C₅×#8798 #11931518

order	1280
nilpotency class	3
isoclinism	isoclinic class of #8794
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₅×#8798

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#382
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂³×C₅	#125
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

220 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 110 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)(6,7,11,15,12,16,10,20)(8,17,18,21,19,14,9,13)(22,23,27,33,28,31,26,32)(24,34,36,48,37,29,42,45)(25,38,40,47,41,49,43,30)(35,50,52,46,39,53,51,44);(6,8,10,9,12,19,11,18)(7,13,15,17,16,21,20,14)(22,24,26,36,28,37,27,42)(23,29,32,45,31,34,33,48)(25,39,40,52,41,35,43,51)(30,46,38,50,47,44,49,53);(6,9,10,19,12,18,11,8)(7,14,15,13,16,17,20,21)(22,25)(23,30,31,47)(24,35)(26,40)(27,43)(28,41)(29,44,34,46)(32,38,33,49)(36,51)(37,39)(42,52)(45,53,48,50)

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, 1786);