C₅×#11970 #11934690

order	1280
nilpotency class	4
isoclinism	isoclinic class of #11858
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#11970

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₄×C₈	C₂³×C₅
prime radical	C₂³×C₅	#102
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

284 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 53
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 3 generators on 53 points:

(1,2,3,4,5)(6,7,9,13,10,12,18,20)(8,14,15,21,16,11,17,19)(22,23,27,33,28,31,43,49,29,34,41,51,25,40,26,32)(24,35,38,48,36,50,37,45,39,52,42,30,44,46,53,47);(6,8,10,16)(7,11,12,14)(9,17,18,15)(13,21,20,19)(22,24)(23,30)(25,36)(26,42)(27,37)(28,44)(29,39)(31,45)(32,50)(33,35)(34,48)(38,43)(40,47)(41,53)(46,49)(51,52);(7,12)(11,14)(13,20)(19,21)(22,25,29,28)(23,31,34,40)(24,36,39,44)(26,41,43,27)(30,45,48,47)(32,33,49,51)(35,46,52,50)(37,42,53,38)

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, 4958);