C₅×#13199 #11935919

order	1280
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#13199

Subgroups

	subgroup	G/H
center	C₅×C₈	#124
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

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

normal subgroups only

This group has 406 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)(6,7,11,17,12,18,33,47,13,19,34,48,10,16,21,36)(8,20,25,49,23,38,50,46,24,44,32,60,30,14,35,39)(9,15,28,42,27,41,56,65,29,43,59,68,31,45,51,62)(22,52,55,67,26,37,57,64,54,40,61,66,53,63,58,69);(6,8,21,50,10,30,34,25,13,24,33,35,12,23,11,32)(7,14,36,49,16,44,48,39,19,38,47,60,18,20,17,46)(9,26,56,55,31,22,28,58,29,53,51,61,27,54,59,57)(15,40,65,64,45,37,42,67,43,52,62,69,41,63,68,66);(6,9)(7,15)(8,22,24,54)(10,31)(11,28)(12,27)(13,29)(14,37,38,63)(16,45)(17,42)(18,41)(19,43)(20,40,44,52)(21,51)(23,26,30,53)(25,55,32,61)(33,56)(34,59)(35,58,50,57)(36,62)(39,64,46,69)(47,65)(48,68)(49,66,60,67)

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, 6187);