C₅×#13513 #11936233

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

Aliases

direct product

C₅×#13513

Subgroups

	subgroup	G/H
center	C₄×C₅	#482
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂×C₅	#2085
prime radical	C₂³×C₅	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#405 ◁ G

228 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 102 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)(6,7,10,28,11,17,13,16)(8,19,21,14,23,39,35,40)(9,15,25,44,27,45,36,42)(12,31,29,46,30,18,34,62)(20,51,52,67,53,49,54,37)(22,47,32,66,56,50,63,38)(24,57,58,41,59,69,55,68)(26,60,33,65,61,43,64,48);(6,8,10,21,11,23,13,35)(7,14,16,19,17,40,28,39)(9,24,25,58,27,59,36,55)(12,32,29,56,30,63,34,22)(15,41,42,57,45,68,44,69)(18,47,46,38,31,50,62,66)(20,33,52,61,53,64,54,26)(37,65,49,60,67,48,51,43);(6,9,10,25,11,27,13,36)(7,15,16,42,17,45,28,44)(8,20,21,52,23,53,35,54)(12,33,29,61,30,64,34,26)(14,37,19,49,40,67,39,51)(18,48,46,43,31,65,62,60)(22,55,32,24,56,58,63,59)(38,57,50,68,66,69,47,41)

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, 6501);