C₅×#13544 #11936264

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13392
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#13544

Subgroups

	subgroup	G/H
center	C₄×C₅	#482
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂×C₅	#2101
prime radical	C₂³×C₅	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#405 ◁ G

212 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 94 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)(8,20,23,48)(9,15,27,42)(10,16)(11,19)(12,17)(13,18)(14,22,38,34)(21,50)(24,44,55,68)(25,57)(26,40,35,49)(28,41,60,67)(29,43)(30,47)(31,62,61,37)(32,45,51,69)(33,46)(36,54)(39,56)(52,64)(53,63)(58,66)(59,65);(6,8,13,34,11,23,10,22)(7,14,16,20,19,38,18,48)(9,25,35,56,27,59,26,58)(12,31,33,24,30,61,29,55)(15,39,40,57,42,66,49,65)(17,44,43,62,47,68,46,37)(21,51,54,60,53,32,52,28)(36,45,50,67,64,69,63,41);(6,9,10,26,11,27,13,35)(7,15,16,40,19,42,18,49)(8,21,22,52,23,53,34,54)(12,32,29,60,30,51,33,28)(14,36,20,50,38,64,48,63)(17,45,43,67,47,69,46,41)(24,56,31,25,55,58,61,59)(37,65,44,39,62,57,68,66)

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, 6532);