C₅×#13459 #11936179

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13459
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#13459

Subgroups

	subgroup	G/H
center	C₄×C₅	#498
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂×C₅	#2058
prime radical	C₂³×C₅	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#97 ◁ G

318 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 118 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)(8,17,19,34)(9,22)(10,15)(11,27)(12,16)(13,20,29,36)(14,26)(18,31,21,33)(23,37)(24,28,35,32)(25,30);(8,18,19,21)(9,23)(13,28,29,32)(14,30)(17,33,34,31)(20,24,36,35)(22,37)(25,26);(6,8,11,18,12,19,10,21)(7,13,15,32,16,29,27,28)(9,24,26,20,23,35,25,36)(14,31,22,17,30,33,37,34)

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, 6447);