C₅×#13441 #11936161

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13313
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#13441

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#466
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂³×C₅	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#405 ◁ G

378 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 73
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 3 generators on 73 points:

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

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6429);