C₅×#13709 #11936429

order	1280
nilpotency class	5
isoclinism	isoclinic class of #13709
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#13709

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#2226
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₅
prime radical	C₅×#508	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₄×C₅×C₈ ◁ C₄×C₅×C₁₆ ◁ G

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

normal subgroups only

This group has 154 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) = 69
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 69 points:

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

Conjugacy classes & character table

185 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6697);