C₅×#13520 #11936240

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13312
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#13520

Subgroups

	subgroup	G/H
center	C₄×C₅	#469
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂×C₅	#2087
prime radical	C₂³×C₅	C₂×Q₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#405 ◁ G

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

normal subgroups only

This group has 238 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)(6,7,10,15,28,39,57,66,12,16,30,40,11,17,29,41)(8,13,19,33,47,62,67,69,20,35,48,64,21,34,49,63)(9,22,25,42,31,58,56,36,24,51,54,37,26,38,27,14)(18,43,44,65,50,68,55,59,23,52,53,60,45,61,46,32);(6,8)(7,13)(9,23,24,18)(10,19)(11,21)(12,20)(14,32,36,59)(15,33)(16,35)(17,34)(22,43,51,52)(25,53,54,44)(26,50,31,45)(27,55,56,46)(28,47)(29,49)(30,48)(37,60,42,65)(38,61,58,68)(39,62)(40,64)(41,63)(57,67)(66,69);(6,9,11,31,12,24,28,26)(7,14,17,42,16,36,39,37)(8,18,21,50,20,23,47,45)(10,27,29,25,30,56,57,54)(13,32,34,65,35,59,62,60)(15,38,41,22,40,58,66,51)(19,46,49,44,48,55,67,53)(33,61,63,43,64,68,69,52)

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, 6508);