C₅×#13716 #11936436

order	1280
nilpotency class	5
isoclinism	isoclinic class of #13661
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#13716

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₅
prime radical	C₂³×C₅	D₁₆
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂²×C₅×C₈ ◁ C₂²×C₅×C₁₆ ◁ G

168 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 77
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 3 generators on 77 points:

(1,2,3,4,5)(6,7)(8,11)(9,10)(12,13)(14,15,20,26)(16,21,30,43)(17,22,34,46)(18,35,37,64)(19,38,36,25)(23,47,49,31)(24,50,51,32)(27,39,56,67)(28,57,59,77)(29,60,58,42)(33,52,61,45)(40,68,70,53)(41,71,72,54)(44,63,48,62)(55,73,74,66)(65,76,69,75);(7,10)(11,13)(14,16,20,30)(15,21,26,43)(17,27,34,56)(18,28,37,59)(19,29,36,58)(22,39,46,67)(23,40,49,70)(24,41,51,72)(25,42,38,60)(31,53,47,68)(32,54,50,71)(33,55,61,74)(35,57,64,77)(44,65,48,69)(45,66,52,73)(62,75,63,76);(6,8,9,12)(7,11,10,13)(15,22)(17,31)(18,32)(19,36)(21,39)(23,24)(25,52)(26,46)(27,53)(28,54)(29,58)(33,63)(34,47)(35,44)(37,50)(38,45)(40,41)(42,73)(43,67)(48,64)(49,51)(55,76)(56,68)(57,65)(59,71)(60,66)(61,62)(69,77)(70,72)(74,75)

Conjugacy classes & character table

245 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6704);