C₅×#13718 #11936438

order	1280
nilpotency class	5
isoclinism	isoclinic class of #13717
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#13718

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#2233
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₅
prime radical	C₂³×C₅	SD₃₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂²×C₅×C₈ ◁ C₂×C₅×M₃₂ ◁ G

160 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 66 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)(6,7,12,17)(8,18,19,47)(9,23,24,57)(10,29,32,65)(11,33,31,16)(13,35,36,20)(14,39,40,27)(15,43,45,26)(21,55,51,69)(22,56,34,50)(25,61,53,44)(28,48,60,59)(30,49,66,41)(37,62,68,64)(38,54,46,67)(42,63,58,52);(9,24)(14,40)(18,47)(20,35)(23,57)(25,53)(27,39)(28,60)(41,49)(42,58)(48,59)(50,56)(52,63)(54,67)(55,69)(62,64);(6,8,11,34,12,19,31,22)(7,13,16,46,17,36,33,38)(9,25,28,64,24,53,60,62)(10,30,26,21,32,66,43,51)(14,41,42,55,40,49,58,69)(15,44,29,37,45,61,65,68)(18,48,56,23,47,59,50,57)(20,52,67,27,35,63,54,39)

Conjugacy classes & character table

155 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6706);