C₅×#11869 #11934589

order	1280
nilpotency class	4
isoclinism	isoclinic class of #11867
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#11869

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1519
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂×M₁₆	C₂³×C₅
prime radical	C₂×C₅×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×#131 ◁ G

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

normal subgroups only

This group has 364 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) = 69
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 69 points:

(1,2,3,4,5)(6,7)(8,19,22,50)(9,25)(10,31,34,48)(11,17)(12,37)(13,18)(14,21)(15,29)(16,35,46,36)(20,51,53,67)(23,44,24,58)(26,61,54,49)(27,45,33,42)(28,43)(30,60)(32,47)(38,62)(39,56)(40,57)(41,55)(52,68)(59,63,65,64)(66,69);(6,8,13,22)(7,14,18,40)(9,26,30,54)(10,32,34,56)(11,24,12,23)(15,41,43,68)(16,44,46,58)(17,47,37,39)(19,48,50,31)(20,29,53,28)(21,35,57,36)(25,38,60,66)(27,52,33,55)(42,49,45,61)(51,59,67,65)(62,63,69,64);(6,9,11,28,13,30,12,29)(7,15,37,60,18,43,17,25)(8,20,23,26,22,53,24,54)(10,33,35,64,34,27,36,63)(14,38,47,68,40,66,39,41)(16,45,48,59,46,42,31,65)(19,49,58,67,50,61,44,51)(21,55,56,69,57,52,32,62)

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, 4857);