C₅×#13472 #11936192

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13472
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#13472

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#2067
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂²×C₅×D₈	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#405 ◁ G

316 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 37
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 3 generators on 37 points:

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

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6460);