C₅×#13511 #11936231

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13459
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#13511

Subgroups

	subgroup	G/H
center	C₄×C₅	#498
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂×C₅	#2084
prime radical	C₂³×C₅	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#97 ◁ G

250 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 110 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)(6,7,11,12,10,15,23,27)(8,16,19,32,18,13,20,29)(9,21,24,25,26,30,28,14)(17,22,34,31,36,37,35,33);(6,8,10,18)(7,13,15,16)(9,22)(11,19,23,20)(12,29,27,32)(14,17)(21,34)(24,31)(25,36)(26,37)(28,33)(30,35);(6,9,23,24,10,26,11,28)(7,14,27,21,15,25,12,30)(8,17,19,35,18,36,20,34)(13,31,29,22,16,33,32,37)

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, 6499);