C₅×#13519 #11936239

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13309
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#13519

Subgroups

	subgroup	G/H
center	C₄×C₅	#469
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂×C₅	#2087
prime radical	C₂³×C₅	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

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

normal subgroups only

This group has 252 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)(6,7,9,13,22,14,33,30,11,16,24,32,10,15,23,31)(8,17,20,26,25,34,37,27,19,35,36,28,18,29,21,12);(8,18,19,25)(12,26,27,28)(17,34,35,29)(20,21,36,37);(6,8,10,25,11,19,22,18)(7,12,15,26,16,27,14,28)(9,21,23,20,24,37,33,36)(13,29,31,17,32,34,30,35)

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, 6507);