C₅×#13198 #11935918

order	1280
nilpotency class	3
isoclinism	isoclinic class of #2010
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#13198

Subgroups

	subgroup	G/H
center	C₅×C₈	#127
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂×C₅	#1955
prime radical	C₅×#575	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₅×C₈ ◁ C₂²×C₅×C₈ ◁ G

410 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 138 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)(6,7,10,15,11,16,26,37,12,17,27,38,28,39,52,63)(8,13,20,31,21,32,46,58,19,30,44,56,45,57,67,69)(9,14,24,35,25,36,41,54,23,34,48,60,51,62,66,64)(18,40,43,65,22,47,50,68,42,29,53,55,49,33,59,61);(6,8,10,20,11,21,26,46,12,19,27,44,28,45,52,67)(7,13,15,31,16,32,37,58,17,30,38,56,39,57,63,69)(9,22,48,59,25,42,66,43,23,49,24,50,51,18,41,53)(14,33,60,68,36,40,64,55,34,47,35,61,62,29,54,65);(6,9)(7,14)(8,18,19,42)(10,24)(11,25)(12,23)(13,29,30,40)(15,35)(16,36)(17,34)(20,43,44,53)(21,22,45,49)(26,41)(27,48)(28,51)(31,55,56,65)(32,33,57,47)(37,54)(38,60)(39,62)(46,50,67,59)(52,66)(58,61,69,68)(63,64)

Conjugacy classes & character table

260 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6186);