C₅×#13523 #11936243

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13523
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#13523

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#2092
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

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

normal subgroups only

This group has 308 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)(6,7,9,14,19,15,35,33,11,17,27,32,10,16,26,13)(8,18,21,36,23,28,37,31,24,29,22,30,20,34,25,12);(6,8,19,20,11,24,10,23)(7,12,15,30,17,31,16,36)(9,25,35,22,27,37,26,21)(13,18,14,34,33,29,32,28);(7,13)(8,20)(9,26)(10,19)(12,28)(14,16)(15,32)(17,33)(18,36)(21,22)(23,24)(27,35)(29,30)(31,34)

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, 6511);