C₅×#13522 #11936242

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13522
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#13522

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#2088
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂³×C₅	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#405 ◁ G

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

normal subgroups only

This group has 240 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)(6,7,13,26,24,9,11,15,12,17,28,23,10,16,25,33)(8,18,27,30,21,32,20,34,22,31,36,37,19,14,29,35);(6,8,10,19,12,22,24,21)(7,14,9,18,17,32,16,31)(11,20,13,27,25,29,28,36)(15,30,23,34,33,37,26,35);(6,9,10,17,12,16,24,7)(8,14,19,18,22,32,21,31)(11,15,13,23,25,33,28,26)(20,35,27,30,29,34,36,37)

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, 6510);