C₅×#10669 #11933389

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#10669

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂²×C₅	#399
prime radical	C₂³×C₅	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ G

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

normal subgroups only

This group has 296 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) = 53
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 53 points:

(1,2,3,4,5)(6,7,10,12,11,15,9,14)(8,16,18,21,19,20,17,13)(22,23,28,32,27,33,26,31)(24,34,42,29,37,45,36,44)(25,30,40,49,41,48,39,47)(35,50,38,53,52,46,51,43);(7,12)(8,17)(9,10)(13,21)(14,15)(18,19)(22,24,26,42,27,37,28,36)(23,29,31,45,33,44,32,34)(25,38,39,52,41,51,40,35)(30,46,47,43,48,50,49,53);(6,8,11,19)(7,13,15,21)(9,18,10,17)(12,20,14,16)(22,25,27,41)(23,30,33,48)(24,35,37,52)(26,39,28,40)(29,43,44,53)(31,47,32,49)(34,46,45,50)(36,51,42,38)

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 3657);