C₅×#13663 #11936383

order	1280
nilpotency class	5
isoclinism	isoclinic class of #13661
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#13663

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₅
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂²×C₅×C₈ ◁ C₂²×C₅×C₁₆ ◁ G

298 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 640

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

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

Conjugacy classes & character table

245 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6651);