C₅×#13313 #11936033

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13313
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#13313

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#466
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#131 ◁ G

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

normal subgroups only

This group has 556 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) = 41
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 41 points:

(1,2,3,4,5)(6,7)(8,9)(10,11)(12,23)(13,17)(14,21)(15,20)(16,22)(18,26)(19,28)(24,37)(25,35)(27,36)(29,32)(30,33)(31,41)(34,40)(38,39);(11,17)(12,24)(13,28)(14,15)(18,34)(19,20)(21,32)(22,33)(23,38)(25,40)(26,27)(29,30)(31,36)(35,37);(7,9)(10,12)(11,18)(13,25)(14,24)(15,26)(16,27)(17,31)(19,35)(20,36)(21,23)(22,37)(28,39)(29,41)(30,40)(32,34)(33,38)

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, 6301);