C₅×#13540 #11936260

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13373
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#13540

Subgroups

	subgroup	G/H
center	C₄×C₅	#486
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂×C₅	#2099
prime radical	C₂³×C₅	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#97 ◁ G

234 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 102 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)(6,7,10,12,11,16,13,17)(8,18,20,31,22,27,21,14)(9,23,25,33,26,15,29,28)(19,24,34,36,32,37,35,30);(6,8,13,20,11,22,10,21)(7,14,17,18,16,31,12,27)(9,24,29,36,26,37,25,30)(15,32,33,35,23,19,28,34);(6,9,10,25,11,26,13,29)(7,15,12,28,16,23,17,33)(8,19,20,34,22,32,21,35)(14,30,18,24,31,36,27,37)

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, 6528);