C₅×#13482 #11936202

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#13482

Subgroups

	subgroup	G/H
center, socle	C₂³×C₅	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂²×C₅×D₈	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

362 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 33
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 3 generators on 33 points:

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

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, 6470);