C₅×#12466 #11935186

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12465
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#12466

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1677
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₅
prime radical	C₅×#135	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₄×C₅×C₈ ◁ G

186 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 70 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)(6,7,14,21)(8,22,29,54)(9,30,34,46)(10,18,35,40)(11,17,23,49)(12,20,13,19)(15,39,45,25)(16,31,48,32)(24,56,59,68)(26,43,36,44)(27,42,28,50)(33,63,38,51)(37,52,55,47)(41,62,66,58)(53,60,65,57)(61,67,64,69);(6,8,23,26,12,27,10,25,14,29,11,36,13,28,35,39)(7,15,40,42,19,43,17,22,21,45,18,50,20,44,49,54)(9,24,55,61,33,58,32,60,34,59,37,64,38,62,31,57)(16,41,63,67,47,56,30,53,48,66,51,69,52,68,46,65);(6,9)(7,16)(8,24)(10,32)(11,37)(12,33)(13,38)(14,34)(15,41)(17,30)(18,51)(19,47)(20,52)(21,48)(22,53)(23,55)(25,60)(26,61)(27,58)(28,62)(29,59)(31,35)(36,64)(39,57)(40,63)(42,67)(43,56)(44,68)(45,66)(46,49)(50,69)(54,65)

Conjugacy classes & character table

155 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5454);