46)(37,53)(38,47,40,61)(43,54)(44,52)(50,68,67,64)(63,69,65,66);(6,8,19,46,11,23,10,21,12,22,31,54,30,52,32,53)(7,13,34,55,17,36,15,18,16,37,28,44,33,43,41,45)(9,20,47,66,27,50,29,51,26,49,61,69,59,67,25,48)(14,35,60,68,39,63,40,42,24,56,58,64,57,65,38,62);(6,9)(7,14)(8,20)(10,29)(11,27)(12,26)(13,35)(15,40)(16,24)(17,39)(18,42)(19,47)(21,51)(22,49)(23,50)(25,32)(28,58)(30,59)(31,61)(33,57)(34,60)(36,63)(37,56)(38,41)(43,65)(44,64)(45,62)(46,66)(48,53)(52,67)(54,69)(55,68)

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, 6063);