C₅×#13473 #11936193

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13472
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#13473

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#2067
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂³×C₅	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#405 ◁ G

296 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 106 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)(6,7,14,19,13,20,12,39)(8,21,41,56,26,44,25,62)(9,27,40,52,32,48,31,51)(10,33,38,18,11,35,22,17)(15,42,45,36,46,57,55,24)(16,34,49,29,50,37,53,30)(23,58,63,64,60,43,59,69)(28,65,67,68,66,54,61,47);(6,8,22,57,12,25,11,36,13,26,38,42,14,41,10,24)(7,15,33,56,19,45,18,44,20,46,35,62,39,55,17,21)(9,28,34,23,31,61,30,59,32,66,37,60,40,67,29,63)(16,47,51,58,49,65,27,64,50,68,52,43,53,54,48,69);(6,9,13,32)(7,16,20,50)(8,23,26,60)(10,34,11,37)(12,40,14,31)(15,43,46,58)(17,51,18,52)(19,53,39,49)(21,54,44,65)(22,29,38,30)(24,61,36,67)(25,63,41,59)(27,33,48,35)(28,42,66,57)(45,64,55,69)(47,56,68,62)

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6461);